├── .github
    └── FUNDING.yml
├── .gitignore
├── .gitmodules
├── .ruby-gemset
├── CONTRIBUTING.md
├── Gemfile
├── Gemfile.lock
├── README.md
├── _bibliography
    └── refs.bib
├── _config.yml
├── _includes
    ├── breadcrumbs.html
    ├── d-pad.html
    ├── head.html
    ├── header.html
    ├── list.html
    ├── macro-loader.html
    ├── search.html
    └── toc.html
├── _layouts
    ├── bib.html
    ├── bibtex.html
    ├── default.html
    ├── page.html
    ├── post.html
    └── sheafy
    │   ├── node
    │       ├── default.md
    │       ├── result.md
    │       └── section.md
    │   └── tree
    │       └── default.html
├── _lectures
    ├── categorical-foundations.md
    └── topos.md
├── _nodes
    ├── 0000.md
    ├── 0001.md
    ├── 0002.md
    ├── 0003.md
    ├── 0004.md
    ├── 0005.md
    ├── 0006.md
    ├── 0007.md
    ├── 0008.md
    ├── 0009.md
    ├── 000A.md
    ├── 000B.md
    ├── 000C.md
    ├── 000D.md
    ├── 000E.md
    ├── 000F.md
    ├── 000G.md
    ├── 000H.md
    ├── 000I.md
    ├── 000J.md
    ├── 000K.md
    ├── 000L.md
    ├── 000N.md
    ├── 000O.md
    ├── 000P.md
    ├── 000Q.md
    ├── 000R.md
    ├── 000S.md
    ├── 000T.md
    ├── 000U.md
    ├── 000V.md
    ├── 000W.md
    ├── 000X.md
    ├── 000Y.md
    ├── 000Z.md
    ├── 0010.md
    ├── 0011.md
    ├── 0012.md
    ├── 0013.md
    ├── 0014.md
    ├── 0015.md
    ├── 0016.md
    ├── 0017.md
    ├── 0018.md
    ├── 0019.md
    ├── 001A.md
    ├── 001B.md
    ├── 001C.md
    ├── 001D.md
    ├── 001E.md
    ├── 001F.md
    ├── 001G.md
    ├── 001H.md
    ├── 001I.md
    ├── 001J.md
    ├── 001K.md
    ├── 001L.md
    ├── 001M.md
    ├── 001N.md
    ├── 001O.md
    ├── 001P.md
    ├── 001Q.md
    ├── 001R.md
    ├── 001S.md
    ├── 001T.md
    ├── 001U.md
    ├── 001V.md
    ├── 001W.md
    ├── 001X.md
    ├── 001Y.md
    ├── 001Z.md
    ├── 0020.md
    ├── 0021.md
    ├── 0022.md
    ├── 0023.md
    ├── 0024.md
    ├── 0025.md
    ├── 0026.md
    ├── 0029.md
    ├── 002A.md
    ├── 002B.md
    ├── 002C.md
    ├── 002D.md
    ├── 002E.md
    ├── 002F.md
    ├── 002G.md
    ├── 002H.md
    ├── 002I.md
    ├── 002J.md
    ├── 002K.md
    ├── 002L.md
    ├── 002M.md
    ├── 002N.md
    ├── 002O.md
    ├── 002P.md
    ├── 002Q.md
    ├── 002R.md
    ├── 002S.md
    ├── 002T.md
    ├── 002U.md
    ├── 002V.md
    ├── 002W.md
    ├── 002X.md
    ├── 002Y.md
    ├── 002Z.md
    ├── 0030.md
    ├── 0031.md
    ├── 0032.md
    ├── 0033.md
    ├── 0034.md
    ├── 0035.md
    ├── 0036.md
    ├── 0037.md
    ├── 0038.md
    ├── 0039.md
    ├── 003A.md
    ├── 003C.md
    ├── 003D.md
    └── 003E.md
├── _plugins
    ├── changelog.rb
    ├── display_title_filter.rb
    ├── nlab_tag.rb
    ├── node_info.rb
    ├── patch-antex.rb
    ├── qed_tag.rb
    ├── ref_tag.rb
    └── to_numbering_filter.rb
├── assets
    ├── macros
    │   ├── jon-tikz.sty
    │   ├── topos.katex.json
    │   ├── topos.mathjax.json
    │   └── topos.sty
    ├── search.js
    ├── simple.css
    └── style.css
├── changelog.md
├── deploy.sh
├── index.md
├── macros
    ├── macro-kit.rkt
    └── topos.rkt
├── mint-node.rkt
└── setup-worktree.sh


/.github/FUNDING.yml:
--------------------------------------------------------------------------------
1 | # These are supported funding model platforms
2 | github: jonsterling
3 | ko_fi: paolobrasolin
4 | 


--------------------------------------------------------------------------------
/.gitignore:
--------------------------------------------------------------------------------
1 | .antex-cache
2 | .DS_store
3 | compiled
4 | _site
5 | _deploy
6 | *~
7 | .sass-cache
8 | .jekyll*
9 | 


--------------------------------------------------------------------------------
/.gitmodules:
--------------------------------------------------------------------------------
https://raw.githubusercontent.com/jonsterling/math/661175965a56e2b94b1f9ed35f993a6f81bb2b82/.gitmodules


--------------------------------------------------------------------------------
/.ruby-gemset:
--------------------------------------------------------------------------------
1 | jon-math
2 | 


--------------------------------------------------------------------------------
/CONTRIBUTING.md:
--------------------------------------------------------------------------------
1 | Contributions are very welcome; however, this book is being written from a specific point of view, so before contributing you should discuss your idea with the maintainer by opening a thread in the [Discussions](https://github.com/jonsterling/math/discussions) tab.
2 | 


--------------------------------------------------------------------------------
/Gemfile:
--------------------------------------------------------------------------------
 1 | source 'https://rubygems.org'
 2 | 
 3 | ruby '2.7.5'
 4 | 
 5 | gem 'jekyll', '~> 4.2.1'
 6 | 
 7 | group :jekyll_plugins do
 8 |   gem 'jekyll-antex', '~> 0.8.1'
 9 |   gem 'jekyll-scholar'
10 |   gem 'jekyll-sheafy', '~> 0.3.0'
11 | end
12 | 
13 | # kramdown v2 ships without the gfm parser by default. If you're using
14 | # kramdown v1, comment out this line.
15 | gem 'kramdown-parser-gfm'
16 | 


--------------------------------------------------------------------------------
/Gemfile.lock:
--------------------------------------------------------------------------------
  1 | GEM
  2 |   remote: https://rubygems.org/
  3 |   specs:
  4 |     addressable (2.8.1)
  5 |       public_suffix (>= 2.0.2, < 6.0)
  6 |     antex (0.1.3)
  7 |       liquid (>= 3, < 6)
  8 |       nokogiri (~> 1)
  9 |     bibtex-ruby (6.0.0)
 10 |       latex-decode (~> 0.0)
 11 |     citeproc (1.0.10)
 12 |       namae (~> 1.0)
 13 |     citeproc-ruby (1.1.14)
 14 |       citeproc (~> 1.0, >= 1.0.9)
 15 |       csl (~> 1.6)
 16 |     colorator (1.1.0)
 17 |     concurrent-ruby (1.1.10)
 18 |     csl (1.6.0)
 19 |       namae (~> 1.0)
 20 |       rexml
 21 |     csl-styles (1.0.1.11)
 22 |       csl (~> 1.0)
 23 |     em-websocket (0.5.3)
 24 |       eventmachine (>= 0.12.9)
 25 |       http_parser.rb (~> 0)
 26 |     eventmachine (1.2.7)
 27 |     ffi (1.15.5)
 28 |     forwardable-extended (2.6.0)
 29 |     http_parser.rb (0.8.0)
 30 |     i18n (1.12.0)
 31 |       concurrent-ruby (~> 1.0)
 32 |     jekyll (4.2.2)
 33 |       addressable (~> 2.4)
 34 |       colorator (~> 1.0)
 35 |       em-websocket (~> 0.5)
 36 |       i18n (~> 1.0)
 37 |       jekyll-sass-converter (~> 2.0)
 38 |       jekyll-watch (~> 2.0)
 39 |       kramdown (~> 2.3)
 40 |       kramdown-parser-gfm (~> 1.0)
 41 |       liquid (~> 4.0)
 42 |       mercenary (~> 0.4.0)
 43 |       pathutil (~> 0.9)
 44 |       rouge (~> 3.0)
 45 |       safe_yaml (~> 1.0)
 46 |       terminal-table (~> 2.0)
 47 |     jekyll-antex (0.8.1)
 48 |       antex (~> 0.1.3)
 49 |       jekyll (>= 3, < 5)
 50 |       kramdown-parser-gfm (~> 1.1.0)
 51 |       nokogiri (~> 1)
 52 |     jekyll-sass-converter (2.2.0)
 53 |       sassc (> 2.0.1, < 3.0)
 54 |     jekyll-scholar (7.1.0)
 55 |       bibtex-ruby (~> 6.0)
 56 |       citeproc-ruby (~> 1.0)
 57 |       csl-styles (~> 1.0)
 58 |       jekyll (~> 4.0)
 59 |     jekyll-sheafy (0.3.0)
 60 |       jekyll (>= 3, < 5)
 61 |     jekyll-watch (2.2.1)
 62 |       listen (~> 3.0)
 63 |     kramdown (2.4.0)
 64 |       rexml
 65 |     kramdown-parser-gfm (1.1.0)
 66 |       kramdown (~> 2.0)
 67 |     latex-decode (0.4.0)
 68 |     liquid (4.0.3)
 69 |     listen (3.7.1)
 70 |       rb-fsevent (~> 0.10, >= 0.10.3)
 71 |       rb-inotify (~> 0.9, >= 0.9.10)
 72 |     mercenary (0.4.0)
 73 |     mini_portile2 (2.8.0)
 74 |     namae (1.1.1)
 75 |     nokogiri (1.13.8)
 76 |       mini_portile2 (~> 2.8.0)
 77 |       racc (~> 1.4)
 78 |     pathutil (0.16.2)
 79 |       forwardable-extended (~> 2.6)
 80 |     public_suffix (5.0.0)
 81 |     racc (1.6.0)
 82 |     rb-fsevent (0.11.2)
 83 |     rb-inotify (0.10.1)
 84 |       ffi (~> 1.0)
 85 |     rexml (3.2.5)
 86 |     rouge (3.30.0)
 87 |     safe_yaml (1.0.5)
 88 |     sassc (2.4.0)
 89 |       ffi (~> 1.9)
 90 |     terminal-table (2.0.0)
 91 |       unicode-display_width (~> 1.1, >= 1.1.1)
 92 |     unicode-display_width (1.8.0)
 93 | 
 94 | PLATFORMS
 95 |   ruby
 96 | 
 97 | DEPENDENCIES
 98 |   jekyll (~> 4.2.1)
 99 |   jekyll-antex (~> 0.8.1)
100 |   jekyll-scholar
101 |   jekyll-sheafy (~> 0.3.0)
102 |   kramdown-parser-gfm
103 | 
104 | RUBY VERSION
105 |    ruby 2.7.5p203
106 | 
107 | BUNDLED WITH
108 |    2.1.4
109 | 


--------------------------------------------------------------------------------
/README.md:
--------------------------------------------------------------------------------
 1 | ***Notice:*** These lecture notes have now been incorporated into my [Forest](https://github.com/jonsterling/forest). This repository will not be updated except in order to fix broken redirects.
 2 | 
 3 | -----
 4 | 
 5 | This repository is based on [Krater](https://github.com/paolobrasolin/krater),
 6 | a tool for building math-rich websites; the purpose of the present code is to
 7 | support "Stacks Project"-style websites, where you create a bunch of files like
 8 | `_nodes/00A4.md` that may import each other, and then specify a root like
 9 | `_lectures/lecture.md` that imports some node.
10 | 
11 | ### Running on your machine
12 | 
13 | You will need a running LaTeX installation, including `latexmk`, `dvisvgm` and whatever engine and packages you plan on using.
14 | Chances are that if you're reading this you already have everything you need.
15 | 
16 | You will need to [install Ruby 2.7.5][ruby-install-url]; you will need to ensure that
17 | you have ghostscript installed, with the `LIBGS` environment variable correctly
18 | set. On macOS with Homebrew, after installing ghostscript you must add the
19 | following line to your `.bashrc` or `.zshrc`:
20 | 
21 | ```bash
22 | export LIBGS=/opt/homebrew/lib/libgs.dylib
23 | ```
24 | 
25 | After that, you can simply
26 | 
27 | ```bash
28 | # install dependencies
29 | bundle
30 | # run Jekyll
31 | bundle exec jekyll serve
32 | ```
33 | 
34 | and your website will be available at `http://127.0.0.1:4000/<REPO_NAME>/`.
35 | 
36 | [ruby-install-url]: https://www.ruby-lang.org/it/documentation/installation/
37 | 
38 | 
39 | ### Node UID policy and drafts
40 | 
41 | 1. Node UIDs should be added in order; so if the greatest node UID is `00X2`,
42 |    then the next node that is committed should be `00X3`.
43 | 
44 | 2. When a new node `00X3` is committed, it is permanent: it is not permitted to
45 |    rename it, however it is permitted to *delete* it.
46 | 
47 | If two branches contain drafted nodes, when one is merged, the node UIDs chosen
48 | on the other branch have to be renamed so that when *they* are committed to
49 | `main`, there are no conflicts and node UIDs are consecutive. This causes
50 | considerable friction, so we will use the following method for drafts.
51 | 
52 | 1. In a draft, name any *new* node with an underscore `_00X3`.
53 | 
54 | 2. Prior to merging, you may acquire a temporary lock on the entire repository
55 |    and rename every node draft node to the next consecutive "permanent" UID.
56 | 
57 | 3. Pull requests that involve draft nodes should be squashed prior to being
58 |    merged; this is because we want to avoid the auto-generated changelog from
59 |    referring to draft nodes that do not exist at merge-time.
60 | 
61 | The script `mint-node.rkt` will assist you in determining what the next
62 | permanent UID is. For now you must handle the renaming manually, but in the
63 | future we can include scripts to automate this process.
64 | 


--------------------------------------------------------------------------------
/_bibliography/refs.bib:
--------------------------------------------------------------------------------
  1 | @string{acm = "Association for Computing Machinery"}
  2 | @string{apal = "Annals of Pure and Applied Logic"}
  3 | @string{cacm = "Communications of the ACM"}
  4 | @string{cup = "Cambridge University Press"}
  5 | @string{entcs = "Electronic Notes in Theoretical Computer Science"}
  6 | @string{eptcs = "Electronic Proceedings in Theoretical Computer Science"}
  7 | @string{i&c = "Information and Computation"}
  8 | @string{ieee = "IEEE Computer Society"}
  9 | @string{jacm = "Journal of the ACM"}
 10 | @string{jal = "Journal of Applied Logic"}
 11 | @string{jems = "Journal of the European Mathematical Society"}
 12 | @string{jfp = "Journal of Functional Programming"}
 13 | @string{jpaa = "Journal of Pure and Applied Algebra"}
 14 | @string{jsc = "Journal of Symbolic Computation"}
 15 | @string{jsl = "The Journal of Symbolic Logic"}
 16 | @string{lmcs = "Logical Methods in Computer Science"}
 17 | @string{mscs = "Mathematical Structures in Computer Science"}
 18 | @string{pacmpl = "Proceedings of the ACM on Programming Languages"}
 19 | @string{tcs = "Theoretical Computer Science"}
 20 | @string{tocl = "ACM Transactions on Computational Logic"}
 21 | @string{toplas = "ACM Transactions on Programming Languages and Systems"}
 22 | 
 23 | @unpublished{streicher:2018,
 24 |   author = {Streicher, Thomas},
 25 |   year = {2018},
 26 |   eprint = {1801.02927},
 27 |   eprinttype = {arXiv},
 28 |   title = {Fibred Categories \`{a} la Jean B\'{e}nabou},
 29 | }
 30 | 
 31 | @article{benabou:1985,
 32 |   author = {B\'{e}nabou, Jean},
 33 |   publisher = cup,
 34 |   year = {1985},
 35 |   doi = {10.2307/2273784},
 36 |   journal = {Journal of Symbolic Logic},
 37 |   number = {1},
 38 |   pages = {10--37},
 39 |   title = {Fibered categories and the foundations of naive category theory},
 40 |   volume = {50},
 41 | }
 42 | 
 43 | @book{jacobs:1999,
 44 |   author = {Jacobs, Bart},
 45 |   address = {Amsterdam},
 46 |   publisher = {North Holland},
 47 |   year = {1999},
 48 |   number = {141},
 49 |   series = {Studies in Logic and the Foundations of Mathematics},
 50 |   title = {Categorical Logic and Type Theory},
 51 | }
 52 | 
 53 | @book{sga:4,
 54 |   author = {Artin, Michael and Grothendieck, Alexander and Verdier, Jean-Louis},
 55 |   address = {Berlin},
 56 |   publisher = {Springer-Verlag},
 57 |   year = {1972},
 58 |   note = {S\'{e}minaire de G\'{e}om\'{e}trie Alg\'{e}brique du Bois-Marie 1963--1964 (SGA 4), Dirig\'{e} par M. Artin, A. Grothendieck, et J.-L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat, Lecture Notes in Mathematics, Vol. 269, 270, 305},
 59 |   title = {Th\'{e}orie des topos et cohomologie \'{e}tale des sch\'{e}mas},
 60 | }
 61 | 
 62 | @book{sga:1,
 63 |   author = {Grothendieck, Alexander},
 64 |   publisher = {Springer-Verlag},
 65 |   year = {1971},
 66 |   series = {Lecture notes in mathematics},
 67 |   title = {Rev\^{e}tements \'{e}tales et groupe fondamental (SGA 1)},
 68 |   volume = {224},
 69 | }
 70 | 
 71 | 
 72 | @article{street:1980,
 73 |   author = {Street, Ross},
 74 |   language = {eng},
 75 |   publisher = {Dunod \'{e}diteur, publi\'{e} avec le concours du CNRS},
 76 |   url = {http://eudml.org/doc/91227},
 77 |   year = {1980},
 78 |   journal = {Cahiers de Topologie et G\'{e}om\'{e}trie Diff\'{e}rentielle Cat\'{e}goriques},
 79 |   number = {2},
 80 |   pages = {111--160},
 81 |   title = {Fibrations in bicategories},
 82 |   volume = {21},
 83 | }
 84 | 
 85 | @article{ahrens-lumsdaine:2019,
 86 |   author = {Ahrens, Benedikt and Lumsdaine, Peter LeFanu},
 87 |   url = {https://lmcs.episciences.org/5252},
 88 |   year = {2019},
 89 |   month = mar,
 90 |   doi = {10.23638/LMCS-15(1:20)2019},
 91 |   eprint = {1705.04296},
 92 |   eprinttype = {arXiv},
 93 |   issue = {1},
 94 |   journal = lmcs,
 95 |   title = {Displayed Categories},
 96 |   volume = {15},
 97 | }
 98 | 
 99 | 
100 | @book{lawvere-schanuel:2009,
101 |   author = {Lawvere, F. William and Schanuel, Stephen H.},
102 |   address = {New York, NY, USA},
103 |   publisher = cup,
104 |   year = {2009},
105 |   edition = {2nd},
106 |   isbn = {0-521-89485-9},
107 |   title = {Conceptual Mathematics: A First Introduction to Categories},
108 | }
109 | 
110 | @book{borceux:1994:vol2,
111 |   author = {Borceux, Francis},
112 |   publisher = cup,
113 |   year = {1994},
114 |   title = {Handbook of Categorical Algebra 2 -- Categories and Structures},
115 | }
116 | 
117 | @misc{streicher:2017:fibered-top,
118 |   author = {Streicher, Thomas},
119 |   url = {https://www2.mathematik.tu-darmstadt.de/~streicher/FIBR/fibtop.pdf},
120 |   date = {2011-06},
121 |   note = {Unpublished manuscript},
122 |   title = {Observations on Fibered Toposes},
123 | }
124 | 
125 | 


--------------------------------------------------------------------------------
/_config.yml:
--------------------------------------------------------------------------------
  1 | title: The Nursery
  2 | email: jon@jonmsterling.com
  3 | description: >-
  4 |   A mathematical web page.
  5 | baseurl: &baseurl "/math"
  6 | url: "https://www.jonmsterling.com/"
  7 | twitter_username: jonmsterling
  8 | github_username: jonsterling
  9 | 
 10 | safe: false
 11 | source: .
 12 | destination: ./_site
 13 | exclude:
 14 |   - .byebug_history
 15 |   - macros/compiled
 16 |   - _site
 17 |   - _bibliography
 18 |   - setup-worktree.sh
 19 |   - deploy.sh
 20 | 
 21 | keep_files:
 22 |   - antex
 23 | 
 24 | markdown: kramdown
 25 | kramdown:
 26 |   input: GFM
 27 |   syntax_highlighter: rouge
 28 |   math_engine: null
 29 |   parse_block_html: true
 30 | 
 31 | antex:
 32 |   inlining: true
 33 |   url_path_prefix: *baseurl
 34 | 
 35 |   preamble: >-
 36 |     \usepackage{jon-tikz}
 37 |     \usepackage{mlmodern,amsfonts,amssymb,amsmath}
 38 | 
 39 |   template: |
 40 |     \documentclass{article}
 41 |     \pagestyle{empty}
 42 |     \usepackage{ {{ macrolib }} }
 43 |     {{ preamble }}
 44 |     \newsavebox\snippet
 45 |     \begin{lrbox}{\snippet}{{ prepend }}{{ snippet }}{{ append }}\end{lrbox}
 46 |     \newwrite\file
 47 |     \immediate\openout\file=\jobname.yml
 48 |     \immediate\write\file{em: \the\dimexpr1em}
 49 |     \immediate\write\file{ex: \the\dimexpr1ex}
 50 |     \immediate\write\file{ht: \the\ht\snippet}
 51 |     \immediate\write\file{dp: \the\dp\snippet}
 52 |     \immediate\write\file{wd: \the\wd\snippet}
 53 |     \closeout\file
 54 |     \begin{document}\usebox{\snippet}\end{document}
 55 | 
 56 |   commands:
 57 |     latexmk:
 58 |       command:
 59 |         - "TEXINPUTS=.:./assets/macros/:${TEXINPUTS}"
 60 |         - "latexmk"
 61 |         - "-output-directory={{ dir.work }}"
 62 |         - "{{ file.tex }}"
 63 |       sources:
 64 |         - "{{ file.tex }}"
 65 |       targets:
 66 |         - "{{ file.dvi }}"
 67 |         - "{{ file.yml }}"
 68 | 
 69 |   aliases: # disable most aliases and use server side AnTeX only for complex LaTeX
 70 |     default:
 71 |       priority: 20
 72 |       regexp: !ruby/regexp |
 73 |         /{%\s*tex\s*(?<markup>.*?)%}
 74 |         (?<code>.*?)
 75 |         {%\s*endtex\s*%}/mx
 76 |       options:
 77 |         classes: [antex, display]
 78 |         prepend: '$\displaystyle '
 79 |         append: $
 80 |     display_math: false
 81 |     inline_math: false
 82 |     inline_macro: false
 83 | 
 84 | plugins:
 85 |   - jekyll-scholar
 86 |   - jekyll/sheafy
 87 | 
 88 | scholar:
 89 |   bibliography: refs.bib
 90 |   bibliography_template: bib
 91 |   details_dir: bibliography
 92 |   details_layout: bibtex.html
 93 |   details_link: Details
 94 | 
 95 | header_pages:
 96 |   - changelog.md
 97 | 
 98 | collections:
 99 |   lectures:
100 |     output: true
101 |   nodes:
102 |     output: true
103 | 
104 | defaults:
105 |   - scope:
106 |       type: nodes
107 |     values:
108 |       taxon: section
109 |       excerpt: ""
110 |   - scope:
111 |       type: lectures
112 |     values:
113 |       taxon: section
114 |       excerpt: ""
115 | 
116 | sheafy:
117 |   inheritable:
118 |     - macrolib
119 |   references:
120 |     matchers:
121 |       - !ruby/regexp /{%\s*[cp]?ref (?<slug>.+?)\s*%}/
122 |   taxa:
123 |     section:
124 |       layout: sheafy/tree/default
125 |       sublayout: sheafy/node/section
126 |       genus: Section
127 |       clicker: section
128 |     theorem:
129 |       layout: sheafy/tree/default
130 |       sublayout: sheafy/node/result
131 |       genus: Theorem
132 |       clicker: result
133 |     lemma:
134 |       layout: sheafy/tree/default
135 |       sublayout: sheafy/node/result
136 |       genus: Lemma
137 |       clicker: result
138 |     corollary:
139 |       layout: sheafy/tree/default
140 |       sublayout: sheafy/node/result
141 |       genus: Corollary
142 |       clicker: result
143 |     definition:
144 |       layout: sheafy/tree/default
145 |       sublayout: sheafy/node/result
146 |       genus: Definition
147 |       clicker: result
148 |     exercise:
149 |       layout: sheafy/tree/default
150 |       sublayout: sheafy/node/result
151 |       genus: Exercise
152 |       clicker: result
153 |     construction:
154 |       layout: sheafy/tree/default
155 |       sublayout: sheafy/node/result
156 |       genus: Construction
157 |       clicker: result
158 |     computation:
159 |       layout: sheafy/tree/default
160 |       sublayout: sheafy/node/result
161 |       genus: Computation
162 |       clicker: result
163 |     example:
164 |       layout: sheafy/tree/default
165 |       sublayout: sheafy/node/result
166 |       genus: Example
167 |       clicker: result
168 |     notation:
169 |       layout: sheafy/tree/default
170 |       sublayout: sheafy/node/result
171 |       genus: Notation
172 |       clicker: result
173 |     remark:
174 |       layout: sheafy/tree/default
175 |       sublayout: sheafy/node/result
176 |       genus: Remark
177 |       clicker: result
178 |     intuition:
179 |       layout: sheafy/tree/default
180 |       sublayout: sheafy/node/result
181 |       genus: Intuition
182 |       clicker: result
183 |     warning:
184 |       layout: sheafy/tree/default
185 |       sublayout: sheafy/node/result
186 |       genus: Warning
187 |       clicker: warning
188 | 
189 | changelog:
190 |   authors:
191 |     jon@jonmsterling.com:
192 |       name: Jon Sterling
193 |       link: https://www.jonmsterling.com/
194 |     carlo@carloangiuli.com:
195 |       link: http://www.cs.cmu.edu/~cangiuli/
196 |     paolo.brasolin@gmail.com:
197 |       hide: true
198 | 


--------------------------------------------------------------------------------
/_includes/breadcrumbs.html:
--------------------------------------------------------------------------------
 1 | <style>
 2 |   .breadcrumbs {
 3 |     font-size: small;
 4 |     margin-bottom: 1.5em;
 5 |   }
 6 | </style>
 7 | 
 8 | {% unless include.list.size == 0 %}
 9 | <nav class="breadcrumbs">
10 |   {% for resource in include.list %}
11 |     <a href="{{resource.url | relative_url}}">
12 |       {% if resource.collection == "nodes" %}
13 |       <span class="numbering">{{ resource.clicks | to_numbering }}.</span>
14 |       {% endif %}
15 |       {{resource.title}}
16 |       {% if resource.collection == "nodes" %}
17 |       <span class="slug">[{{resource.slug}}]</span>
18 |       {% endif %}
19 |     </a>
20 |     /
21 |   {% endfor %}
22 | </nav>
23 | {% endunless %}
24 | 


--------------------------------------------------------------------------------
/_includes/d-pad.html:
--------------------------------------------------------------------------------
 1 | <style>
 2 |   nav.d-pad {
 3 |     display: flex;
 4 |     flex-direction: row;
 5 |     justify-content: space-around;
 6 |     flex-wrap: wrap;
 7 |     margin: 3rem 0rem 4rem 0rem;
 8 |     gap: 0.6rem;
 9 |   }
10 | 
11 |   nav.d-pad i.arrow {
12 |     display: inline-block;
13 |     font-style: normal;
14 |     position: relative;
15 |     width: 0.6em;
16 |     height: 0.6em;
17 |     border: 0.2em solid black;
18 |     border-bottom: unset;
19 |     border-left: unset;
20 |     box-sizing: border-box;
21 |   }
22 |   nav.d-pad i.arrow.left {
23 |     transform: rotate(-135deg);
24 |   }
25 |   nav.d-pad i.arrow.right {
26 |     transform: rotate(45deg);
27 |   }
28 |   nav.d-pad a.prev { text-align: right; }
29 |   nav.d-pad a.next { text-align: left; }
30 | </style>
31 | 
32 | <nav class="d-pad">
33 | 
34 | {% assign prev = page.predecessors | last %}
35 | {% unless prev %}
36 |   {% assign fallback = page.ancestors | reverse | find_exp: "node", "node.predecessors.size > 0" %}
37 |   {% assign prev = fallback.predecessors | last %}
38 | {% endunless %}
39 | 
40 | {% assign next = page.successors | first %}
41 | {% unless next %}
42 |   {% assign fallback = page.ancestors | reverse | find_exp: "node", "node.successors.size > 0" %}
43 |   {% assign next = fallback.successors | first %}
44 | {% endunless %}
45 | 
46 | {% if prev %}
47 |   <a class="prev" href="{{prev.url | relative_url}}">
48 |     <i class="arrow left"></i>
49 |     {{ prev | display_title }}
50 |     <span class="slug">[{{prev.slug}}]</span>
51 |   </a>
52 | {% endif %}
53 | 
54 | {% if next %}
55 |   <a class="next" href="{{next.url | relative_url}}">
56 |     {{ next | display_title }}
57 |     <span class="slug">[{{next.slug}}]</span>
58 |     <i class="arrow right"></i>
59 |   </a>
60 | {% endif %}
61 | 
62 | </nav>
63 | 


--------------------------------------------------------------------------------
/_includes/head.html:
--------------------------------------------------------------------------------
 1 | <head>
 2 | 
 3 |   <!-- Setting up a redirect to the new version of these notes on my Forest. -->
 4 |   {% if  page.collection == "nodes" %}
 5 |   <meta http-equiv="Refresh" content="0; url=https://www.jonmsterling.com/forest/trees/frct-{{page.slug | downcase}}" />
 6 |   {% endif %}
 7 | 
 8 | 
 9 | 
10 |   <meta charset="utf-8">
11 |   <meta http-equiv="X-UA-Compatible" content="IE=edge">
12 |   <meta name="viewport" content="width=device-width, initial-scale=1">
13 |   <link rel="stylesheet" href="{{ "/assets/style.css" | relative_url }}">
14 |   <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.15.1/dist/katex.min.css"
15 |   integrity="sha384-R4558gYOUz8mP9YWpZJjofhk+zx0AS11p36HnD2ZKj/6JR5z27gSSULCNHIRReVs" crossorigin="anonymous">
16 |   <script defer src="https://cdn.jsdelivr.net/npm/katex@0.15.1/dist/katex.min.js"
17 |     integrity="sha384-z1fJDqw8ZApjGO3/unPWUPsIymfsJmyrDVWC8Tv/a1HeOtGmkwNd/7xUS0Xcnvsx" crossorigin="anonymous"></script>
18 |   <script defer src="https://cdn.jsdelivr.net/npm/katex@0.15.1/dist/contrib/auto-render.min.js"
19 |     integrity="sha384-+XBljXPPiv+OzfbB3cVmLHf4hdUFHlWNZN5spNQ7rmHTXpd7WvJum6fIACpNNfIR" crossorigin="anonymous"></script>
20 | 
21 |   <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/microtip@0.2.2/microtip.min.css"
22 |     integrity="sha256-RUSUTTyNeNKfP+mJISBiUTVElF6MSDsJcwYZrcSyUII=" crossorigin="anonymous">
23 |   <script src="https://cdn.jsdelivr.net/npm/mark.js@8.11.1/dist/mark.min.js"
24 |     integrity="sha256-IdYuEFP3WJ/mNlzM18Y20Xgav3h5pgXYzl8fW4GnuPo=" crossorigin="anonymous"></script>
25 | 
26 |   <script src="https://cdn.jsdelivr.net/npm/lunr@2.3.9/lunr.min.js"
27 |     integrity="sha256-DFDZACuFeAqEKv/7Vnu1Tt5ALa58bcWZegGGFNgET8g=" crossorigin="anonymous"></script>
28 | 
29 |   <title>{{page.title}}</title>
30 | 
31 |   <script>
32 |     const katexMacroLibCache = {};
33 |     function loadKaTeXMacroLibrary(macrolib) {
34 |       const cached = katexMacroLibCache[macrolib];
35 |       if (cached) { return cached; }
36 |       const promise = fetch("{{site.baseurl}}/assets/macros/" + macrolib + ".katex.json").then(response => response.json());
37 |       katexMacroLibCache[macrolib] = promise;
38 |       return promise;
39 |     }
40 | 
41 |     function renderMacrosInSection(macrolib, sectionId) {
42 |       const macros_promise = loadKaTeXMacroLibrary(macrolib);
43 |       const sectionElement = document.getElementById(sectionId);
44 |       document.addEventListener("DOMContentLoaded", function () {
45 |         macros_promise.then(function (macros) {
46 |           renderMathInElement(sectionElement, {
47 |             delimiters: [
48 |               { left: '$$', right: '$$', display: true },
49 |               { left: '$', right: '$', display: false },
50 |               { left: '\\[', right: '\\]', display: true }
51 |             ],
52 |             macros: macros,
53 |             throwOnError: false
54 |           });
55 |         });
56 |       });
57 |     }
58 |   </script>
59 | 
60 | </head>
61 | 


--------------------------------------------------------------------------------
/_includes/header.html:
--------------------------------------------------------------------------------
 1 | <header>
 2 | 
 3 | 
 4 |     {%- assign default_paths = site.pages | map: "path" -%}
 5 |     {%- assign page_paths = site.header_pages | default: default_paths -%}
 6 |     {%- assign titles_size = site.pages | map: 'title' | join: '' | size -%}
 7 | 
 8 |     {%- if titles_size > 0 -%}
 9 |       <nav class="site-nav">
10 |         <a class="site-title" rel="author" href="{{ "/" | relative_url }}">{{ site.title | escape }}</a>
11 |         <div class="trigger">
12 |           <ul>
13 |             {%- for path in page_paths -%}
14 |               {%- assign my_page = site.pages | where: "path", path | first -%}
15 |               {%- if my_page.title -%}
16 |               <li><a class="page-link" href="{{ my_page.url | relative_url }}">{{ my_page.title | escape }}</a></li>
17 |               {%- endif -%}
18 |             {%- endfor -%}
19 |             {% if page.taxon %}<li><a class="page-link" id="search-toggle">Search</a></li>{% endif %}
20 |           </ul>
21 |         </div>
22 |       </nav>
23 |     {%- endif -%}
24 | 
25 | </header>
26 | 


--------------------------------------------------------------------------------
/_includes/list.html:
--------------------------------------------------------------------------------
 1 | <ul>
 2 |   {% assign sorted_resources = include.list | sort: "clicks" -%}
 3 |   {% for resource in sorted_resources %}
 4 |   <li>
 5 |     <a href="{{resource.url | relative_url}}">
 6 |       {{ resource | display_title }}
 7 |       {% if resource.collection == "nodes" %}
 8 |       <span class="slug">[{{resource.slug}}]</span>
 9 |       {% endif %}
10 |     </a>
11 |   </li>
12 |   {% endfor %}
13 | </ul>
14 | 


--------------------------------------------------------------------------------
/_includes/macro-loader.html:
--------------------------------------------------------------------------------
1 | <script>
2 |   {% if page.macrolib %}
3 |   renderMacrosInSection("{{page.macrolib}}", "{{page.slug}}");
4 |   {% endif %}
5 | </script>
6 | 


--------------------------------------------------------------------------------
/_includes/search.html:
--------------------------------------------------------------------------------
 1 | <style>
 2 |   ul.toc li.ellipsis {
 3 |     height: 0;
 4 |     overflow: hidden;
 5 |   }
 6 |   section.ellipsis > header {
 7 |     opacity: 0.5;
 8 |     border-bottom: 3px dashed black;
 9 |   }
10 |   section.ellipsis > :not(header) {
11 |     display: none;
12 |   }
13 |   #search-form {
14 |     margin-bottom: 2rem;
15 |   }
16 |   #search-form fieldset {
17 |     display: flex;
18 |     flex-wrap: wrap;
19 |   }
20 |   #search-form input {
21 |     flex-grow: 1;
22 |   }
23 |   #search-form .hint {
24 |     flex-basis: 100%;
25 |     margin: 0;
26 |   }
27 |   #search-form .hint summary {
28 |     font-size: small;
29 |   }
30 | </style>
31 | 
32 | <script src="{{ 'assets/search.js' | relative_url }}"></script>
33 | 
34 | <form id="search-form" hidden>
35 |   <fieldset>
36 |     <legend>Search</legend>
37 |     <input autocomplete="off" type="search" id="search-query" name="q" placeholder="Search...."/>
38 |     <button type="submit">Search</button>
39 |     <button type="reset">Clear</button>
40 |     <details class="hint">
41 |       <summary>Click here for help.</summary>
42 |       <ul>
43 |         <li>You can type your query in the input box, click "Search" to start the search, and click "Clear" to clear the page.</li>
44 |         <li>The search happens on the current page. If you want to extend it, navigate the book upwards. A search on the root note will search the whole book.</li>
45 |         <li>Once the search is completed, matches will be highlighted and nodes with no matches will be folded. You can expand them by clicking.</li>
46 |         <li>Multiple terms are allowed in the query and the default behaviour is <code>OR</code>ing them. You can <code>AND</code> terms making them mandatory, see <code>+foo</code> below.</li>
47 |         <li><code>*foo</code>, <code>f*o</code>, and <code>fo*</code> are all allowed wildcards.</li>
48 |         <li><code>+foo</code> makes a term mandatory, while <code>-foo</code> makes a term forbidden.</li>
49 |         <li><code>foo~n</code> fuzzy matches within edit distance <code>n</code> (a positive number) from the term.</li>
50 |         <li>Note that the URL changes after a search. You can copy and paste it to share your search results. They will be displayed immediately upon visiting the URL.</li>
51 |       </ul>
52 |     </details>
53 |   </fieldset>
54 | </form>
55 | 


--------------------------------------------------------------------------------
/_includes/toc.html:
--------------------------------------------------------------------------------
 1 | <ul class="toc">
 2 |   {% assign filtered_resources = include.page.children | where: "clicker", "section" %}
 3 |   {% for resource in filtered_resources %}
 4 |   <li>
 5 |     {{ resource.clicks | to_numbering }}.
 6 |     <a href="#{{resource.slug}}">
 7 |       {{ resource.title }}
 8 |     </a>
 9 |     <a class="slug" href="{{resource.url | relative_url}}">[{{resource.slug}}]</a>
10 |     {% include toc.html page=resource %}
11 |   </li>
12 |   {% endfor %}
13 | </ul>
14 | 


--------------------------------------------------------------------------------
/_layouts/bib.html:
--------------------------------------------------------------------------------
 1 | ---
 2 | ---
 3 | <!-- _layouts/bib.html -->
 4 | <div class="text-justify">
 5 |     {{reference}}
 6 |     {% if entry.eprinttype == "arXiv" %}
 7 |     <span><a href="{{ entry.eprint | prepend: 'http://arxiv.org/abs/' }}">arXiv:{{entry.eprint}}</a>.</span>
 8 |     {% endif %}
 9 |     {% if entry.hal_id %}
10 |     <span><a href="{{ entry.hal_id | prepend: 'http://hal.inria.fr/' }}">hal:{{entry.hal_id}}</a>.</span>
11 |     {% endif %}
12 | </div>
13 | 


--------------------------------------------------------------------------------
/_layouts/bibtex.html:
--------------------------------------------------------------------------------
 1 | ---
 2 | layout: page
 3 | ---
 4 | <div class="text-justify">
 5 |     {{page.reference}}
 6 |     {% if page.entry.eprinttype == "arXiv" %}
 7 |     <span><a href="{{ page.entry.eprint | prepend: 'http://arxiv.org/abs/' }}">arXiv:{{page.entry.eprint}}</a>.</span>
 8 |     {% endif %}
 9 |     {% if page.entry.hal_id %}
10 |     <span><a href="{{ page.entry.hal_id | prepend: 'http://hal.inria.fr/' }}">hal:{{page.entry.hal_id}}</a>.</span>
11 |     {% endif %}
12 | </div>
13 | 
14 | <div class="bibtex-code">
15 |   <pre>{{page.entry.bibtex}}</pre>
16 | </div>
17 | 


--------------------------------------------------------------------------------
/_layouts/default.html:
--------------------------------------------------------------------------------
 1 | ---
 2 | ---
 3 | 
 4 | <!DOCTYPE html>
 5 | <html>
 6 |   {%include head.html %}
 7 | 
 8 |   <body>
 9 |     {%include header.html %}
10 |     {{ content }}
11 |   </body>
12 | 
13 | </html>
14 | 


--------------------------------------------------------------------------------
/_layouts/page.html:
--------------------------------------------------------------------------------
 1 | ---
 2 | layout: default
 3 | ---
 4 | <article class="post">
 5 | 
 6 |   <header class="post-header">
 7 |     <h1 class="post-title">{{ page.title | escape }}</h1>
 8 |   </header>
 9 | 
10 |   <div class="post-content">
11 |     {{ content }}
12 |   </div>
13 | 
14 | </article>
15 | 


--------------------------------------------------------------------------------
/_layouts/post.html:
--------------------------------------------------------------------------------
 1 | ---
 2 | layout: default
 3 | ---
 4 | 
 5 | <article class="post h-entry" itemscope itemtype="http://schema.org/BlogPosting">
 6 | 
 7 |   <header class="post-header">
 8 |     <h1 class="post-title p-name" itemprop="name headline">{{ page.title | escape }}</h1>
 9 | 
10 | 
11 |     <p class="post-meta">
12 |       {%- assign date_format = site.minima.date_format | default: "%b %-d, %Y" -%}
13 |       <time class="dt-published" datetime="{{ page.date | date_to_xmlschema }}" itemprop="datePublished">
14 |         {{ page.date | date: date_format }}
15 |       </time>
16 |       {%- if page.modified_date -%}
17 |         ~
18 |         {%- assign mdate = page.modified_date | date_to_xmlschema -%}
19 |         <time class="dt-modified" datetime="{{ mdate }}" itemprop="dateModified">
20 |           {{ mdate | date: date_format }}
21 |         </time>
22 |       {%- endif -%}
23 |       {%- if page.author -%}
24 |         • {% for author in page.author %}
25 |           <span itemprop="author" itemscope itemtype="http://schema.org/Person">
26 |             <span class="p-author h-card" itemprop="name">{{ author }}</span></span>
27 |             {%- if forloop.last == false %}, {% endif -%}
28 |         {% endfor %}
29 |       {%- endif -%}</p>
30 |   </header>
31 | 
32 |   <div class="post-content e-content" itemprop="articleBody">
33 |     {{ content }}
34 |   </div>
35 | 
36 |   {%- if site.disqus.shortname -%}
37 |     {%- include disqus_comments.html -%}
38 |   {%- endif -%}
39 | 
40 |   <a class="u-url" href="{{ page.url | relative_url }}" hidden></a>
41 | </article>
42 | 


--------------------------------------------------------------------------------
/_layouts/sheafy/node/default.md:
--------------------------------------------------------------------------------
1 | <section>
2 |   {{ content }}
3 | </section>
4 | 


--------------------------------------------------------------------------------
/_layouts/sheafy/node/result.md:
--------------------------------------------------------------------------------
 1 | 
 2 | <section id="{{ page.slug }}">
 3 | 
 4 | {::nomarkdown}
 5 | 
 6 | <header class="inline">
 7 |   <a class="slug" href="{{ page.url | relative_url }}">[{{ page.slug }}]</a>
 8 |   {{page | display_title_parenthetical -}}.
 9 | </header>
10 | 
11 | {:/}
12 | 
13 | {{ content }}
14 | 
15 | </section>
16 | 
17 | {% include macro-loader.html page=page %}
18 | 


--------------------------------------------------------------------------------
/_layouts/sheafy/node/section.md:
--------------------------------------------------------------------------------
 1 | <section id="{{ page.slug }}">
 2 | 
 3 | {::nomarkdown}
 4 | 
 5 | <header>
 6 |   <h{{ page.depth | plus: 1 }}>
 7 |     {% if page.depth == 0 %}
 8 |       {{ page.title }}
 9 |     {% else %}
10 |       <span class="numbering">{{ page.clicks | to_numbering }}.</span>
11 |       {{ page.title }}
12 |       <a class="slug" href="{{ page.url | relative_url }}">[{{ page.slug }}]</a>
13 |     {% endif %}
14 |   </h{{ page.depth | plus: 1 }}>
15 |   {%- if page.author -%}
16 |   <p>
17 |     by {% for author in page.author %}
18 |       <span itemprop="author" itemscope itemtype="http://schema.org/Person">
19 |       <span class="p-author h-card" itemprop="name">{{ author }}</span></span>
20 |       {%- if forloop.last == false %}, {% endif -%}
21 |     {% endfor %}
22 |   </p>
23 |   {%- endif -%}
24 | </header>
25 | 
26 | {% if page == page.subroot %}
27 |   {% assign filtered_resources = page.children | where: "clicker", "section" %}
28 |   {% if filtered_resources.size != 0 %}
29 |     <nav>
30 |       <h4>Table of Contents</h4>
31 |       {% include toc.html page=page %}
32 |     </nav>
33 |   {% endif %}
34 | {% endif %}
35 | 
36 | {:/}
37 | 
38 | {{ content }}
39 | 
40 | </section>
41 | 
42 | {% include macro-loader.html page=page %}
43 | 


--------------------------------------------------------------------------------
/_layouts/sheafy/tree/default.html:
--------------------------------------------------------------------------------
 1 | ---
 2 | layout: default
 3 | ---
 4 | 
 5 | {% include search.html %}
 6 | 
 7 | <article class="post">
 8 |   <nav>
 9 |     {% include breadcrumbs.html list=page.ancestors %}
10 |   </nav>
11 | 
12 |   <div class="post-content">
13 |     {{ content }}
14 |   </div>
15 | 
16 |   {% include d-pad.html page=page %}
17 | 
18 |   {% if page.referrers.size != 0 %}
19 |   <nav>
20 |     <h4>Referrers</h4>
21 |     {% include list.html list=page.referrers %}
22 |   </nav>
23 |   {% endif %}
24 | 
25 |   {% capture bibliography_count_cited %}{% bibliography_count --cited %}{% endcapture %}
26 |   {% if bibliography_count_cited != "0" %}
27 |     <h4>Bibliography</h4>
28 |     {% bibliography --cited %}
29 |   {% endif %}
30 | 
31 | </article>
32 | 


--------------------------------------------------------------------------------
/_lectures/categorical-foundations.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Foundations of Relative Category Theory
 3 | date: 2022-01-01
 4 | author:
 5 | - Jonathan Sterling
 6 | - Carlo Angiuli
 7 | macrolib: topos
 8 | ---
 9 | 
10 | **The current version of these notes is now located on my <a href="https://www.jonmsterling.com/forest/trees/frct-003i/">Forest</a>. Permalinks from the existing website are redirected to the corresponding nodes in the Forest.**
11 | 
12 | We assume knowledge of basic categorical concepts such as categories, functors,
13 | and natural transformations. The purpose of this lecture is to develop the
14 | notion of a category *over* another category.
15 | 
16 | We will draw on the following materials:
17 | 
18 | > - Ahrens and Lumsdaine. [Displayed Categories](https://arxiv.org/abs/1705.04296).
19 | > - Benabou. Fibered categories and the foundations of naive category theory.
20 | > - Borceux. Handbook of Categorical Algebra 2: Categories and Structures.
21 | > - Jacobs.  [Categorical Logic and Type Theory](https://people.mpi-sws.org/~dreyer/courses/catlogic/jacobs.pdf) .
22 | > - Streicher. [Fibered Categories à la Jean Bénabou](https://www2.mathematik.tu-darmstadt.de/~streicher/FIBR/FiBo.pdf).
23 | 
24 | This material is deeply influenced by conversations with Thomas Streicher, as well as his writings.
25 | 
26 | @include{000R}
27 | @include{0008}
28 | @include{0009}
29 | @include{000E}
30 | @include{000N}
31 | @include{0012}
32 | 


--------------------------------------------------------------------------------
/_lectures/topos.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Geometric universes and topoi (draft)
 3 | date: 2022-01-08
 4 | author: Jonathan Sterling
 5 | macrolib: topos
 6 | ---
 7 | 
 8 | @include{0023}
 9 | @include{0024}
10 | 
11 | 


--------------------------------------------------------------------------------
/_nodes/0000.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Displayed category
 3 | taxon: definition
 4 | ---
 5 | 
 6 | Let $B$ be a category. A *displayed category* $E$ over $B$ is defined by the
 7 | following data {%cite ahrens-lumsdaine:2019%}:
 8 | 1. for each object $x\in B$, a collection of *displayed objects* $E\Sub{x}$,
 9 | 2. for each morphism $f : x \to y\in B$ and displayed objects $\bar{x}\in E\Sub{x}$ and
10 |    $\bar{y}\in E\Sub{y}$, a family of collections of *displayed morphisms* $E\Sub{f}(\bar{x},\bar{y})$,
11 |    which we may also write $\bar{f}:\bar{x}\to\Sub{f} \bar{y}$,
12 | 3. for each $x\in B$ and $\bar{x}\in E\Sub{x}$, a morphism $\Idn{\bar{x}} \in
13 |    E\Sub{\Idn{x}}(\bar{x},\bar{x})$,
14 | 4. for each $f : x \to y$ and $g:y \to z$ in $B$ and objects $\bar{x}\in E\Sub{x}, \bar{y}\in
15 |    E\Sub{y}, \bar{z}\in E\Sub{z}$, a function
16 |     \\[
17 |       E\Sub{f}(\bar{x},\bar{y}) \times E\Sub{g}(\bar{y},\bar{z}) \to E\Sub{f;g}(\bar{x},\bar{z})
18 |     \\]
19 |    that we will denote like ordinary (diagrammatic) function composition,
20 | 5. such that the following equations hold:
21 |   \\[
22 |       \Idn{\bar{x}};\bar{h} = \bar{h}\qquad
23 |       \bar{h};\Idn{\bar{y}} = \bar{h}\qquad
24 |       \bar{f};(\bar{g};\bar{h}) = (\bar{f};\bar{g});\bar{h}
25 |   \\]
26 | Note that these are well-defined because of the corresponding
27 | laws for the base category $B$.
28 | 
29 | **Notation**. When we have too many subscripts, we will write $E[x]$ instead of $E\Sub{x}$.
30 | 


--------------------------------------------------------------------------------
/_nodes/0001.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Cartesian morphisms
 3 | taxon: definition
 4 | ---
 5 | 
 6 | Let $E$ be displayed over $B$, and let $f:x\to y \in B$; a morphism
 7 | $\bar{f}:\bar{x}\to\Sub{f} \bar{y}$ in $E$ is called *cartesian* over $f$ when for
 8 | any $m:u\to x$ and $\bar{h}:\bar{u}\to\Sub{m;f} \bar{y}$ there exists a unique
 9 | $\bar{m} : \bar{u}\to\Sub{m} \bar{x}$ with $\bar{m};\bar{f} = \bar{h}$. We visualize
10 | this unique factorization of $\bar{h}$ through $\bar{f}$ over $m$ as follows:
11 | {%tex macrolib: topos%}
12 |   \begin{tikzpicture}[diagram]
13 |     \SpliceDiagramSquare{
14 |       west/style = lies over,
15 |       east/style = lies over,
16 |       north/node/style = upright desc,
17 |       height = 1.5cm,
18 |       nw = \bar{x},
19 |       ne = \bar{y},
20 |       sw = x,
21 |       north = \bar{f},
22 |       south = f,
23 |       se = y,
24 |       nw/style = pullback,
25 |     }
26 |     \node (u') [above left = 1.5cm of nw,xshift=-.5cm] {$\bar{u}$};
27 |     \node (u) [above left = 1.5cm of sw,xshift=-.5cm] {$u$};
28 |     \draw[lies over] (u') to (u);
29 |     \draw[->,bend left=30] (u') to node [sloped,above] {$\bar{h}$} (ne);
30 |     \draw[->] (u) to node [sloped,below] {$m$} (sw);
31 |     \draw[->,exists] (u') to node [desc] {$\bar{m}$} (nw);
32 |   \end{tikzpicture}
33 | {%endtex%}
34 | 
35 | Above we have used the "pullback corner" to indicate $\bar{x}\to\bar{y}$ as a
36 | cartesian map. We return to this in our discussion of the self-indexing {%ref 0003 %}
37 | of a category.
38 | 


--------------------------------------------------------------------------------
/_nodes/0002.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Cartesian fibration
 3 | taxon: definition
 4 | ---
 5 | 
 6 | A displayed category $E$ over $B$ is said to be a *cartesian fibration*, when
 7 | for each morphism $f : x \to y$ and displayed object $\bar{y}\in E\Sub{y}$, there
 8 | exists a displayed object $\bar{x}\in E\Sub{x}$ and a *cartesian* morphism
 9 | $\bar{f} : \bar{x}\to\Sub{f} \bar{y}$ in the sense of {%cref 0001%}. Note that the pair $(\bar{x},\bar{f})$ is unique up to
10 | unique isomorphism, so being a cartesian fibration is a *property* of a displayed category.
11 | 
12 | There are other variations of fibration. For instance, $E$ is said to be an
13 | *isofibration* when the condition above holds just for isomorphisms $f : x
14 | \cong y$ in the base.
15 | 


--------------------------------------------------------------------------------
/_nodes/0003.md:
--------------------------------------------------------------------------------
1 | ---
2 | title: The fundamental self-indexing
3 | ---
4 | 
5 | @include{001X}
6 | @include{001Y}
7 | 


--------------------------------------------------------------------------------
/_nodes/0004.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Displayed and fibered functors
 3 | ---
 4 | 
 5 | Let $E$ be displayed over $B$ and let $F$ be displayed over $C$. If $U:B \to C$
 6 | is an ordinary functor, than a *displayed functor* from $E$ to $F$ over $U$ is
 7 | given by the following data:
 8 | 
 9 | 1. for each displayed object $\bar{x}\in E\Sub{x}$, a displayed object $\bar{U}\bar{x}\in F\Sub{Ux}$,
10 | 2. for each displayed morphism $\bar{f} : \bar{x}\to\Sub{f}\bar{y}$, a displayed morphism $\bar{U}\bar{f} : \bar{U}\bar{x}\to\Sub{Uf}\bar{U}\bar{y}$,
11 | 3. such that the assignment $\bar{U}f$ preserves displayed identities and displayed composition.
12 | 
13 | From this notion, we can see the variation of displayed categories over their
14 | base categories itself has a "displayed categorical" structure; up to size
15 | issues, we could speak of the displayed *bicategory* of displayed categories.
16 | 
17 | **Note.** The correct notion of morphism between cartesian fibrations is given
18 | by displayed functors that preserve cartesian maps. We will call these *fibered
19 | functors*.
20 | 
21 | 


--------------------------------------------------------------------------------
/_nodes/0005.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Fiber categories and vertical maps
 3 | ---
 4 | 
 5 | Let $E$ be a category displayed over $B$. A *vertical map* in $E$ is defined to be one that
 6 | lies over the identity map in $B$.
 7 | For every $b\in B$, there the collection $E\Sub{b}$ of displayed objects has the
 8 | structure of a category; in particular, we set $E\Sub{b}(u,v)$ to be the collection
 9 | of vertical maps $u\to\Sub{\Idn{b}}v$.
10 | 


--------------------------------------------------------------------------------
/_nodes/0006.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: "Example: the family fibration"
 3 | ---
 4 | 
 5 | Any ordinary category $C$ can be viewed as a displayed category $\FAM{C}$ over $\SET$:
 6 | 
 7 | 1. For $S\in \SET$, an object in $\FAM{C}[S]$ is specified by a functor $C^S$
 8 |    where $S$ is regarded as a discrete category.
 9 | 2. Given $f : S \to T$ in $\SET$ and $x\in C^S$ and $y\in C^T$, a morphism
10 |    $x \to\Sub{f} y$ is given by a morphism $x\to \InvImg{f}y$ in $C^S$ where
11 |    $\InvImg{f} : C^T \to C^S$ is precomposition with $f$.
12 | 
13 | The displayed category $\FAM{C}$ is in fact a cartesian fibration. This family
14 | fibration is the starting point for developing a *relative* form of category
15 | theory, the purpose of this lecture. By analogy with viewing an ordinary
16 | category $C$ as a fibration $\FAM{C}$ over $\SET$, we may reasonably define a
17 | "relative category" over another base $B$ to be a fibration over $B$.
18 | 
19 | This story for relative category theory reflects the way that ordinary
20 | categories are "based on" $\SET$ in some sense in spite of the fact that they
21 | do not necessarily have sets of objects or even sets of morphisms between
22 | objects. Being small and locally small respectively will later be seen to be
23 | properties of a family fibration over an arbitrary base $B$, strictly
24 | generalizing the classical notions.
25 | 


--------------------------------------------------------------------------------
/_nodes/0007.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Change of base
 3 | ---
 4 | 
 5 | Suppose that $E$ is displayed over $B$ and $F : X\to B$ is a
 6 | functor; then we may define a displayed category $\InvImg{F}E$ as over $X$ follows:
 7 | 
 8 | 1. An object of $(\InvImg{F}E)\Sub{x}$ is an object of $E\Sub{Fx}$.
 9 | 
10 | 2. Given $\bar{x}\in (\InvImg{F}E)\Sub{x}$, $\bar{y}\in (\InvImg{F}E)\Sub{y}$ and $f : x
11 |    \to y$, a morphism $\bar{x}\to\Sub{f}\bar{y}$ in $\InvImg{F}E$ is given by a morphism $\bar{x}\to\Sub{Ff}\bar{y}$ in $E$.
12 | 
13 | We visualize the change of base scenario as follows:
14 | {%tex macrolib: topos%}
15 | \DiagramSquare{
16 |   nw/style = pullback,
17 |   west/style = lies over,
18 |   east/style = lies over,
19 |   height = 1.5cm,
20 |   nw = \InvImg{F}{E},
21 |   ne = E,
22 |   sw = X,
23 |   se = B,
24 |   south = F,
25 | }
26 | {%endtex%}
27 | 


--------------------------------------------------------------------------------
/_nodes/0008.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Displayed categories and fibrations
 3 | ---
 4 | 
 5 | @include{0000}
 6 | @include{0001}
 7 | @include{0002}
 8 | @include{0003}
 9 | @include{0014}
10 | @include{0029}
11 | @include{0004}
12 | @include{0005}
13 | @include{000Q}
14 | @include{0006}
15 | @include{0007}
16 | 
17 | 
18 | @include{002J}
19 | 


--------------------------------------------------------------------------------
/_nodes/0009.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: The Grothendieck construction
 3 | ---
 4 | 
 5 | @include{000A}
 6 | @include{000B}
 7 | @include{000D}
 8 | 
 9 | {{page.refs}}
10 | 


--------------------------------------------------------------------------------
/_nodes/000A.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: The total category and its projection
 3 | ---
 4 | 
 5 | Note that any displayed category $E$ over $B$ can be viewed as an undisplayed
 6 | category $\TotCat{E}$ equipped with a projection functor $p\Sub{E}: \TotCat{E}\to
 7 | B$; in this case $\TotCat{E}$ is called the *total category* of $E$.
 8 | 
 9 | 1. An object of $\TotCat{E}$ is given by a pair $(x,\bar{x})$ where $x\in B$ and
10 |    $\bar{x}\in E\Sub{x}$.
11 | 2. A morphism $(x,\bar{x})\to (y,\bar{y})$ in $\TotCat{E}$ is given by a pair $(f,\bar{f})$ where
12 |    $f:x\to y$ and $\bar{f}:\bar{x}\to\Sub{f}\bar{y}$.
13 | 
14 | The construction of the total category of a displayed category is called the *Grothendieck construction.*
15 | 
16 | 
17 | @include{001T}
18 | 


--------------------------------------------------------------------------------
/_nodes/000B.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Displayed categories from functors
 3 | ---
 4 | 
 5 | In many cases, one starts with a functor $P:E\to B$; if it were meaningful to
 6 | speak of *equality* of objects in an arbitrary category then there would be an
 7 | obvious construction of a displayed category $P\Sub{\bullet}$ from $P$; we would
 8 | simply set $P\Sub{x}$ to be the collection of objects $u\in E$ such that $Pu=x$. As
 9 | it stands there is a more subtle version that will coincide up to categorical
10 | equivalence with the naïve one in all cases that the latter is meaningful.
11 | 
12 | 1. We define an object of $P\Sub{x}$ to be a pair $(u,\phi\Sub{u})$ where $u\in E$ and
13 |    $\phi\Sub{u} : Pu\cong x$. It is good to visualize such a pair as a "crooked
14 |    leg" like so:
15 | {%tex macrolib: topos%}
16 | \begin{tikzpicture}[diagram]
17 | \node (u) {$u$};
18 | \node (Pu) [below = 1cm of u] {$Pu$};
19 | \node (x) [right = 1.5cm of Pu] {$x$};
20 | \draw[lies over] (u) to (Pu);
21 | \draw[->] (Pu) to node [below] {$\phi\Sub{u}$} (x);
22 | \end{tikzpicture}
23 | {%endtex%}
24 | 
25 | 2. A morphism $(u,\phi\Sub{u})\to\Sub{f} (v,\phi\Sub{v})$ over $f : x \to y$ is given by
26 |    a morphism $h : u\to v$ that lies over $f$ modulo the isomorphisms
27 |    $\phi\Sub{u},\phi\Sub{v}$ in sense depicted below:
28 | {%tex macrolib: topos%}
29 | \begin{tikzpicture}[diagram]
30 | \node (pu) {$Pu$};
31 | \node (pv) [right = of pu] {$Pv$};
32 | \node (x) [below left = 1.5cm of pu] {$x$};
33 | \node (y) [below right = 1.5cm of pv] {$y$};
34 | \node (u) [above = 1.5cm of pu] {$u$};
35 | \node (v) [above = 1.5cm of pv] {$v$};
36 | \draw[lies over] (u) to (pu);
37 | \draw[lies over] (v) to (pv);
38 | \draw[->] (u) to node [above] {$h$} (v);
39 | \draw[->] (x) to node [sloped,above] {$\phi\Sub{u}\Sup{-1}$} (pu);
40 | \draw[->] (pu) to node [upright desc] {$Ph$} (pv);
41 | \draw[->] (pv) to node [sloped,above] {$\phi\Sub{v}$} (y);
42 | \draw[->,bend right=30] (x) to node [below] {$f$} (y);
43 | \end{tikzpicture}
44 | {%endtex%}
45 | 
46 | @include{001U}
47 | 
48 | We have a functor $\TotCat{P\Sub{\bullet}}\to E$ taking a pair $(x,(u,\phi\Sub{u}))$ to
49 | $u$.
50 | 
51 | @include{001V}
52 | @include{001W}
53 | 
54 | 
55 | @include{000C}
56 | 


--------------------------------------------------------------------------------
/_nodes/000C.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Relationship to Street's fibrations
 3 | ---
 4 | 
 5 | In classical category theory, cartesian fibrations are defined by
 6 | Grothendieck {%cite sga:1 -A%} to be certain functors $E\to B$ such that any morphism $f:x\to Pv$
 7 | in $B$ lies strictly underneath a cartesian morphism in $E$. As we have
 8 | discussed, this condition cannot be formulated unless equality is meaningful
 9 | for the collection of objects of $B$.
10 | 
11 | There is an alternative definition of cartesian fibration {%cite street:1980%} that avoids
12 | equality of objects; here we require for each $f:x\to Pv$ a cartesian morphism
13 | $h:\InvImg{f}v \to v$ together with an isomorphism $\phi : P(\InvImg{f}v)\cong x$
14 | such that $\phi^{-1};Ph = f$.
15 | 
16 | By unrolling definitions, it is not difficult to see that the displayed
17 | category $P\Sub{\bullet}$ is a cartesian fibration in our sense if and only if the functor
18 | $P:E\to B$ was a fibration in Street's sense. Moreover, it can be seen that the
19 | Grothendieck construction yields a *Grothendieck* fibration
20 | $\TotCat{P\Sub{\bullet}}\to B$; hence we have introduced by accident a convenient
21 | destription of the *strictification* of Street fibrations into equivalent
22 | Grothendieck fibrations.
23 | 


--------------------------------------------------------------------------------
/_nodes/000D.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Iteration and pushforward
 3 | ---
 4 | 
 5 | It also makes sense to speak of categories displayed over other displayed
 6 | categories; one way to formalize this notion is as follows. Let $E$ be
 7 | displayed over $B$; we define a category displayed over $E$ to be simply a
 8 | category displayed over the total category $\TotCat{E}$.
 9 | 
10 | Now let $F$ be displayed over $E$ over $B$. Then we may regard $F$ as a
11 | displayed category $B\Sub{!}F$ over $B$ as follows:
12 | 
13 | 1. An object of $(B\Sub{!}F)\Sub{x}$ is a pair $(\bar{x},{\ddot{x}})$ with $\bar{x}\in E\Sub{x}$ and ${\ddot{x}}\in F\Sub{\bar{x}}$.
14 | 2. A morphism $(\bar{x},{\ddot{x}})\to\Sub{f}(\bar{y},{\ddot{y}})$ is given by a pair $(\bar{f},{\ddot{f}})$ where $\bar{f}:\bar{x}\to\Sub{f}\bar{y}$ in $E$ and ${\ddot{f}}:{\ddot{x}}\to\Sub{\bar{f}} {\ddot{y}}$ in $F$.
15 | 
16 | By virtue of {%cref 000B%}, we may define the *pushforward* of a displayed category along a functor. In particular, let $E$
17 | be displayed over $B$ and let $U:B\to C$ be an ordinary functor; then we may
18 | obtain a displayed category $U\Sub{!}E$ over $C$ as follows:
19 | 
20 | 1. First we construct the displayed category $U\Sub{\bullet}$ corresponding to the
21 |    functor $U:B \to C$.
22 | 2. We recall that there is a canonical equivalence of categories
23 |    $\TotCat{U\Sub{\bullet}}\to B$.
24 | 3. Because $E$ is displayed over $B$, we may regard it as displayed over the
25 |    equivalent total category $\TotCat{U\Sub{\bullet}}$ by
26 |    change of base {%pref 0007%}.
27 | 4. Hence we may define the pushforward $U\Sub{!}E$ to be the displayed category $(U\Sub{\bullet})\Sub{!}E$ as defined above.
28 | 
29 | 


--------------------------------------------------------------------------------
/_nodes/000E.md:
--------------------------------------------------------------------------------
1 | ---
2 | title: Properties of fibrations
3 | ---
4 | 
5 | @include{000F}
6 | @include{000J}
7 | @include{002E}
8 | @include{002L}
9 | 


--------------------------------------------------------------------------------
/_nodes/000F.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Locally small fibrations
 3 | ---
 4 | 
 5 | There are a number of (equivalent) variations on the definition of a locally
 6 | small fibration. We attempt to provide
 7 | some intuition for these definitions.
 8 | 
 9 | @include{000G}
10 | @include{000H}
11 | @include{000I}
12 | 


--------------------------------------------------------------------------------
/_nodes/000G.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: "Warmup: locally small family fibrations"
 3 | ---
 4 | 
 5 | An ordinary category $E$ is called *locally small* when for any $x,y\in E$ the
 6 | collection of morphisms $x\to y$ is a set.  This property of $E$ can be
 7 | rephrased in terms of its *family fibration* {% pref 0006 %} $\FAM{E}$ over
 8 | $\SET$ as follows.
 9 | 
10 | Consider an index set $I\in \SET$ and two families $u,v\in C^I$. We may define
11 | an $I$-indexed collection $[u,v]\Sub{i\in I}$ consisting (pointwise) of all the
12 | morphisms $u\Sub{i}\to v\Sub{i}$ in $C$:
13 | {%tex macrolib: topos%}[u,v]\Sub{i} = \{ f \mid f: u\Sub{i}\to v\Sub{i} \}{%endtex%}
14 | 
15 | If $C$ is locally small, $[u,v]\Sub{i\in I}$ is in fact a family of sets for
16 | any $I\in\SET$ as each $[u,v]\Sub{i}$ is a set. Conversely, if $[u,v]\Sub{i\in I}$
17 | is a family of sets for any $I\in \SET$, then $C$ is locally small as we may
18 | consider in particular the case that $I=\mathbf{1}$.
19 | 


--------------------------------------------------------------------------------
/_nodes/000H.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: A more abstract formulation
 3 | ---
 4 | 
 5 | We will reformulate the above in a way that uses only the language that makes
 6 | sense for an arbitrary cartesian fibration, though for now we stick with $\FAM{C}$. Given
 7 | $u,v\in \FAM{C}[I]$, we have a "relative hom family" $[u,v]\in\Sl{\SET}{I}$,
 8 | defined in {%cref 000G%}. The fact that each $[u,v]\Sub{i}$ is the set of all
 9 | morphisms $u\Sub{i}\to v\Sub{i}$ can be rephrased more abstractly.
10 | 
11 | First we consider the restriction of $u\in \FAM{C}[I]$ to $\FAM{C}[[u,v]]$ as follows:
12 | {%tex macrolib: topos%}
13 | \DiagramSquare{
14 |   nw/style = pullback,
15 |   west/style = lies over,
16 |   east/style = lies over,
17 |   north/style = {->,exists},
18 |   height = 1.5cm,
19 |   nw = \InvImg{[u,v]}u,
20 |   ne = u,
21 |   sw = {[u,v]},
22 |   se = I,
23 |   south = p\Sub{[u,v]},
24 |   north = \bar{p}\Sub{[u,v]}
25 | }
26 | {%endtex%}
27 | Explicitly the family $\InvImg{[u,v]}u$ is indexed in a pair of an element $i\in I$ and a morphism $u\Sub{i}\to v\Sub{i}$.
28 | We can think of $\InvImg{[u,v]}u$ as the object of elements of $u\Sub{i}$ indexed in pairs $(i,u\Sub{i}\to v\Sub{i})$.
29 | 
30 | There is a canonical map
31 | $\epsilon\Sub{[u,v]}:\InvImg{[u,v]}u\to\Sub{p\Sub{[u,v]}} v$ that
32 | "evaluates" each indexing morphism $u\Sub{i}\to v\Sub{i}$.
33 | 
34 | That each $[u,v]\Sub{i}$ is the set of all morphisms $u\Sub{i}\to v\Sub{i}$ can be
35 | rephrased as a universal property: for any family $h\in\Sl{\SET}{I}$ and
36 | morphism $\epsilon\Sub{h} : \InvImg{h}u\to\Sub{h} v$ in $\FAM{C}$, there is a
37 | unique cartesian map $\InvImg{h}u\to \InvImg{[u,v]}u$ factoring $\epsilon\Sub{h}$ through $\epsilon\Sub{[u,v]}$
38 | in the sense depicted below:
39 | {%tex macrolib: topos%}
40 | \begin{tikzpicture}[diagram]
41 |   \SpliceDiagramSquare{
42 |     height = 1.5cm,
43 |     width = 3cm,
44 |     west/style = lies over,
45 |     east/style = lies over,
46 |     north/node/style = upright desc,
47 |     south/node/style = upright desc,
48 |     nw = \InvImg{[u,v]}u,
49 |     ne = v,
50 |     sw = {[u,v]},
51 |     se = I,
52 |     south = p\Sub{[u,v]},
53 |     north = \epsilon\Sub{[u,v]},
54 |   }
55 |   \node (h/u) [pullback,left = of nw] {$\InvImg{h}u$};
56 |   \node (h) [left = of sw] {$h$};
57 |   \draw[bend left,->] (h/u) to node [sloped,above] {$\epsilon\Sub{h}$} (ne);
58 |   \draw[lies over] (h/u) to (h);
59 |   \draw[->,exists] (h) to (sw);
60 |   \draw[->,exists] (h/u) to (nw);
61 |   \draw[->,bend right=30] (h) to node [below] {$p\Sub{h}$} (se);
62 | \end{tikzpicture}
63 | {%endtex%}
64 | 
65 | To convince ourselves of this, we note that the family $H \coloneqq \brc{u\Sub{i}\to v\Sub{i}}\Sub{i\in I}$ itself satisfies the universal property above. Indeed, fix a candidate $h \in \Sl{\SET}{I}$ equipped with a map $\epsilon\Sub{h} : \InvImg{h}u \to_h v$. Unfolding the meaning of this map in set theoretical notation, we see that it amounts to a family of maps $\epsilon\Sub{h}[i] : \prod\Sub{x\in h_i} \brc{u_i\to v_i}$ for each $i\in I$; such a family immediately induces the desired map $h\to H$.
66 | {%qed%}
67 | 


--------------------------------------------------------------------------------
/_nodes/000I.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: The definition of local smallness
 3 | ---
 4 | 
 5 | Based on our explorations above, we are now prepared to write down (and
 6 | understand) the proper definition of local smallness for an arbitrary cartesian fibration
 7 | $E$ over $B$, which should be thought of as a (potentially large) category
 8 | relative to $B$.
 9 | 
10 | 
11 | @include{001C}
12 | @include{001B}
13 | 
14 | 


--------------------------------------------------------------------------------
/_nodes/000J.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Globally small fibrations
 3 | ---
 4 | 
 5 | In ordinary category theory, a category $C$ is called *small* when the objects
 6 | of $C$ can be arranged into a set, and so can for every $x,y\in C$ the
 7 | collection of morphisms $x\to y$. It is useful to separate these two conditions
 8 | when we generalize them to fibrations. The latter is called *local smallness*
 9 | and is discussed and defined in {%cref 000F%}; the former is called *global
10 | smallness* by Jacobs {%cite jacobs:1999 -A%} and factors through an important concept: the *generic object*.
11 | 
12 | @include{001D}
13 | 
14 | @include{000K}
15 | 
16 | @include{000P}
17 | 


--------------------------------------------------------------------------------
/_nodes/000K.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Generic objects
 3 | ---
 4 | 
 5 | Up to equivalence of categories, we may detect global smallness of a category $C$ from the perspective of the family fibration $\FAM{C}$ {%pref 0006%}. In particular, a category is equivalent to a globally small category when its family fibration has a *generic object* in the following sense.
 6 | 
 7 | @include{001E}
 8 | 
 9 | @include{000L}
10 | 


--------------------------------------------------------------------------------
/_nodes/000L.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Globally small categories
 3 | taxon: theorem
 4 | ---
 5 | 
 6 | An ordinary category $C$ is equivalent to a globally small category if and only if the fibration $\FAM{C}$ has a generic object.
 7 | 
 8 | **Proof.**
 9 | To see that this is the case, suppose that $C$ has a set of objects. Then
10 | $C\in\SET$ and we define $\lfloor{C}\rfloor$ to be the displayed object
11 | $\brc{x}\Sub{x\in C}\in \FAM{C}[C]$. Fixing $I\in \SET$ and $z\in C^I$, we
12 | consider the cartesian map displayed over $z : I \to C$:
13 | {%tex macrolib: topos%}
14 | \DiagramSquare{
15 |   height = 1.5cm,
16 |   width = 3cm,
17 |   west/style = lies over,
18 |   east/style = lies over,
19 |   sw = I,
20 |   se = C,
21 |   south = z,
22 |   ne = \brc{x}\Sub{x\in C},
23 |   nw = {z = \brc{z\,i}\Sub{i\in I} }
24 | }
25 | {%endtex%}
26 | 
27 | Conversely assume that $\FAM{C}$ has a generic object $\bar{u}\in\FAM{C}[U]$
28 | for some $U\in \SET$; then we may equip $U$ with the structure of a globally
29 | small category such that $U$ is equivalent to $C$, using a construction that is similar to our implementation of the opposite fibration {%pref 000Q%}. In particular we define a
30 | morphism $x\to y\in U$ to be given by the following data:
31 | 
32 | 1. a cartesian map $a\to\Sub{x} \bar{u}$ over $x : 1\to U$,
33 | 2. a cartesian map $b\to\Sub{y} \bar{u}$ over $y : 1\to U$,
34 | 3. and a vertical map $h:a\to b$ in $\FAM{C}[1]\simeq C$,
35 | 
36 | such that $(a\Sub{1},b\Sub{1},h\Sub{1})$ is identified with $(a\Sub{2},b\Sub{2},h\Sub{2})$ when $h_1$ and $h_2$ are equal modulo the (unique) vertical isomorphisms between the cartesian lifts in the sense depicted below:
37 | {%tex macrolib: topos%}
38 | \begin{tikzpicture}[diagram]
39 |   \SpliceDiagramSquare{
40 |     nw = a_1,
41 |     ne = a_2,
42 |     sw = b_1,
43 |     se = b_2,
44 |     north = \cong,
45 |     south = \cong,
46 |     west = h_1,
47 |     east = h_2,
48 |     east/node/style = upright desc,
49 |   }
50 |   \node (u) [between = ne and se, xshift = 2cm] {$\bar{u}$};
51 |   \draw[->] (ne) to node [sloped,above] {cart.}(u);
52 |   \draw[->] (se) to node [sloped,below] {cart.} (u);
53 | \end{tikzpicture}
54 | {%endtex%}
55 | 
56 | Remember that a cartesian map $a\to\Sub{x}\bar{u}$ is standing for a *choice*
57 | of an object of $C$ encoded by $x\in U$. Because such choices are unique only
58 | up to isomorphism, we must include them explicitly in the data.
59 | {%qed%}
60 | 


--------------------------------------------------------------------------------
/_nodes/000N.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Small fibrations and internal categories
 3 | ---
 4 | 
 5 | The purpose of this section is to develop the relationship between *internal
 6 | categories* (categories defined in the internal language of a category $B$) and
 7 | cartesian fibrations over $B$, generalizing the relationship between categories internal
 8 | to $\SET$ (i.e. small categories) and their family fibrations over $\SET$.
 9 | 
10 | @include{001Q}
11 | 
12 | We have already seen in {%cref 000G%} and {%cref 000L%} that smallness in the
13 | sense of the definition above appropriately generalizes the ordinary notion of
14 | smallness for categories over $\SET$. Another perspective on smallness is given
15 | by the *internal language*, in which a category is viewed as an algebra for the
16 | "theory of categories" computed in another category with enough structure. The
17 | notion of internal categories is credited (independently) to Alexander
18 | Grothendieck and Charles Ehresmann.
19 | 
20 | @include{000O}
21 | @include{000V}
22 | @include{0011}
23 | @include{000Z}
24 | 


--------------------------------------------------------------------------------
/_nodes/000O.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Internal categories
 3 | ---
 4 | 
 5 | The notion of a (meta-)category is an {%nlab essentially algebraic theory%}. As such it is
 6 | possible to compute *models* of this theory in any category that has finite
 7 | limits.
 8 | 
 9 | @include{001A}
10 | 


--------------------------------------------------------------------------------
/_nodes/000P.md:
--------------------------------------------------------------------------------
1 | ---
2 | taxon: definition
3 | title: Globally small fibration
4 | ---
5 | 
6 | A cartesian fibration $E$ over $B$ is called *globally small* when it has a generic object in the sense of {%cref 001E%}.
7 | 


--------------------------------------------------------------------------------
/_nodes/000Q.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Opposite categories
 3 | ---
 4 | 
 5 | We adapt Bénabou's construction as reported by Streicher {%cite streicher:2018 -A%}.
 6 | 
 7 | @include{001Z}
 8 | @include{000T}
 9 | @include{000U}
10 | 
11 | 
12 | @include{000S}
13 | 


--------------------------------------------------------------------------------
/_nodes/000R.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Foundational assumptions
 3 | ---
 4 | 
 5 | @include{001F}
 6 | 
 7 | In our definition above, we have not imposed any restrictions on what kinds of
 8 | things the objects and morphisms are; our definition is pre-mathematical, so we
 9 | do not assume beforehand that there is a such thing as a collection of "all"
10 | meta-categories.
11 | 
12 | **Remark.** We may define analogous notions of meta-functor, etc. But we do not
13 | assume that the notion of "all meta-functors $\mathfrak{C}\to\mathfrak{D}$" is
14 | well-defined; the notion is entirely schematic.
15 | 
16 | 
17 | **Assumption.** We assume a meta-category $\BoldSymbol{\mathfrak{Coll}}$ whose
18 | objects we will refer to as "collections". We assume that the meta-category of all collections satisfies the axioms of Lawvere's {%nlab ETCS%}.
19 | 
20 | @include{001G}
21 | 
22 | Consequently there exists a meta-category $\BoldSymbol{\mathfrak{Cat}}$ of all
23 | categories. Following [Lawvere](http://rfcwalters.blogspot.com/2010/10/old-post-why-are-we-concerned-fw.html) (but deviating from some other authors that
24 | ground the notion of meta-categories in *classes*) we notice that
25 | $\BoldSymbol{\mathfrak{Cat}}$ is cartesian closed; in other words, all functor
26 | categories exist regardless of size.
27 | 
28 | **Assumption.** At times we may assume that there exists a *category*
29 | $\SET\subseteq\BoldSymbol{\mathfrak{Coll}}$ of collections that we will refer
30 | to as sets, such that $\SET$ is closed under the axioms of {%nlab ETCS%}. Rather than work with $\SET$ at all
31 | times, our approach is to use the tools of relative category theory to
32 | objectify the notions of "small" and "locally small" category over any category
33 | $B$, generalizing the role of $\SET$ from classical category theory.
34 | 


--------------------------------------------------------------------------------
/_nodes/000S.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Exegesis of opposite categories
 3 | ---
 4 | 
 5 | The construction of fibered opposite categories {%pref 001Z%} does appear quite
 6 | involved, but it can be seen to be inevitable from the perspective of the fiber
 7 | categories $\OpCat{E}\Sub{x}$ {%pref 0005%}. Indeed, let $u,v\in
 8 | \OpCat{E}\Sub{x}$ and fix a *vertical* map $h : u \to v\in \OpCat{E}\Sub{x}$;
 9 | by unfolding definitions, we see that the vertical map $h : u \to v$ is
10 | uniquely determined by a morphism $v\to u\in E\Sub{x}$.
11 | 
12 | **Proof.** A displayed morphism $u\to\Sub{\Idn{x}} v\in \OpCat{E}$ is
13 | determined by a span $\brc{u\leftarrow v\Sub{x} \to v}\in E$ where the right-hand
14 | map is cartesian over $\Idn{x} : x\to x$ and the left-hand map is vertical, taken up to the identification of
15 | different cartesian lifts $v\Sub{x}\to x$. A displayed morphism that is cartesian over the identity is an isomorphism; hence, displayed morphisms $u\to\Sub{\Idn{x}} v\in\OpCat{E}$ are equivalently determined by vertical maps $v\to u \in E$.
16 | {%qed%}
17 | 


--------------------------------------------------------------------------------
/_nodes/000T.md:
--------------------------------------------------------------------------------
1 | ---
2 | title: Characterization of cartesian maps
3 | ---
4 | 
5 | There is a simple characterization of cartesian maps in $\OpCat{E}$.
6 | 
7 | @include{0020}
8 | 


--------------------------------------------------------------------------------
/_nodes/000U.md:
--------------------------------------------------------------------------------
1 | ---
2 | title: Cartesian lifts in the opposite category
3 | ---
4 | 
5 | The foregoing characterization of cartesian maps in $\OpCat{E}$
6 | immediately implies that $\OpCat{E}$ is fibered over $B$.
7 | 
8 | @include{0021}
9 | 


--------------------------------------------------------------------------------
/_nodes/000V.md:
--------------------------------------------------------------------------------
1 | ---
2 | title: The externalization of an internal category
3 | ---
4 | 
5 | @include{001R}
6 | @include{000W}
7 | @include{000X}
8 | @include{000Y}
9 | 


--------------------------------------------------------------------------------
/_nodes/000W.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | taxon: lemma
 3 | ---
 4 | 
 5 | The externalization {%ref 000V%} is a cartesian fibration.
 6 | 
 7 | **Proof.**
 8 | Given an object $v\in \brk{C}\Sub{y}$ and a
 9 | morphism $f:x\to y$ in $E$, we may define a cartesian lift
10 | $\InvImg{f}v\to\Sub{f} v$ by setting $\InvImg{f}v = v\circ f : x \to C\Sub{0}$.
11 | {%qed%}
12 | 


--------------------------------------------------------------------------------
/_nodes/000X.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | taxon: lemma
 3 | ---
 4 | 
 5 | The externalization {%ref 000V%} is globally small
 6 | 
 7 | **Proof.**
 8 | We may choose a generic object {%ref 000K%} for $\brk{C}$, namely the identity element $(C\Sub{0},\Idn{C\Sub{0}})\in \TotCat{\brk{C}}$. Given any object $(x,u)\in \TotCat{\brk{C}}$ the cartesian map $(x,u)\to (C\Sub{0},\Idn{C\Sub{0}})$ is given as follows:
 9 | {%tex macrolib: topos%}
10 | \begin{tikzpicture}[diagram]
11 |   \node (x) {$x$};
12 |   \node (y) [below = of x] {$C\Sub{0}$};
13 |   \node (C/1) [right = of x] {$C\Sub{1}$};
14 |   \node (C/0) [below = of C/1] {$C\Sub{0}$};
15 |   \node (C/0') [above = of C/1] {$C\Sub{0}$};
16 |   \draw[->] (x) to node [upright desc] {$\Con{idn}\Sub{u}$} (C/1);
17 |   \draw[->] (C/1) to node [right] {$t$} (C/0);
18 |   \draw[->] (C/1) to node [right] {$s$} (C/0');
19 |   \draw[->] (y) to node [below] {$\Idn{C\Sub{0}}$} (C/0);
20 |   \draw[->] (x) to node [left] {$u$} (y);
21 |   \draw[->] (x) to node [sloped,above] {$u$} (C/0');
22 | \end{tikzpicture}
23 | {%endtex%}
24 | {%qed%}
25 | 


--------------------------------------------------------------------------------
/_nodes/000Y.md:
--------------------------------------------------------------------------------
  1 | ---
  2 | taxon: lemma
  3 | ---
  4 | 
  5 | The externalization {%ref 000V%} is locally small
  6 | 
  7 | **Proof.**
  8 | Fix $x\in E$ and $u,v\in
  9 | \brk{C}\Sub{x}$, we must exhibit a terminal object to the (total) category
 10 | $\TotCat{\CandHom{x}{u}{v}}$ of "hom candidates" defined in {%ref 000I%}.
 11 | First we define $\brk{u,v}$ to be the following pullback in $E$:
 12 | {%tex macrolib: topos%}
 13 | \DiagramSquare{
 14 |   nw/style = pullback,
 15 |   ne = C\Sub{1},
 16 |   se = C\Sub{0}\times C\Sub{0},
 17 |   east = {(s,t)},
 18 |   south = {(u,v)},
 19 |   sw = x,
 20 |   nw = \brk{u,v},
 21 |   west = p,
 22 |   north = q,
 23 |   west/style = {->,exists},
 24 |   north/style = {->,exists},
 25 | }
 26 | {%endtex%}
 27 | 
 28 | We define $\overline{p}:\InvImg{\brk{u,v}}{u}\to\Sub{p} u\in \brk{C}\Sub{\brk{u,v}}$ to be
 29 | the cartesian lift of $u\in \brk{C}\Sub{x}$ along $p:\brk{u,v}\to x$:
 30 | {%tex macrolib: topos%}
 31 | \DiagramSquare{
 32 |   height = 1.5cm,
 33 |   nw/style = pullback,
 34 |   west/style = lies over,
 35 |   east/style = lies over,
 36 |   north/style = {->,exists},
 37 |   ne = u,
 38 |   nw = \InvImg{\brk{u,v}}u,
 39 |   sw = \brk{u,v},
 40 |   se = x,
 41 |   south = p,
 42 |   north = \overline{p},
 43 | }
 44 | {%endtex%}
 45 | 
 46 | We need to define a displayed evaluation map
 47 | $\epsilon : \InvImg{\brk{u,v}}u\to\Sub{p} v$; unraveling the definition of a displayed
 48 | morphism in the externalization of $C$, we choose the following diagram:
 49 | 
 50 | {%tex macrolib: topos%}
 51 |   \begin{tikzpicture}[diagram]
 52 |     \SpliceDiagramSquare{
 53 |       ne = C\Sub{1},
 54 |       se = C\Sub{0},
 55 |       east = t,
 56 |       nw = \brk{u,v},
 57 |       sw = x,
 58 |       west = p,
 59 |       south = v,
 60 |       north = q,
 61 |       north/style = {magenta,->},
 62 |       north/node/style = upright desc,
 63 |     }
 64 |     \node (nne) [above = of ne] {$C\Sub{0}$};
 65 |     \node (nnw) [above = of nw] {$x$};
 66 |     \draw[->] (ne) to node [right] {$s$} (nne);
 67 |     \draw[->] (nw) to node [left] {$p$} (nnw);
 68 |     \draw[->] (nnw) to node [above] {$u$} (nne);
 69 |     \draw[->] (nw) to node [desc] {$\InvImg{\brk{u,v}}{u}$} (nne);
 70 |   \end{tikzpicture}
 71 | {%endtex%}
 72 | 
 73 | 
 74 | Putting all this together, we assert that the terminal object of
 75 | $\TotCat{\CandHom{x}{u}{v}}$ is the following span in $\brk{C}$:
 76 | {%tex macrolib: topos%}
 77 | \begin{tikzpicture}[diagram]
 78 | \SpliceDiagramSquare<l/>{
 79 |   height = 1.5cm,
 80 |   west/style = lies over,
 81 |   east/style = lies over,
 82 |   north/style = <-,
 83 |   south/style = <-,
 84 |   ne/style = ne pullback,
 85 |   ne = \InvImg{\brk{u,v}}{u},
 86 |   se = \brk{u,v},
 87 |   sw = x,
 88 |   nw = u,
 89 |   south = p,
 90 |   north = \bar{p},
 91 | }
 92 | \SpliceDiagramSquare<r/>{
 93 |   height = 1.5cm,
 94 |   west/style = lies over,
 95 |   east/style = lies over,
 96 |   glue = west,
 97 |   glue target = l/,
 98 |   ne = v,
 99 |   se = x,
100 |   north = \epsilon,
101 |   south = p,
102 | }
103 | \end{tikzpicture}
104 | {%endtex%}
105 | 
106 | Fixing another such candidate hom span $\brc{u \leftarrow \bar{h}\rightarrow v}\in\TotCat{\CandHom{x}{u}{v}}$, according to {%ref 000I%} we must exhibit a unique cartesian morphism $\bar\alpha : \bar{h}\to \InvImg{\brk{u,v}}{u}$ making the following diagram commute:
107 |    {%tex macrolib: topos%}
108 |    \begin{tikzpicture}[diagram]
109 |      \node (u) {$u$};
110 |      \node (h) [above right = 1.5cm of u,xshift=.5cm] {$\bar{h}$};
111 |      \node (hom) [below right = 1.5cm of u,xshift=.5cm] {$\InvImg{\brk{u,v}}{u}$};
112 |      \node (v) [below right = 1.5cm of h,xshift=.5cm] {$v$};
113 |      \draw[->] (hom) to node [sloped,below] {$\bar{p}$} (u);
114 |      \draw[->] (h) to node [sloped,above] {$\bar{p}\Sub{h}$} (u);
115 |      \draw[->] (hom) to node [sloped,below] {$\epsilon$} (v);
116 |      \draw[->] (h) to node [sloped,above] {$\epsilon\Sub{h}$} (v);
117 |      \draw[->] (h) to node [upright desc] {$\bar\alpha$} (hom);
118 |    \end{tikzpicture}
119 |    {%endtex%}
120 | 
121 | First we note that the evaluation map $\epsilon\Sub{h} : \bar{h}\to v$ amounts
122 | to an internal morphism $h\to C\Sub{1}$ satisfying the appropriate
123 | compatibility conditions. Therefore we may define the base $\alpha:h\to \brk{u,v}$ of
124 | the universal map using the universal property of the pullback that defines $\brk{u,v}$:
125 | {%tex macrolib: topos%}
126 | \begin{tikzpicture}[diagram]
127 |   \SpliceDiagramSquare{
128 |     nw/style = pullback,
129 |     ne = C\Sub{1},
130 |     se = C\Sub{0}\times C\Sub{0},
131 |     east = {(s,t)},
132 |     south = {(u,v)},
133 |     sw = x,
134 |     nw = \brk{u,v},
135 |     west = p,
136 |     north = q,
137 |     north/node/style = upright desc,
138 |     west/node/style = upright desc,
139 |   }
140 |   \node (h) [above left = of nw] {$h$};
141 |   \draw[->,bend right=30] (h) to node [left] {$p_h$} (sw);
142 |   \draw[->,bend left=30] (h) to node [sloped,above] {$\epsilon_h$} (ne);
143 |   \draw[->,exists] (h) to node [desc] {$\alpha$} (nw);
144 | \end{tikzpicture}
145 | {%endtex%}
146 | 
147 | The morphism $\alpha:h\to \brk{u,v}$ defined above is the *unique* map in $E$
148 | satisfying the conditions required of the base for $\bar\alpha$; therefore, it
149 | suffices to show that there exists a cartesian morphism
150 | $\bar\alpha:\bar{h}\to\Sub{\alpha}\InvImg{\brk{u,v}}u$ since it will be unique if it
151 | exists. We define $\bar\alpha$ using the universal property of the cartesian lift:
152 | {%tex macrolib: topos%}
153 | \begin{tikzpicture}[diagram]
154 |   \SpliceDiagramSquare<l/>{
155 |     ne/style = pullback,
156 |     nw/style = dotted pullback,
157 |     west/style = lies over,
158 |     east/style = lies over,
159 |     height = 1.5cm,
160 |     se = \brk{u,v},
161 |     ne = \InvImg{\brk{u,v}}u,
162 |     sw = h,
163 |     nw = \bar{h},
164 |     south = \alpha,
165 |     north = \bar\alpha,
166 |     north/style = {->,exists},
167 |     north/node/style = upright desc,
168 |     south/node/style = upright desc,
169 |   }
170 |   \SpliceDiagramSquare<r/>{
171 |     glue = west,
172 |     glue target = l/,
173 |     east/style = lies over,
174 |     height = 1.5cm,
175 |     ne = u,
176 |     se = x,
177 |     south = p,
178 |     north = \bar{p},
179 |     north/node/style = upright desc,
180 |     south/node/style = upright desc,
181 |   }
182 |   \draw[->,bend left=30] (l/nw) to node [above] {$\bar{p}\Sub{h}$} (r/ne);
183 |   \draw[->,bend right=30] (l/sw) to node [below] {$p\Sub{h}$} (r/se);
184 | \end{tikzpicture}
185 | {%endtex%}
186 | 
187 | That $\bar{\alpha}:\bar{h}\to\Sub{\alpha}\InvImg{\brk{u,v}}u$ is cartesian follows from the
188 | "generalized pullback lemma" for cartesian morphisms {%ref 0014 %}: it suffices
189 | to observe that both $\bar{p}\Sub{h}:\bar{h}\to u$ and its second factor
190 | $\bar{p}:\InvImg{\brk{u,v}}u\to u$ are cartesian.
191 | {%qed%}
192 | 


--------------------------------------------------------------------------------
/_nodes/000Z.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: The internalization of a small fibration
 3 | ---
 4 | 
 5 | Let $C$ be a small cartesian fibration {%pref 001Q %} over $B$ a category with finite
 6 | limits, i.e. a cartesian fibration that is both locally small {%pref 001B%} and globally
 7 | small {%pref 000P%}. We will show that $C$ is equivalent to the externalization
 8 | {%ref 000V%} of an internal category {%ref 000O%} $\underline{C}$ in $B$,
 9 | namely the full internal subcategory {%ref 0011%} associated to the generic
10 | object $\bar{u}\in C$.
11 | 
12 | By {%cref 001S%} we know that the externalization of $\underline{C}$ so-defined
13 | is equivalent to the full subfibration $\FullSubfib{\bar{u}}$ of $C$ spanned by
14 | objects that are "classified" by $\bar{u}$. Because $\bar{u}$ is generic, we
15 | know that *every* object of $C$ is classified by $\bar{u}$, so we are done.
16 | 


--------------------------------------------------------------------------------
/_nodes/0010.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: The full subfibration associated to a displayed object
 3 | taxon: construction
 4 | ---
 5 | 
 6 | Let $E$ be a cartesian fibration
 7 | over $B$; then any displayed object $\bar{x} \in E\Sub{x}$ induces a *full subfibration* $\FullSubfib{\bar{u}}\subseteq E$ spanned by displayed objects that are classified by $\bar{u}$, i.e. arise from $\bar{u}$ by cartesian lift.
 8 | 
 9 | 1. An object of $\FullSubfib{\bar{u}}\Sub{x}$ is specified by an object $\bar{x}\in E\Sub{x}$ together with a cartesian morphism $\bar{x}\to \bar{u}$.
10 | 
11 | 2. Given $f:x\to y\in B$, a morphism from $\bar{x}\to \bar{u}$ to $\bar{y}\to\bar{u}\in$ over $f$ is given by any displayed morphism $\bar{x}\to\Sub{f}\bar{y}$.
12 | 
13 | 
14 | 


--------------------------------------------------------------------------------
/_nodes/0011.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: The full internal subcategory associated to a displayed object
 3 | ---
 4 | 
 5 | The full internal subfibration {%pref 0010%} associated to a displayed object $\bar{u}$ of a locally small cartesian fibration $E$ over $B$ can be seen to be equivalent to the externalization
 6 | of an internal category $\gl{\bar{u}}$ in $B$. In particular, we let the object of objects $\gl{\bar{u}}\Sub{0}$ be $u$ itself; defining the object of arrows $\gl{\bar{u}}\Sub{1}$ is more complex, making critical use of the local smalness of $E$ over $B$.
 7 | 
 8 | We will think of the fiber $E\Sub{u\times u}$ as the category of
 9 | objects indexed in the *boundary* (source and target) of a morphism. Restricting $\bar{u}$ along the source and target projections, we obtain the displayed objects of "points of the source" and "points of the target" respectively:
10 | {%tex macrolib: topos%}
11 | \DiagramSquare{
12 |   height = 1.5cm,
13 |   nw/style = pullback,
14 |   west/style = lies over,
15 |   east/style = lies over,
16 |   nw = \bar{\partial}\Sub{1},
17 |   sw = u\times u,
18 |   se = u,
19 |   ne = \bar{u},
20 |   south = \pi\Sub{1},
21 |   north = \bar{\pi}\Sub{1},
22 | }
23 | \qquad
24 | \DiagramSquare{
25 |   height = 1.5cm,
26 |   nw/style = pullback,
27 |   west/style = lies over,
28 |   east/style = lies over,
29 |   nw = \bar{\partial}\Sub{2},
30 |   sw = u\times u,
31 |   se = u,
32 |   ne = \bar{u},
33 |   south = \pi\Sub{2},
34 |   north = \bar{\pi}\Sub{2},
35 | }
36 | {%endtex%}
37 | 
38 | Because $E$ is locally small, there is an object
39 | $\brk{\bar\partial\Sub{1},\bar\partial\Sub{2}}\in \Sl{B}{u\times u}$ that
40 | behaves as the "generic hom set". We define $\gl{\bar{u}}\Sub{1}\in B$ and its
41 | source and target projections to be this very object.
42 | 
43 | @include{001S}
44 | 


--------------------------------------------------------------------------------
/_nodes/0012.md:
--------------------------------------------------------------------------------
1 | ---
2 | title: Other kinds of fibrations
3 | ---
4 | 
5 | @include{0015}
6 | @include{0013}
7 | 


--------------------------------------------------------------------------------
/_nodes/0013.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Right fibrations
 3 | ---
 4 | 
 5 | @include{001O}
 6 | 
 7 | Recall from {%cref 0005 %} that for every $b\in B$, the collection of displayed
 8 | objects $E\Sub{b}$ and vertical maps $E\Sub{1\Sub{b}}$ forms a category. When $E$ is
 9 | a right fibration over $B$, this category is in fact a *groupoid*.
10 | 
11 | @include{001P}
12 | 


--------------------------------------------------------------------------------
/_nodes/0014.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: The generalized pullback lemma
 3 | ---
 4 | 
 5 | In light of {%ref 0003 %}, the following result for displayed categories generalizes
 6 | the ordinary "pullback lemma."
 7 | 
 8 | @include{001H}
 9 | 
10 | 


--------------------------------------------------------------------------------
/_nodes/0015.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Cocartesian fibrations
 3 | ---
 4 | 
 5 | Cocartesian fibrations are a dual notion to cartesian fibrations, in which the
 6 | variance of indexing is reversed.
 7 | 
 8 | @include{0016}
 9 | 
10 | We use a "pushout corner" to indicate $\bar{x}\to\bar{y}$ as a cocartesian morphism,
11 | a notation justified by {%cref 0019 %}.
12 | 
13 | @include{0017}
14 | 
15 | **Remark.** These are also known as *opfibrations*.
16 | 
17 | @include{0018}
18 | @include{002X}
19 | @include{0019}
20 | 


--------------------------------------------------------------------------------
/_nodes/0016.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Cocartesian morphism
 3 | taxon: definition
 4 | ---
 5 | 
 6 | Let $E$ be displayed over $B$, and let $f:x\to y \in B$; a morphism
 7 | $\bar{f}:\bar{x}\to\Sub{f} \bar{y}$ in $E$ is called *cocartesian* over $f$ when for
 8 | any $m:y\to u$ and $\bar{h}:\bar{x}\to\Sub{f;m} \bar{u}$ there exists a unique
 9 | $\bar{m} : \bar{y}\to\Sub{m} \bar{u}$ with $\bar{f};\bar{m} = \bar{h}$:
10 | {%tex macrolib: topos%}
11 |   \begin{tikzpicture}[diagram]
12 |     \SpliceDiagramSquare{
13 |       west/style = lies over,
14 |       east/style = lies over,
15 |       north/node/style = upright desc,
16 |       height = 1.5cm,
17 |       nw = \bar{x},
18 |       ne = \bar{y},
19 |       sw = x,
20 |       north = \bar{f},
21 |       south = f,
22 |       se = y,
23 |       ne/style = ne pullback,
24 |     }
25 |     \node (u') [above right = 1.5cm of ne,xshift=.5cm] {$\bar{u}$};
26 |     \node (u) [above right = 1.5cm of se,xshift=.5cm] {$u$};
27 |     \draw[lies over] (u') to (u);
28 |     \draw[->,bend left=30] (nw) to node [sloped,above] {$\bar{h}$} (u');
29 |     \draw[->] (se) to node [sloped,below] {$m$} (u);
30 |     \draw[->,exists] (ne) to node [desc] {$\bar{m}$} (u');
31 |   \end{tikzpicture}
32 | {%endtex%}
33 | 


--------------------------------------------------------------------------------
/_nodes/0017.md:
--------------------------------------------------------------------------------
1 | ---
2 | title: Cocartesian fibration
3 | taxon: definition
4 | ---
5 | 
6 | A displayed category $E$ over $B$ is a *cocartesian fibration* when for each
7 | $f : x \to y\in B$ and $\bar{x}\in E\Sub{x}$, there exists a displayed object
8 | $\bar{y}\in E\Sub{y}$ and a *cocartesian* morphism $\bar{f} : \bar{x}\to\Sub{f} \bar{y}$.
9 | 


--------------------------------------------------------------------------------
/_nodes/0018.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: The total opposite of a displayed category
 3 | ---
 4 | 
 5 | @include{001I}
 6 | 
 7 | **Warning.** Do not confuse this construction with {%cref 001Z%}, which produces a
 8 | displayed category over $B$ and not $\OpCat{B}$.
 9 | 
10 | @include{001J}
11 | @include{001K}
12 | @include{001L}
13 | 


--------------------------------------------------------------------------------
/_nodes/0019.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: "Example: The dual self-indexing"
 3 | ---
 4 | 
 5 | Dually to the fundamental self-indexing {%pref 001X %}, every category $B$ can
 6 | also be displayed over itself via its *coslices* $\CoSl{x}{B}$.
 7 | 
 8 | @include{002Z}
 9 | 
10 | @include{001M}
11 | @include{001N}
12 | @include{0030}
13 | 


--------------------------------------------------------------------------------
/_nodes/001A.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Internal category
 3 | taxon: definition
 4 | ---
 5 | 
 6 | Let $E$ be a category with finite limits; then an _internal category_
 7 | in $E$ is defined by the following data:
 8 | 
 9 | 1. an object of objects $C\Sub{0}\in E$,
10 | 2. an object of morphisms $C\Sub{1}\in E$,
11 | 3. source and target morphisms $s,t:C\Sub{1}\to C\Sub{0}$,
12 | 4. a generic identity morphism $C\Sub{0}\to C\Sub{1}$,
13 | 5. a generic composition morphism $C\Sub{1}\times\Sub{C\Sub{0}}C\Sub{1}\to C\Sub{1}$,
14 | 6. satisfying a number of laws corresponding to those of a category.
15 | 
16 | For the details of these laws, we refer to the nLab ({%nlab internal category%}).
17 | 


--------------------------------------------------------------------------------
/_nodes/001B.md:
--------------------------------------------------------------------------------
1 | ---
2 | title: Locally small fibration
3 | taxon: definition
4 | ---
5 | 
6 | A cartesian fibration $E$ over $B$ is *locally small* if and only if for
7 | each $x\in B$ and $u,v\in E\Sub{x}$, the total category $\TotCat{\CandHom{x}{u}{v}}$
8 | has a terminal object.
9 | 


--------------------------------------------------------------------------------
/_nodes/001C.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Hom candidates
 3 | taxon: definition
 4 | ---
 5 | 
 6 | For any $x\in B$ and displayed objects $u,v\in E\Sub{x}$, we define a *hom
 7 | candidate* for $u,v$ to be a span $u\leftarrow \bar{h} \rightarrow v$ in $E$ in which the left-hand leg is cartesian:
 8 | {%tex macrolib: topos%}
 9 | \begin{tikzpicture}[diagram]
10 | \SpliceDiagramSquare<l/>{
11 |   height = 1.5cm,
12 |   west/style = lies over,
13 |   east/style = lies over,
14 |   north/style = <-,
15 |   south/style = <-,
16 |   ne/style = ne pullback,
17 |   ne = \bar{h},
18 |   se = h,
19 |   sw = x,
20 |   nw = u,
21 |   south = p\Sub{h},
22 |   north = \bar{p}\Sub{h}
23 | }
24 | \SpliceDiagramSquare<r/>{
25 |   height = 1.5cm,
26 |   west/style = lies over,
27 |   east/style = lies over,
28 |   glue = west,
29 |   glue target = l/,
30 |   ne = v,
31 |   se = x,
32 |   north = \epsilon\Sub{h},
33 |   south = p\Sub{h},
34 | }
35 | \end{tikzpicture}
36 | {%endtex%}
37 | 
38 | In the above, $h$ should be thought of as a candidate for the "hom object" of $u,v$,
39 | and $\epsilon\Sub{h}$ should be viewed as the structure of an "evaluation map" for $h$.
40 | This structure can be rephrased in terms of a displayed category $\CandHom{x}{u}{v}$ over $\Sl{B}{x}$:
41 | 
42 | 1. Given $h\in \Sl{B}{x}$, an object of $\CandHom{x}{u}{v}\Sub{h}$ is given by a hom candidate
43 |    whose apex in the base is $h$ itself. We will write $\bar{h}$ metonymically
44 |    for the entire hom candidate over $h$.
45 | 
46 | 2. Given $\alpha:l\to h\in\Sl{B}{x}$ and hom candidates $\bar{l}\in \CandHom{x}{u}{v}\Sub{l}$ and
47 |    $\bar{h}\in \CandHom{x}{u}{v}\Sub{h}$, a morphism $\bar{h}\to\Sub{\alpha} \bar{l}$ is given by a
48 |    cartesian morphism $\bar\alpha:\bar{l}\to\Sub{\alpha}\bar{h}$ in $E$ such that the
49 |    following diagram commutes:
50 |    {%tex macrolib: topos%}
51 |    \begin{tikzpicture}[diagram]
52 |      \node (u) {$u$};
53 |      \node (l) [above right = 1.5cm of u,xshift=.5cm] {$\bar{l}$};
54 |      \node (h) [below right = 1.5cm of u,xshift=.5cm] {$\bar{h}$};
55 |      \node (v) [below right = 1.5cm of l,xshift=.5cm] {$v$};
56 |      \draw[->] (h) to node [sloped,below] {$\bar{p}\Sub{h}$} (u);
57 |      \draw[->] (l) to node [sloped,above] {$\bar{p}\Sub{l}$} (u);
58 |      \draw[->] (h) to node [sloped,below] {$\epsilon\Sub{h}$} (v);
59 |      \draw[->] (l) to node [sloped,above] {$\epsilon\Sub{l}$} (v);
60 |      \draw[->] (l) to node [upright desc] {$\bar\alpha$} (h);
61 |    \end{tikzpicture}
62 |    {%endtex%}
63 | 


--------------------------------------------------------------------------------
/_nodes/001D.md:
--------------------------------------------------------------------------------
1 | ---
2 | taxon: definition
3 | ---
4 | 
5 | An ordinary category is called *globally small* when it has a set of objects.
6 | 


--------------------------------------------------------------------------------
/_nodes/001E.md:
--------------------------------------------------------------------------------
1 | ---
2 | taxon: definition
3 | title: Generic object
4 | ---
5 | 
6 | Let $E$ be a cartesian fibration over $B$; a *generic object* for $E$ is defined to be an object $\bar{u}\in \TotCat{E}$  such that for any $\bar{z}\in \TotCat{E}$ there exists a cartesian map $\bar{z}\to \bar{u}$.
7 | 
8 | **Warning.** Our terminology differs from that of Jacobs {%cite jacobs:1999 -A%}; what we refer to as a *generic object* here is Jacobs' *weak generic object*. We prefer the unqualified terminology, as generic objects in the stronger sense are very rare.
9 | 


--------------------------------------------------------------------------------
/_nodes/001F.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | taxon: definition
 3 | ---
 4 | 
 5 | A *meta-category* $\mathfrak{C}$ is defined by explaining what
 6 | an object of $\mathfrak{E}$ is, and, given two objects $x,y\in \mathfrak{E}$,
 7 | what a morphism from $x$ to $y$ is, together with the following operations:
 8 | 
 9 | 1. for each object $x\in \mathfrak{E}$, an *identity* map $\Idn{x} : x \to x$,
10 | 2. for any two maps $f:x\to y$ and $g:y\to z$, a *composite map* $f;g : x \to z$,
11 | 3. such that the following equations hold:
12 | \\[
13 |   \Idn{x};h = h\qquad
14 |   h;\Idn{y} = h\qquad
15 |   f;(g;h) = (f;g);{h}
16 | \\]
17 | 
18 | 


--------------------------------------------------------------------------------
/_nodes/001G.md:
--------------------------------------------------------------------------------
1 | ---
2 | taxon: definition
3 | ---
4 | 
5 | A *category* $E$ is defined to be a meta-category whose objects
6 | are defined to be the elements of some collection, and for any two objects
7 | $x,y\in E$ the morphisms $x\to y$ are defined to be the elements of some
8 | collection.
9 | 


--------------------------------------------------------------------------------
/_nodes/001H.md:
--------------------------------------------------------------------------------
  1 | ---
  2 | taxon: lemma
  3 | title: Generalized pullback lemma
  4 | ---
  5 | 
  6 | Let $\bar{f} : \bar{x}\to\Sub{f}\bar{y}$, and suppose that
  7 | $\bar{g} : \bar{y}\to\Sub{g}\bar{z}$ is cartesian over $g$. Then
  8 | $\bar{f};\bar{g}$ is cartesian over $f;g$ if and only if $\bar{f}$ is cartesian
  9 | over $f$.
 10 | {%tex macrolib: topos%}
 11 |   \begin{tikzpicture}[diagram]
 12 |     \SpliceDiagramSquare{
 13 |       height = 1.5cm,
 14 |       west/style = lies over,
 15 |       east/style = lies over,
 16 |       nw/style = pullback,
 17 |       sw = y,
 18 |       nw = \bar{y},
 19 |       se = z,
 20 |       ne = \bar{z},
 21 |       south = g,
 22 |       north = \bar{g},
 23 |     }
 24 |     \node (x*) [dotted pullback, left = of nw] {$\bar{x}$};
 25 |     \node (x) [left = of sw] {$x$};
 26 |     \draw[->] (x*) to node [above] {$\bar{f}$} (nw);
 27 |     \draw[lies over] (x*) to (x);
 28 |     \draw[->] (x) to node [below] {$f$} (sw);
 29 |   \end{tikzpicture}
 30 | {%endtex%}
 31 | 
 32 | **Proof.** Suppose first that $\bar{f}$ is cartesian. To see that $\bar{f};\bar{g}$
 33 | is cartesian, we must construct a unique factorization as follows:
 34 | {%tex macrolib: topos%}
 35 |   \begin{tikzpicture}[diagram]
 36 |     \SpliceDiagramSquare{
 37 |       west/style = lies over,
 38 |       east/style = lies over,
 39 |       north/node/style = upright desc,
 40 |       height = 1.5cm,
 41 |       nw = \bar{x},
 42 |       ne = \bar{y},
 43 |       sw = x,
 44 |       north = \bar{f},
 45 |       south = f,
 46 |       se = y,
 47 |       nw/style = pullback,
 48 |       ne/style = pullback,
 49 |     }
 50 |     \node (z') [right = 2cm of ne] {$\bar{z}$};
 51 |     \node (z) [right = 2cm of se] {$z$};
 52 |     \draw[lies over] (z') to (z);
 53 |     \draw[->] (se) to node [below] {$g$} (z);
 54 |     \draw[->] (ne) to node [desc] {$\bar{g}$} (z');
 55 | 
 56 |     \node (u') [above left = 1.5cm of nw,xshift=-.5cm] {$\bar{u}$};
 57 |     \node (u) [above left = 1.5cm of sw,xshift=-.5cm] {$u$};
 58 |     \draw[lies over] (u') to (u);
 59 |     \draw[->,bend left=30] (u') to node [sloped,above] {$\bar{h}$} (z');
 60 |     \draw[->] (u) to node [sloped,below] {$m$} (sw);
 61 |     \draw[->,exists] (u') to (nw);
 62 |   \end{tikzpicture}
 63 | {%endtex%}
 64 | Because $\bar{g}$ is cartesian, we can factor $\bar{h} = i;\bar{g}$ for a unique
 65 | $i:\bar{u}\to\Sub{m;f}\bar{y}$. Then, because $\bar{f}$ is cartesian, we can further
 66 | factor $i = j;\bar{f}$ for a unique $j:\bar{u}\to\Sub{m}\bar{x}$. We conclude that
 67 | there is a unique $j:\bar{u}\to\Sub{m}\bar{x}$ for which
 68 | $\bar{h} = j;\bar{f};\bar{g}$, as required.
 69 | 
 70 | Conversely, suppose that $\bar{f};\bar{g}$ is cartesian. To see that $\bar{f}$ is
 71 | cartesian, we must construct a unique factorization as follows:
 72 | {%tex macrolib: topos%}
 73 |   \begin{tikzpicture}[diagram]
 74 |     \SpliceDiagramSquare{
 75 |       west/style = lies over,
 76 |       east/style = lies over,
 77 |       north/node/style = upright desc,
 78 |       height = 1.5cm,
 79 |       nw = \bar{x},
 80 |       ne = \bar{y},
 81 |       sw = x,
 82 |       north = \bar{f},
 83 |       south = f,
 84 |       se = y,
 85 |       ne/style = pullback,
 86 |     }
 87 |     \node (z') [right = 2cm of ne] {$\bar{z}$};
 88 |     \node (z) [right = 2cm of se] {$z$};
 89 |     \draw[lies over] (z') to (z);
 90 |     \draw[->] (se) to node [below] {$g$} (z);
 91 |     \draw[->] (ne) to node [desc] {$\bar{g}$} (z');
 92 | 
 93 |     \node (u') [above left = 1.5cm of nw,xshift=-.5cm] {$\bar{u}$};
 94 |     \node (u) [above left = 1.5cm of sw,xshift=-.5cm] {$u$};
 95 |     \draw[lies over] (u') to (u);
 96 |     \draw[->,bend left=30] (u') to node [sloped,above] {$\bar{h}$} (ne);
 97 |     \draw[->] (u) to node [sloped,below] {$m$} (sw);
 98 |     \draw[->,exists] (u') to (nw);
 99 |   \end{tikzpicture}
100 | {%endtex%}
101 | Because $\bar{f};\bar{g}$ is cartesian, we can factor
102 | $\bar{h};\bar{g} = i;\bar{f};\bar{g}$ for a unique $i:\bar{u}\to\Sub{m}\bar{x}$.
103 | On the other hand, because $\bar{g}$ is cartesian, there is a unique
104 | $j:\bar{u}\to\Sub{m;f}\bar{y}$ for which $\bar{h};\bar{g} = j;\bar{g}$; as both
105 | $\bar{h}$ and $i;\bar{f}$ satisfy this condition, we conclude $\bar{h}=i;\bar{f}$.
106 | Therefore, there is a unique $i:\bar{u}\to\Sub{m}\bar{x}$ for which
107 | $\bar{h} = i;\bar{f}$, as required.
108 | {%qed%}
109 | 


--------------------------------------------------------------------------------
/_nodes/001I.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | taxon: construction
 3 | ---
 4 | 
 5 | Let $E$ be displayed over $B$; we define its *total opposite* $\TotOpCat{E}$
 6 | displayed over $\OpCat{B}$ as follows:
 7 | 
 8 | 1. An object of $\TotOpCat{E}\Sub{x}$ is given by an object of $E\Sub{x}$.
 9 | 
10 | 2. Given $f : x \to y\in \OpCat{B}$, a displayed morphism $\bar{x}\to\Sub{f} \bar{y}$ in $\TotOpCat{E}$
11 |    is given by a displayed morphism $\bar{y}\to\Sub{f} \bar{x}$ in $E$.
12 | 
13 | 
14 | 


--------------------------------------------------------------------------------
/_nodes/001J.md:
--------------------------------------------------------------------------------
1 | ---
2 | taxon: exercise
3 | ---
4 | 
5 | Let $E$ be displayed over $B$. Prove that the total category {%pref 000A %}
6 | $\TotCat{\TotOpCat{E}}$ is $\OpCat{\prn{\TotCat{E}}}$, and its projection functor is
7 | $\OpCat{\prn{p\Sub{E}}} : \OpCat{\TotCat{E}}\to\OpCat{B}$.
8 | 


--------------------------------------------------------------------------------
/_nodes/001K.md:
--------------------------------------------------------------------------------
1 | ---
2 | taxon: exercise
3 | ---
4 | 
5 | Let $E$ be displayed over $B$, and let $f:x\to y\in B$. Prove that a
6 | morphism $\bar{f}:\bar{x}\to\Sub{f}\bar{y}$ is cartesian over $f$ in $E$ if and only
7 | if $\bar{f}:\bar{y}\to\Sub{f}\bar{x}$ is cocartesian over $f$ in $\TotOpCat{E}$.
8 | 
9 | 


--------------------------------------------------------------------------------
/_nodes/001L.md:
--------------------------------------------------------------------------------
1 | ---
2 | taxon: exercise
3 | ---
4 | 
5 | Prove that a displayed category $E$ is a cartesian fibration over $B$
6 | if and only if $\TotOpCat{E}$ is a cocartesian fibration over $\OpCat{B}$.
7 | 
8 | 


--------------------------------------------------------------------------------
/_nodes/001M.md:
--------------------------------------------------------------------------------
1 | ---
2 | taxon: exercise
3 | ---
4 | 
5 | Prove that $\overline{B}$ is a cocartesian fibration if and only if
6 | $B$ has pushouts.
7 | 


--------------------------------------------------------------------------------
/_nodes/001N.md:
--------------------------------------------------------------------------------
1 | ---
2 | taxon: exercise
3 | ---
4 | 
5 | Prove that the total category {%pref 000A%} of $\overline{B}$ is the
6 | arrow category $B^{\to}$, and the projection is the *domain* functor.
7 | 


--------------------------------------------------------------------------------
/_nodes/001O.md:
--------------------------------------------------------------------------------
1 | ---
2 | taxon: definition
3 | ---
4 | 
5 | A cartesian fibration $E$ over $B$ is said to be a *right fibration*
6 | when all displayed morphisms in $E$ are cartesian.
7 | 


--------------------------------------------------------------------------------
/_nodes/001P.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | taxon: theorem
 3 | title: Characterization of right fibrations
 4 | ---
 5 | 
 6 | A cartesian fibration $E$ over $B$ is a right fibration if and only if
 7 | all its vertical maps are isomorphisms.
 8 | 
 9 | **Proof.** Suppose that $E$ is a right fibration over $B$, and fix $b\in B$,
10 | $\bar{b}\in E\Sub{b}$, and a vertical map $f:\bar{b}\to\Sub{1\Sub{b}} \bar{b}$.
11 | Using the hypothesis that $f$ is cartesian, it has a unique section
12 | $g:\bar{b}\to\Sub{1\Sub{b}} \bar{b}$ as follows:
13 | {%tex macrolib: topos%}
14 |   \begin{tikzpicture}[diagram]
15 |     \SpliceDiagramSquare{
16 |       west/style = lies over,
17 |       east/style = lies over,
18 |       north/node/style = upright desc,
19 |       height = 1.5cm,
20 |       nw = \bar{b},
21 |       ne = \bar{b},
22 |       north = f,
23 |       sw = b,
24 |       se = b,
25 |       south/style = double,
26 |       nw/style = pullback,
27 |     }
28 |     \node (u') [above left = 1.5cm of nw,xshift=-.5cm] {$\bar{b}$};
29 |     \node (u) [above left = 1.5cm of sw,xshift=-.5cm] {$b$};
30 |     \draw[lies over] (u') to (u);
31 |     \draw[double,bend left=30] (u') to (ne);
32 |     \draw[double] (u) to (sw);
33 |     \draw[->,exists] (u') to node [desc] {$g$} (nw);
34 |   \end{tikzpicture}
35 | {%endtex%}
36 | Likewise, because $g$ is cartesian, $f$ is the unique section of $g$; thus $f$ is an
37 | isomorphism in $E\Sub{b}$.
38 | 
39 | Conversely, suppose that $E$ is a cartesian fibration whose vertical maps are
40 | isomorphisms. Fix $f:x\to y \in B$ and an arbitrary displayed morphism
41 | $\bar{g}:\bar{x}\to\Sub{f}\bar{y}$. Then $\bar{g}$ is the precomposition of a
42 | cartesian lift $\bar{f}:\bar{x}\tick\to\Sub{f}\bar{y}$ with a vertical map:
43 | {%tex macrolib: topos%}
44 |   \begin{tikzpicture}[diagram]
45 |     \SpliceDiagramSquare{
46 |       west/style = lies over,
47 |       east/style = lies over,
48 |       north/node/style = upright desc,
49 |       height = 1.5cm,
50 |       nw = \bar{x}',
51 |       ne = \bar{y},
52 |       sw = x,
53 |       north = \bar{f},
54 |       south = f,
55 |       se = y,
56 |       nw/style = pullback,
57 |     }
58 |     \node (u') [above left = 1.5cm of nw,xshift=-.5cm] {$\bar{x}$};
59 |     \node (u) [above left = 1.5cm of sw,xshift=-.5cm] {$x$};
60 |     \draw[lies over] (u') to (u);
61 |     \draw[->,bend left=30] (u') to node [sloped,above] {$\bar{g}$} (ne);
62 |     \draw[double] (u) to (sw);
63 |     \draw[->,exists] (u') to node [desc] {$i$} (nw);
64 |   \end{tikzpicture}
65 | {%endtex%}
66 | Because vertical maps are isomorphisms and $\bar{f}$ is cartesian, we can observe
67 | that $\bar{g}$ is cartesian as follows, writing $\bar{m} : \bar{u}\to\Sub{m}
68 | \bar{x}\tick$ for the unique factorization of $\bar{h}$ through $\bar{f}$ over $m$:
69 | {%tex macrolib: topos%}
70 |   \begin{tikzpicture}[diagram]
71 |     \SpliceDiagramSquare{
72 |       west/style = lies over,
73 |       east/style = lies over,
74 |       north/node/style = upright desc,
75 |       height = 1.5cm,
76 |       nw = \bar{x},
77 |       ne = \bar{y},
78 |       sw = x,
79 |       north = \bar{g},
80 |       south = f,
81 |       se = y,
82 |       nw/style = pullback,
83 |     }
84 |     \node (u') [above left = 1.5cm of nw,xshift=-1cm] {$\bar{u}$};
85 |     \node (u) [above left = 1.5cm of sw,xshift=-1cm] {$u$};
86 |     \draw[lies over] (u') to (u);
87 |     \draw[->,bend left=30] (u') to node [sloped,above] {$\bar{h}$} (ne);
88 |     \draw[->] (u) to node [sloped,below] {$m$} (sw);
89 |     \draw[->,exists] (u') to node [desc] {$\bar{m};i\Sup{-1}$} (nw);
90 |   \end{tikzpicture}
91 | {%endtex%}
92 | {%qed%}
93 | 


--------------------------------------------------------------------------------
/_nodes/001Q.md:
--------------------------------------------------------------------------------
1 | ---
2 | taxon: definition
3 | title: Small fibration
4 | ---
5 | 
6 | A cartesian fibration is called *small* when it is both locally small
7 | {%ref 000I%} and globally small {%ref 000J%}.
8 | 


--------------------------------------------------------------------------------
/_nodes/001R.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | taxon: construction
 3 | ---
 4 | 
 5 | Let $C$ be an internal category in $E$. We may define a fibered category $\brk{C}$ over $E$ called the *externalization* of $C$.
 6 | 
 7 | 1. Given $x\in E$, an object of $\brk{C}\Sub{x}$ is defined to be a morphism $x\to C\Sub{0}$ in $E$.
 8 | 
 9 | 2. Given $x,y\in E$ and $f:x\to y$ and $u \in \brk{C}\Sub{x}$ and $v\in
10 |    \brk{C}\Sub{y}$, a morphism $u\to\Sub{f} v$ is defined to be a morphism $h :
11 |    x\to C\Sub{1}$ in $E$ such that the following diagram commutes:
12 |    {%tex macrolib: topos%}
13 |    \begin{tikzpicture}[diagram]
14 |    \node (x) {$x$};
15 |    \node (y) [below = of x] {$y$};
16 |    \node (C/1) [right = of x] {$C\Sub{1}$};
17 |    \node (C/0) [below = of C/1] {$C\Sub{0}$};
18 |    \node (C/0') [above = of C/1] {$C\Sub{0}$};
19 |    \draw[->] (x) to node [upright desc] {$h$} (C/1);
20 |    \draw[->] (C/1) to node [right] {$t$} (C/0);
21 |    \draw[->] (C/1) to node [right] {$s$} (C/0');
22 |    \draw[->] (y) to node [below] {$v$} (C/0);
23 |    \draw[->] (x) to node [left] {$f$} (y);
24 |    \draw[->] (x) to node [sloped,above] {$u$} (C/0');
25 |    \end{tikzpicture}
26 |    {%endtex%}
27 | 


--------------------------------------------------------------------------------
/_nodes/001S.md:
--------------------------------------------------------------------------------
  1 | ---
  2 | taxon: theorem
  3 | ---
  4 | 
  5 | 
  6 | The externalization $\brk{\gl{\bar{u}}}$ of the internal category $\gl{\bar{u}}$ is equivalent to $\FullSubfib{\bar{u}}$.
  7 | 
  8 | **Proof.** We will define a fibred equivalence $F : \brk{\gl{\bar{u}}}\to \FullSubfib{\bar{u}}$ over $B$.
  9 | 
 10 | 1. Fix $x\in B$ and $\chi\Sub{x} \in \brk{\gl{\bar{u}}}\Sub{x}$, i.e. $\chi\Sub{x} : x\to u$; we define $F\prn{\chi\Sub{x}}$ to be an arbitrary cartesian map $\phi\Sub{x} : \bar{x}\to\Sub{\chi\Sub{x}} \bar{u}$. (Here we have used the axiom of choice for collections {%ref 000R%}.)
 11 | 
 12 | 2. Fix $f : x\to y\in B$ and $\chi\Sub{x} :x\to u$ and $\chi\Sub{y}:y\to u$ and a diagram representing a displayed morphism $h$ from $\chi\Sub{x}$ to $\chi\Sub{y}$ over $f$ as below:
 13 |    {%tex macrolib: topos%}
 14 |    \begin{tikzpicture}[diagram]
 15 |    \node (x) {$x$};
 16 |    \node (y) [below = of x] {$y$};
 17 |    \node (C/1) [right = of x] {$\brk{\bar\partial\Sub{1},\bar\partial\Sub{2}}$};
 18 |    \node (C/0) [below = of C/1] {$u$};
 19 |    \node (C/0') [above = of C/1] {$u$};
 20 |    \draw[->,magenta] (x) to node [upright desc] {$h$} (C/1);
 21 |    \draw[->] (C/1) to node [right] {$t$} (C/0);
 22 |    \draw[->] (C/1) to node [right] {$s$} (C/0');
 23 |    \draw[->] (y) to node [below] {$\chi\Sub{y}$} (C/0);
 24 |    \draw[->] (x) to node [left] {$f$} (y);
 25 |    \draw[->] (x) to node [sloped,above] {$\chi\Sub{x}$} (C/0');
 26 |    \end{tikzpicture}
 27 |    {%endtex%}
 28 | 
 29 |    We must define $F\prn{h}:\bar{x}\to\Sub{f} \bar{y}$, fixing arbitrary
 30 |    cartesian maps $\bar\chi\Sub{x}:\bar{x}\to\Sub{\chi\Sub{x}}\bar{u}$ and
 31 |    $\bar\chi\Sub{y}:\bar{y}\to\Sub{\chi\Sub{y}}\bar{u}$. First we lift $h:x\to \brk{\bar\partial\Sub{1},\bar\partial\Sub{2}}$
 32 |    into $E$ using the universal property of the cartesian lift:
 33 |    {%tex macrolib: topos%}
 34 |     \begin{tikzpicture}[diagram]
 35 |       \SpliceDiagramSquare<0/>{
 36 |         height = 1.5cm,
 37 |         west/style = lies over,
 38 |         east/style = lies over,
 39 |         width = 2.5cm,
 40 |         north/node/style = upright desc,
 41 |         ne/style = pullback,
 42 |         north/style = {->,exists},
 43 |         nw = \bar{x},
 44 |         sw = x,
 45 |         se = \brk{\bar\partial\Sub{1},\bar\partial\Sub{2}},
 46 |         ne = \InvImg{\brk{\bar\partial\Sub{1},\bar\partial\Sub{2}}}\bar\partial\Sub{1},
 47 |         south = h,
 48 |         north = \bar{h},
 49 |       }
 50 |       \SpliceDiagramSquare<1/>{
 51 |         glue = west,
 52 |         glue target = 0/,
 53 |         height = 1.5cm,
 54 |         width = 2.5cm,
 55 |         north/node/style = upright desc,
 56 |         south/node/style = upright desc,
 57 |         east/style = lies over,
 58 |         ne/style = pullback,
 59 |         ne = \bar\partial\Sub{1},
 60 |         se = u\times u,
 61 |         south = p,
 62 |         north = \bar{p},
 63 |       }
 64 |       \SpliceDiagramSquare<2/>{
 65 |         glue = west,
 66 |         glue target = 1/,
 67 |         height = 1.5cm,
 68 |         width = 2.5cm,
 69 |         north/node/style = upright desc,
 70 |         south/node/style = upright desc,
 71 |         east/style = lies over,
 72 |         ne = \bar{u},
 73 |         se = u,
 74 |         south = \pi\Sub{1},
 75 |         north = \bar\pi\Sub{1},
 76 |       }
 77 |       \draw[->,bend left=30] (0/nw) to node [above] {$\bar\chi\Sub{x}$} (2/ne);
 78 |       \draw[->,bend right=30] (0/se) to node [below] {$s$} (2/se);
 79 |     \end{tikzpicture}
 80 |    {%endtex%}
 81 | 
 82 |    By composition with the "evaluation map" for our hom object, we have a map $\bar{x}\to\Sub{f;\chi\Sub{y}}\bar{u}$:
 83 |    {%tex macrolib: topos%}
 84 |     \begin{tikzpicture}[diagram]
 85 |       \SpliceDiagramSquare<0/>{
 86 |         height = 1.5cm,
 87 |         west/style = lies over,
 88 |         east/style = lies over,
 89 |         south/node/style = upright desc,
 90 |         width = 2.5cm,
 91 |         nw = \bar{x},
 92 |         sw = x,
 93 |         se = \brk{\bar\partial\Sub{1},\bar\partial\Sub{2}},
 94 |         ne = \InvImg{\brk{\bar\partial\Sub{1},\bar\partial\Sub{2}}}\bar\partial\Sub{1},
 95 |         south = h,
 96 |         north = \bar{h},
 97 |       }
 98 |       \SpliceDiagramSquare<1/>{
 99 |         glue = west,
100 |         glue target = 0/,
101 |         height = 1.5cm,
102 |         width = 2.5cm,
103 |         south/node/style = upright desc,
104 |         east/style = lies over,
105 |         ne/style = pullback,
106 |         ne = \bar\partial\Sub{2},
107 |         se = u\times u,
108 |         south = p,
109 |         north = \bar{\epsilon},
110 |       }
111 |       \SpliceDiagramSquare<2/>{
112 |         glue = west,
113 |         glue target = 1/,
114 |         height = 1.5cm,
115 |         width = 2.5cm,
116 |         south/node/style = upright desc,
117 |         east/style = lies over,
118 |         ne = \bar{u},
119 |         se = u,
120 |         south = \pi\Sub{2},
121 |         north = \bar\pi\Sub{2},
122 |       }
123 |       \draw[->,bend right=30] (0/se) to node [upright desc] {$t$} (2/se);
124 |       \node (y) [between = 0/sw and 2/se, yshift=-2cm] {$y$};
125 |       \draw[bend right=30,->] (0/sw) to node [sloped,below] {$f$} (y);
126 |       \draw[bend right=30,->] (y) to node [sloped,below] {$\chi\Sub{y}$} (2/se);
127 |     \end{tikzpicture}
128 |    {%endtex%}
129 | 
130 |    Next we define $F\prn{h}:\bar{x}\to\Sub{f}\bar{y}$ using the universal property of (another) cartesian lift:
131 |    {%tex macrolib: topos%}
132 |     \begin{tikzpicture}[diagram]
133 |       \SpliceDiagramSquare<l/>{
134 |         height = 1.5cm,
135 |         east/style = lies over,
136 |         west/style = lies over,
137 |         ne/style = pullback,
138 |         north/style = {->,exists},
139 |         north/node/style = upright desc,
140 |         ne = \bar{y},
141 |         nw = \bar{x},
142 |         se = y,
143 |         sw = x,
144 |         south = f,
145 |         north = F\prn{h},
146 |         width = 2.5cm,
147 |       }
148 |       \SpliceDiagramSquare<r/>{
149 |         glue = west,
150 |         glue target = l/,
151 |         height = 1.5cm,
152 |         east/style = lies over,
153 |         north/node/style = upright desc,
154 |         ne = \bar{u},
155 |         se = u,
156 |         south = \chi\Sub{y},
157 |         north = \bar\chi\Sub{y},
158 |       }
159 |       \draw[->,bend left=30] (l/nw) to node [above] {$\bar{h};\bar{\epsilon};\bar\pi\Sub{2}$} (r/ne);
160 |     \end{tikzpicture}
161 |    {%endtex%}
162 | {%qed%}
163 | 


--------------------------------------------------------------------------------
/_nodes/001T.md:
--------------------------------------------------------------------------------
1 | ---
2 | taxon: exercise
3 | ---
4 | 
5 | Prove that the total category $\TotCat{\SelfIx{B}}$ of the fundamental self-indexing {%pref 0003%} is the arrow category $B^{\to}$, and the projection is the codomain functor.
6 | 


--------------------------------------------------------------------------------
/_nodes/001U.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | taxon: exercise
 3 | ---
 4 | 
 5 | Suppose that $B$ is an internal category in $\mathbf{Set}$, i.e.
 6 | it has a set of objects. Exhibit an equivalence of displayed categories between
 7 | $P\Sub{\bullet}$ as described above, and the naïve definition which $E\Sub{x}$ is the
 8 | collection of objects $u\in E$ such that $Pu = x$.
 9 | 
10 | 


--------------------------------------------------------------------------------
/_nodes/001V.md:
--------------------------------------------------------------------------------
1 | ---
2 | taxon: exercise
3 | ---
4 | 
5 | Explicitly construct the functorial action of $\TotCat{P\Sub{\bullet}}\to E$.
6 | 


--------------------------------------------------------------------------------
/_nodes/001W.md:
--------------------------------------------------------------------------------
1 | ---
2 | taxon: exercise
3 | ---
4 | 
5 | Verify that $\TotCat{P\Sub{\bullet}}\to E$ is a categorical equivalence.
6 | 


--------------------------------------------------------------------------------
/_nodes/001X.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: The fundamental self-indexing
 3 | taxon: construction
 4 | ---
 5 | 
 6 | Let $B$ be an ordinary category; there is an important displayed category
 7 | $\SelfIx{B}$ over $B$ given fiberwise by the *slices* of $B$.
 8 | 1. For $x\in B$, we define $\SelfIx{B}\Sub{x}$ to be the collection $\Sl{B}{x}$
 9 |    of pairs $(\bar{x}\in B,p\Sub{x}:\bar{x}\to x)$.
10 | 2. For $f : x\to y\in B$, we define $\SelfIx{B}\Sub{f}$ to be the collection of
11 |    commuting squares in the following configuration:
12 | {%tex macrolib: topos%}
13 |   \DiagramSquare{
14 |     height = 1.7cm,
15 |     nw = \bar{x},
16 |     ne = \bar{y},
17 |     sw = x,
18 |     se = y,
19 |     west = p\Sub{x},
20 |     east = p\Sub{y},
21 |     south = f,
22 |     north = \bar{f},
23 |     west/style = {->,exists},
24 |     east/style = {->,exists},
25 |     north/style = {->,exists},
26 |   }
27 | {%endtex%}
28 | 


--------------------------------------------------------------------------------
/_nodes/001Y.md:
--------------------------------------------------------------------------------
1 | ---
2 | taxon: exercise
3 | ---
4 | 
5 | Prove that $\SelfIx{B}$ from {%ref 001X%} is a cartesian fibration if and only if $B$ has pullbacks.
6 | 
7 | 


--------------------------------------------------------------------------------
/_nodes/001Z.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | taxon: construction
 3 | ---
 4 | 
 5 | Let $E$ be fibered over $B$; we may define the *opposite* fibered category
 6 | $\OpCat{E}$ over $B$ like so:
 7 | 
 8 | 1. An object of $\OpCat{E}\Sub{x}$ is given by an object of $E\Sub{x}$.
 9 | 
10 | 2. Given $f : x \to y\in B$, a morphism $\bar{x}\to_f \bar{y}$ in $\OpCat{E}$
11 |    is given in terms of $E$ by a cartesian map $\bar{y}\Sub{f} : \bar{y}\Sub{x} \to\Sub{f} \bar{y}$ together
12 |    with a *vertical* map $h : \bar{y}\Sub{x}\to\Sub{\Idn{x}} \bar{y}$ as depicted below:
13 |    {%tex macrolib: topos%}
14 |     \begin{tikzpicture}[diagram]
15 |       \SpliceDiagramSquare{
16 |         height = 1.5cm,
17 |         nw/style = pullback,
18 |         west/style = lies over,
19 |         east/style = lies over,
20 |         north = \bar{y}\Sub{f},
21 |         nw = \bar{y}\Sub{x},
22 |         ne = \bar{y},
23 |         sw = x,
24 |         se = y
25 |       }
26 |       \node (x*) [left = of nw] {$\bar{x}$};
27 |       \node (x) [left = of sw] {$x$};
28 |       \draw[->] (nw) to node [above] {$h$} (x*);
29 |       \draw[lies over] (x*) to (x);
30 |       \draw[double] (x) to (sw);
31 |     \end{tikzpicture}
32 |    {%endtex%}
33 |    such that $\brc{\bar{x} \leftarrow \bar{y}\Sub{x}\Sup{1}\to \bar{y}}$ is
34 |    identified with $\brc{\bar{x}\leftarrow\bar{y}\Sub{x}\Sup{2}\to \bar{y}}$
35 |    when they agree up to the unique vertical isomorphism $\bar{y}\Sub{x}\Sup{1}\cong\bar{y}\Sub{x}\Sup{2}$
36 |    induced by the universal property of cartesian maps in the sense that the following diagram commutes:
37 |    {%tex macrolib: topos%}
38 |    \begin{tikzpicture}[diagram]
39 |     \node (x) {$\bar{x}$};
40 |     \node (yx/1) [above right = of x] {$\bar{y}\Sub{x}\Sup{1}$};
41 |     \node (y) [below right = of yx/1] {$\bar{y}$};
42 |     \node (yx/2) [below right = of x] {$\bar{y}\Sub{x}\Sup{2}$};
43 |     \draw[->] (yx/1) to node [sloped,above] {vert.} (x);
44 |     \draw[->] (yx/2) to node [sloped,below] {vert.} (x);
45 |     \draw[->] (yx/1) to node [sloped,above] {cart.} (y);
46 |     \draw[->] (yx/2) to node [sloped,below] {cart.} (y);
47 |     \draw[->] (yx/1) to node [upright desc] {$\cong$} (yx/2);
48 |    \end{tikzpicture}
49 |    {%endtex%}
50 | 


--------------------------------------------------------------------------------
/_nodes/0020.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | taxon: lemma
 3 | ---
 4 | 
 5 | A morphism $\bar{f}:\bar{x}\to\Sub{f} \bar{y}\in\OpCat{E}$ is cartesian
 6 | over $f:x\to y$ if and only if the vertical leg of $f$ is an isomorphism.
 7 | 
 8 | **Proof.** Suppose that $\bar{f} : \bar{x}\to\Sub{f}\bar{y}$ is represented by a span $\brc{\bar{x}\leftarrow\bar{y}\Sub{x}\to\bar{y}}$ in $E$ in which the vertical leg $\bar{x}\leftarrow\bar{y}\Sub{x}$ is an isomorphism. We must show that $\bar{x}\to_f\bar{y}$ is cartesian in $\OpCat{E}$. We fix a morphism $\bar{h}:\bar{w}\to\Sub{m;f}\bar{y}\in \OpCat{E}$ where $m:w\to x$, depicted below in terms of $\OpCat{E}$:
 9 | {%tex macrolib: topos%}
10 | \begin{tikzpicture}[diagram]
11 |   \SpliceDiagramSquare{
12 |     west/style = lies over,
13 |     east/style = lies over,
14 |     north/node/style = upright desc,
15 |     height = 1.5cm,
16 |     nw = \bar{x},
17 |     ne = \bar{y},
18 |     sw = x,
19 |     se = y,
20 |     south = f,
21 |     north = \bar{f},
22 |   }
23 |   \node (w) [left = of sw] {$w$};
24 |   \node (w') [left = of nw] {$\bar{w}$};
25 |   \draw[lies over] (w') to (w);
26 |   \draw[->] (w) to node [below] {$m$} (sw);
27 |   \draw[bend left=30] (w') to node [above] {$\bar{h}$} (ne);
28 | \end{tikzpicture}
29 | {%endtex%}
30 | 
31 | We must define the unique intervening map $\bar{w}\to_m \bar{x}$ in $\OpCat{E}$.
32 | We first translate the above into the language of $E$ by unfolding definitions:
33 | {%tex macrolib: topos%}
34 | \begin{tikzpicture}[diagram]
35 |   \SpliceDiagramSquare<0/>{
36 |     west/style = lies over,
37 |     east/style = lies over,
38 |     north/style = {<-},
39 |     south/style = double,
40 |     height = 1.5cm,
41 |     nw = \bar{w},
42 |     sw = w,
43 |     se = w,
44 |     ne = \bar{y}\Sub{w},
45 |     north = \bar{h},
46 |     ne/style = pullback,
47 |   }
48 |   \SpliceDiagramSquare<1/>{
49 |     glue = west,
50 |     glue target = 0/,
51 |     east/style = lies over,
52 |     height = 1.5cm,
53 |     ne/style = pullback,
54 |     se = x,
55 |     ne = \bar{y}\Sub{x}\cong \bar{x},
56 |     south = m,
57 |   }
58 |   \SpliceDiagramSquare<2/>{
59 |     glue = west,
60 |     glue target = 1/,
61 |     east/style = lies over,
62 |     height = 1.5cm,
63 |     se = y,
64 |     ne = \bar{y},
65 |     south = f,
66 |     north = \bar{f},
67 |   }
68 | \end{tikzpicture}
69 | {%endtex%}
70 | 
71 | The desired intervening map $\bar{w}\to\Sub{m} \bar{x}\in \OpCat{E}$ shakes out in the language of $E$ to be a span $\brc{\bar{w}\leftarrow \bar{x}\Sub{w}\to\Sub{m} \bar{x}}$ in which the left-hand leg is vertical and the right-hand leg is cartesian over $m:w\to x$. But the left-hand span $\brc{\bar{w}\leftarrow\bar{y}\Sub{w}\to \bar{y}\Sub{x}\cong \bar{x}}$ in the diagram above is exactly what we need.
72 | 
73 | We leave the converse to the reader.
74 | {%qed%}
75 | 


--------------------------------------------------------------------------------
/_nodes/0021.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | taxon: corollary
 3 | ---
 4 | 
 5 | The displayed category $\OpCat{E}$ is a cartesian fibration.
 6 | 
 7 | **Proof.**
 8 | Fixing $\bar{y}\in \OpCat{E}\Sub{y}$ and $f:x\to y\in B$, we must
 9 | exhibit a cartesian lift $\bar{f} : \bar{x}\to\Sub{f}\bar{y}\in \OpCat{E}$.
10 | By the characterization {%ref 000T%} it suffices to find *any* map over $f$ whose vertical component is an isomorphism. Writing $\bar{y}\Sub{x}\to\Sub{f}\bar{y}$ for the cartesian lift of $f$ in $E$, consider the map in $\OpCat{E}$ presented by the following span in $E$:
11 | {%tex macrolib: topos%}
12 | \begin{tikzpicture}[diagram]
13 |   \SpliceDiagramSquare<l/>{
14 |     nw = \bar{y}\Sub{x},
15 |     ne = \bar{y}\Sub{x},
16 |     sw = x,
17 |     se = x,
18 |     north/style = double,
19 |     south/style = double,
20 |     west/style = lies over,
21 |     east/style = lies over,
22 |     ne/style = pullback,
23 |     height = 1.5cm,
24 |   }
25 |   \SpliceDiagramSquare<r/>{
26 |     glue = west,
27 |     glue target = l/,
28 |     height = 1.5cm,
29 |     east/style = lies over,
30 |     ne = \bar{y},
31 |     se = y,
32 |     south = f,
33 |   }
34 | \end{tikzpicture}
35 | {%endtex%}
36 | {%qed%}
37 | 


--------------------------------------------------------------------------------
/_nodes/0022.md:
--------------------------------------------------------------------------------
1 | ---
2 | title: Geometric universe
3 | taxon: definition
4 | ---
5 | 
6 | A *geometric universe* $E$ is defined to be a finitely complete cartesian
7 | closed category with a subobject classifier and a natural numbers object.
8 | 


--------------------------------------------------------------------------------
/_nodes/0023.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Geometric universes
 3 | ---
 4 | 
 5 | @include{0022}
 6 | 
 7 | 
 8 | @include{0036}
 9 | @include{0038}
10 | @include{0037}
11 | 
12 | @include{0039}
13 | 


--------------------------------------------------------------------------------
/_nodes/0024.md:
--------------------------------------------------------------------------------
1 | ---
2 | title: Topoi in a geometric universe
3 | ---
4 | 
5 | @include{0025}
6 | @include{0026}
7 | @include{0035}
8 | 


--------------------------------------------------------------------------------
/_nodes/0025.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Topos
 3 | taxon: definition
 4 | ---
 5 | 
 6 | Let $E$ be a geometric universe. A $E$-*topos* $X$ or a *topos in
 7 | $E$* is defined to be a geometric universe $\Sh{X}$ equipped with an algebraic morphism $X:E\to\Sh{X}$ whose gluing fibration $\GL{X}$ has a
 8 | small separator {%pref 002H%}. A morphism of $E$-topoi $f:{X}\to{Y}$ is defined by an algebraic
 9 | morphism $\Sh{f}:\Sh{Y}\to\Sh{X}$ of geometric universes equipped with
10 | a 2-isomorphism $\varphi_{f}:X \cong Y;\Sh{f}$ in $[E,\Sh{X}]$.
11 | {%tex macrolib: topos%}
12 |     \begin{tikzpicture}[diagram]
13 |       \node (E) {$E$};
14 |       \node (S/Y) [below left = of E] {$\Sh{Y}$};
15 |       \node (S/X) [below right = of E] {$\Sh{X}$};
16 |       \draw[->] (E) to node [sloped,above] {$Y$} (S/Y);
17 |       \draw[->] (S/Y) to node (S/f) [below] {$\Sh{f}$} (S/X);
18 |       \draw[->] (E) to node [sloped,above] {$X$} (S/X);
19 |       \node [between = E and S/f] {$\varphi_{f}$};
20 |     \end{tikzpicture}
21 | {%endtex%}
22 | 
23 | It will often be convenient to display pasting diagrams as *string diagrams*,
24 | which we read in diagrammatic order from top left to bottom right. A string
25 | diagram views 2-cell as a "transformer" through which information passes
26 | along a wire; the input/output interface of such a transformer is written on
27 | the wires. Regions of whitespace denote objects of the ambient 2-category. Our pasting diagram above is depicted as a string diagram in $\AlgGU$ below:
28 | {%tex macrolib: topos%}
29 | \begin{ZX}
30 | \zxN{} &\dar[close]{X}\\
31 | \zxN{} &\zxX[pilr]{\varphi_f}\dlar[swap,-o,close]{Y}\drar[N,close]{\Sh{f}}\\[3mm]
32 | \zxN{} & \zxN{} & \zxN{}
33 | \end{ZX}
34 | {%endtex%}
35 | 
36 | A 2-morphism $\alpha:{f}\to{g}$ in $[X,Y]$ is defined to be a
37 | 2-morphism $\Sh{\alpha}:\Sh{f}\to{\Sh{g}}$ compatible with $\varphi_{f},\varphi_{g}$
38 | in the sense that the following pasting diagrams are equal:
39 | {%tex macrolib: topos%}
40 |   \begin{tikzpicture}[diagram,baseline=(S/X.base)]
41 |     \node (E) {$E$};
42 |     \node (S/Y) [below left = of E] {$\Sh{Y}$};
43 |     \node (S/X) [below right = of E] {$\Sh{X}$};
44 |     \draw[->] (E) to node [sloped,above] {$Y$} (S/Y);
45 |     \draw[->] (S/Y) to node (S/f) [upright desc] {$\Sh{f}$} (S/X);
46 |     \draw[->] (E) to node [sloped,above] {$X$} (S/X);
47 |     \draw[->,bend right=60] (S/Y) to node (S/g) [below] {$\Sh{g}$} (S/X);
48 |     \node [between = E and S/f] {$\varphi_{f}$};
49 |     \node [between = S/g.north and S/f] {$\Sh{\alpha}$};
50 |   \end{tikzpicture}
51 |   =
52 |   \begin{tikzpicture}[diagram,baseline=(S/X.base)]
53 |     \node (E) {$E$};
54 |     \node (S/Y) [below left = of E] {$\Sh{Y}$};
55 |     \node (S/X) [below right = of E] {$\Sh{X}$};
56 |     \draw[->] (E) to node [sloped,above] {$Y$} (S/Y);
57 |     \draw[->] (E) to node [sloped,above] {$X$} (S/X);
58 |     \draw[->,bend right=60] (S/Y) to node (S/g) [below] {$\Sh{g}$} (S/X);
59 |     \node [between = S/Y and S/X] {$\varphi_g$};
60 |   \end{tikzpicture}
61 | {%endtex%}
62 | 
63 | We may likewise represent the equation above in terms of string diagrams.
64 | {%tex macrolib: topos%}
65 | \begin{ZX}
66 |   \zxN{} &\dar[close]{X}\\
67 |    \zxN{} & \zxX[pilr]{\varphi_f} \arrow[-N', near end, close, swap]{ddl}{Y} \drar[N, close, pos=.6]{\Sh{f}}\\
68 |    \zxN{} & \zxN{} & \zxX[pilr]{\Sh{\alpha}} \dar{\Sh{g}}\\
69 |    \zxN{} & \zxN{} & \zxN{}
70 | \end{ZX}
71 | =
72 | \begin{ZX}
73 |   \zxN{} &\dar[close]{X}\\
74 |   \zxN{} &\zxX[pilr]{\varphi_g}\dlar[swap,-o,close]{Y}\drar[N,close]{\Sh{g}}\\[3mm]
75 |   \zxN{} & \zxN{} & \zxN{}
76 | \end{ZX}
77 | {%endtex%}
78 | 


--------------------------------------------------------------------------------
/_nodes/0026.md:
--------------------------------------------------------------------------------
1 | ---
2 | taxon: notation
3 | ---
4 | 
5 | We will write $\TOP{E}$ for the meta-2-category of $E$-topoi.
6 | 


--------------------------------------------------------------------------------
/_nodes/0029.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: An alternative definition of fibration
 3 | ---
 4 | 
 5 | **Warning.** Some authors including Grothendieck {%cite sga:1 -A%} give an
 6 | equivalent definition of cartesian fibration that factors through a
 7 | *nonequivalent* definition of cartesian morphisms. Such authors refer to our
 8 | notion of cartesian morphism as *hypercartesian* {%cite streicher:2018%}.
 9 | 
10 | @include{002A}
11 | 
12 | Cartesian morphisms are clearly hypocartesian (setting $u=x$ and $m=\Idn{x}$),
13 | but the converse does not hold. The problem is that in an arbitrary displayed
14 | category, hypocartesian morphisms may not be closed under composition.
15 | 
16 | @include{002C}
17 | 
18 | Grothendieck {%cite sga:1 -A%} defines a fibration in terms of (what we refer
19 | to as) hypocartesian morphisms rather than (what we refer to as) cartesian
20 | morphisms, and therefore imposes the additional constraint that the
21 | hypocartesian morphisms be closed under composition. In {%cref 002B %} below, we
22 | verify that these two definitions of cartesian fibration coincide.
23 | 
24 | @include{002B}
25 | @include{002D}
26 | 
27 | 


--------------------------------------------------------------------------------
/_nodes/002A.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Hypocartesian morphisms
 3 | taxon: definition
 4 | ---
 5 | 
 6 | Let $E$ be displayed over $B$, and let $f:x\to y \in B$; a morphism
 7 | $\bar{f}:\bar{x}\to\Sub{f} \bar{y}$ in $E$ is called *hypocartesian* over $f$ when
 8 | for any $\bar{u}\in E\Sub{x}$ and $\bar{h}:\bar{u}\to\Sub{f} \bar{y}$ there exists a
 9 | unique $i : \bar{u}\to\Sub{\Idn{x}} \bar{x}$ with $i;\bar{f} = \bar{h}$ as follows:
10 | {%tex macrolib: topos%}
11 |   \begin{tikzpicture}[diagram]
12 |     \SpliceDiagramSquare{
13 |       west/style = lies over,
14 |       east/style = lies over,
15 |       north/node/style = upright desc,
16 |       height = 1.5cm,
17 |       nw = \bar{x},
18 |       ne = \bar{y},
19 |       sw = x,
20 |       north = \bar{f},
21 |       south = f,
22 |       se = y,
23 |     }
24 |     \node (u') [above left = 1.5cm of nw,xshift=-.5cm] {$\bar{u}$};
25 |     \node (u) [above left = 1.5cm of sw,xshift=-.5cm] {$x$};
26 |     \draw[lies over] (u') to (u);
27 |     \draw[->,bend left=30] (u') to node [sloped,above] {$\bar{h}$} (ne);
28 |     \draw[double] (u) to (sw);
29 |     \draw[->,exists] (u') to node [desc] {$i$} (nw);
30 |   \end{tikzpicture}
31 | {%endtex%}
32 | 


--------------------------------------------------------------------------------
/_nodes/002B.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | taxon: lemma
 3 | title: "Equivalence with Grothendieck's fibrations"
 4 | ---
 5 | 
 6 | Let $E$ be displayed over $B$. Then $E$ is a cartesian fibration in the sense of
 7 | {%cref 0002%} if and only if the following two conditions hold:
 8 | 
 9 | 1. *Hypocartesian lifts.* For each $f:x\to y\in B$ and $\bar{y}\in E\Sub{y}$ there
10 |    exists a displayed object $\bar{x}\in E\Sub{x}$ and hypocartesian morphism
11 |    $\bar{f}:\bar{x}\to\Sub{f}\bar{y}$.
12 | 2. *Closure under composition.* If $\bar{f}:\bar{x}\to\Sub{f}\bar{y}$ and
13 |    $\bar{g}:\bar{y}\to\Sub{g}\bar{z}$ are hypocartesian, then $\bar{f};\bar{g}$
14 |    is hypocartesian.
15 | 
16 | **Proof.** Suppose first that $E$ is a cartesian fibration in our sense. Then
17 | $E$ has hypocartesian lifts because it has cartesian lifts. For closure under
18 | composition, fix hypocartesian $\bar{f},\bar{g}$; by {%cref 002C%} we know that
19 | $\bar{f},\bar{g}$ are also cartesian and hence by {%cref 001H%} so is the composite $\bar{f};\bar{g}$; therefore it follows
20 | that $\bar{f};\bar{g}$ is also hypocartesian.
21 | 
22 | Conversely, suppose that $E$ is a cartesian fibration in the sense of
23 | Grothendieck, and let $\bar{f}:\bar{x}\to\Sub{f}\bar{y}$ be the hypocartesian
24 | lift of $f:x\to y$ at $\bar{y}\in E\Sub{y}$; we shall see that $\bar{f}$ is also
25 | a *cartesian* lift of $f$ at $\bar{y}$ by constructing a unique factorization as
26 | follows:
27 | {%tex macrolib: topos%}
28 |   \begin{tikzpicture}[diagram]
29 |     \SpliceDiagramSquare{
30 |       west/style = lies over,
31 |       east/style = lies over,
32 |       north/node/style = upright desc,
33 |       height = 1.5cm,
34 |       nw = \bar{x},
35 |       ne = \bar{y},
36 |       sw = x,
37 |       north = \bar{f},
38 |       south = f,
39 |       se = y,
40 |     }
41 |     \node (u') [above left = 1.5cm of nw,xshift=-.5cm] {$\bar{u}$};
42 |     \node (u) [above left = 1.5cm of sw,xshift=-.5cm] {$u$};
43 |     \draw[lies over] (u') to (u);
44 |     \draw[->,bend left=30] (u') to node [sloped,above] {$\bar{h}$} (ne);
45 |     \draw[->] (u) to node [sloped,below] {$m$} (sw);
46 |     \draw[->,exists] (u') to (nw);
47 |   \end{tikzpicture}
48 | {%endtex%}
49 | Let $\bar{m}:\bar{u}\tick\to\Sub{m}\bar{x}$ be the hypocartesian lift of
50 | $m$ at $\bar{x}$, where $\bar{u}\tick\in E\Sub{u}$. By hypothesis, the
51 | composite $\bar{m};\bar{f} : \bar{u}\tick\to\Sub{m;f}\bar{y}$ is
52 | hypocartesian, so $\bar{h}$ factors uniquely through $\bar{m};\bar{f}$ over
53 | $\Idn{u}$:
54 | {%tex macrolib: topos%}
55 |   \begin{tikzpicture}[diagram]
56 |     \SpliceDiagramSquare{
57 |       west/style = lies over,
58 |       east/style = lies over,
59 |       north/node/style = upright desc,
60 |       height = 1.5cm,
61 |       nw = \bar{u}',
62 |       ne = \bar{y},
63 |       sw = u,
64 |       north = \bar{m};\bar{f},
65 |       south = m;f,
66 |       se = y,
67 |     }
68 |     \node (u') [above left = 1.5cm of nw,xshift=-.5cm] {$\bar{u}$};
69 |     \node (u) [above left = 1.5cm of sw,xshift=-.5cm] {$u$};
70 |     \draw[lies over] (u') to (u);
71 |     \draw[->,bend left=30] (u') to node [sloped,above] {$\bar{h}$} (ne);
72 |     \draw[double] (u) to (sw);
73 |     \draw[->,exists] (u') to node [desc] {$i$} (nw);
74 |   \end{tikzpicture}
75 | {%endtex%}
76 | The composite $i;\bar{m} : \bar{u}\to\Sub{m}\bar{x}$ is the required (cartesian)
77 | factorization of $\bar{h}$ through $\bar{f}$ over $m$. To see that $i;\bar{m}$
78 | is the unique such map, we observe that all morphisms $\bar{u}\to\Sub{m}\bar{x}$
79 | factor uniquely through $\bar{m}$ over $\Idn{u}$ as a consequence of $\bar{m}$
80 | being hypocartesian.
81 | {%qed %}
82 | 


--------------------------------------------------------------------------------
/_nodes/002C.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | taxon: lemma
 3 | title: Hypocartesian = cartesian in a cartesian fibration
 4 | ---
 5 | 
 6 | Let $E$ be a cartesian fibration in the sense of {%cref 0002%}, and let
 7 | $\bar{f} : \bar{x}\to\Sub{f}\bar{y}$ be displayed over $f:x\to y$. The
 8 | displayed morphism $\bar{f}$ is cartesian if and only if it is hypocartesian.
 9 | 
10 | **Proof.**
11 | Any cartesian map is clearly hypocartesian. To see that a hypocartesian map
12 | $\bar{f}:\bar{x}\to\Sub{f}\bar{y}$ in a cartesian fibration is cartesian, we
13 | consider the cartesian lift of $f:x\to y$ under $\bar{y}$:
14 | {%tex macrolib: topos%}
15 | \DiagramSquare{
16 |   height = 1.5cm,
17 |   west/style = lies over,
18 |   east/style = lies over,
19 |   nw/style = pullback,
20 |   nw = \bar{x}\tick,
21 |   ne = \bar{y},
22 |   se = y,
23 |   sw = x,
24 |   south = f,
25 |   north = \bar{f}\tick,
26 |   north/style = {->,exists},
27 | }
28 | {%endtex%}
29 | 
30 | As the cartesian lift $\bar{x}\tick\to \bar{y}$ is also hypocartesian, it
31 | follows that there is a unique vertical isomorphism identifying $\bar{x}$ with
32 | $\bar{x}\tick$ factoring $\bar{f} : \bar{x}\to\Sub{f}\bar{y}$ through
33 | $\bar{f}\tick : \bar{x}\tick\to\Sub{f}\bar{y}$. Being cartesian over $f$ is clearly
34 | stable under isomorphism, hence we conclude that $\bar{f}$ is cartesian from
35 | the fact that $\bar{f}\tick$ is cartesian.
36 | {%qed%}
37 | 


--------------------------------------------------------------------------------
/_nodes/002D.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Two ways to generalize pullbacks
 3 | taxon: remark
 4 | ---
 5 | 
 6 | Hypocartesian {%ref 002A%} and cartesian {%ref 0001%} morphisms can be thought
 7 | of as two distinct ways to generalize the concept of a pullback, depending on
 8 | what one considers the essential properties of pullbacks. Hypocartesian
 9 | morphisms more directly generalize the "little picture" universal property of
10 | pullbacks as limiting cones, whereas cartesian morphisms generalize the "big
11 | picture" dynamics of the pullback pasting lemma. As we have seen in {%cref
12 | 002C%} these two notions coincide in any cartesian fibration; the instance of
13 | this result for the fundamental self-indexing {%pref 001X%} verifies that
14 | pullbacks can be equivalently presented in terms of cartesian morphisms, as we
15 | have pointed out in {%cref 001Y%}.
16 | 


--------------------------------------------------------------------------------
/_nodes/002E.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Separators for cartesian fibrations
 3 | ---
 4 | 
 5 | Let $E$ be an ordinary category. In general,
 6 | to compare two morphisms $f,g:x\to y$ in $E$, it is not enough to see if they
 7 | agree on global points $u:1\to x$, because the behavior of $u,v$ may differ
 8 | only on generalized elements. In some cases, however, there is a family of
 9 | objects $\prn{s\Sub{i}}\Sub{i\in I}\in E$ are together *adequate* for comparing
10 | morphisms of $E$ in the sense of {%cref 002G%} below.
11 | 
12 | @include{002G}
13 | 
14 | The intuition of {%cref 002G%} is that to compare two morphisms $f,g:x\to y\in E$,
15 | it suffices to check that they behave the same on all $s\Sub{i}$-shaped figures
16 | when $\prn{s\Sub{i}}\Sub{i\in I}$ is a separating family for $E$.
17 | 
18 | @include{002F}
19 | 
20 | We will now generalize {%cref 002G%} to the case of a cartesian fibration.
21 | 
22 | @include{002I}
23 | @include{002H}
24 | 


--------------------------------------------------------------------------------
/_nodes/002F.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Well-pointedness of the category of sets
 3 | taxon: example
 4 | ---
 5 | 
 6 | In the category of sets, to compare two morphisms it is enough to check
 7 | that they agree on global points. This means that the unary family
 8 | $\brc{\ObjTerm{\SET}}$ is a separator for $\SET$, a property referred to more generally as *well-pointedness*.
 9 | 
10 | 


--------------------------------------------------------------------------------
/_nodes/002G.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Separating family for a category
 3 | taxon: definition
 4 | ---
 5 | 
 6 | Given an ordinary category $E$, a set-indexed family
 7 | $\prn{s\Sub{i}}\Sub{i\in I}$ of $E$-objects is called a *small separating
 8 | family* for $E$ when, assuming that for all $i\in I$ and all $u:s_i\to x$ we have
 9 | $u;f=u;g$, we then have $f=g$.
10 | 


--------------------------------------------------------------------------------
/_nodes/002H.md:
--------------------------------------------------------------------------------
1 | ---
2 | title: Small separator for a fibration
3 | taxon: definition
4 | ---
5 | 
6 | Let $E$ be a cartesian fibration over $B$ such that $B$ has
7 | binary products. A displayed object $\bar{s}\in E\Sub{s}$ is said to be a
8 | *small separator* for $E$ when any two vertical maps $f,g:\bar{u}\to\Sub{\Idn{x}}\bar{v}\in E\Sub{x}$ are equal when they are agree on $\bar{s}$-figures in the sense of {%cref 002I%}.
9 | 


--------------------------------------------------------------------------------
/_nodes/002I.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Agreement on a class of figure shapes
 3 | taxon: definition
 4 | ---
 5 | 
 6 | Let $\bar{s}$ be a displayed object in a cartesian fibration $E$ over $B$.
 7 | A pair of displayed morphisms $f,g:\bar{x}\to \bar{y}\in E$ are said to
 8 | *agree on $\bar{s}$-figures* when for any $\bar{s}$-figure
 9 | $h : \bar{z}\to \bar{x}$ in the sense of {%cref 002K%}, we have $h;f = h;g : \bar{z}\to \bar{y}$.
10 | 


--------------------------------------------------------------------------------
/_nodes/002J.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Full subfibrations and figure shapes
 3 | ---
 4 | 
 5 | In a category $E$, a morphism $f : x\to y$ can be thought of as a "figure"
 6 | of shape $x$ drawn in $y$. For instance, if $x$ is the point (i.e.
 7 | $x=\ObjTerm{E}$) then a morphism $x\to y$ is a "point" of the "space" $y$.
 8 | We refer to $x$ as the figure-shape in any such scenario.
 9 | The perspective of morphisms as figures is developed in more detail by Lawvere and Schanuel {%cite lawvere-schanuel:2009 -A%}.
10 | 
11 | It often happens that a useful class of figure shapes can be arranged
12 | into a *set*-indexed family $\prn{u\Sub{i}}\Sub{i\in I}$; viewed from the
13 | perspective of the family fibration $\FAM{E}$ {%pref 0006%}, this family is just a
14 | displayed object $\bar{u}$ over $I$ and then a figure shape "in"
15 | this family is given by any cartesian morphism $\bar{z}\to\bar{u}$. We will
16 | generalize this situation to the case of an arbitrary fibration, by
17 | constructing the full subfibration spanned by displayed objects equipped
18 | with a cartesian morphism into $\bar{u}$ in {%cref 0010%} below.
19 | 
20 | @include{0010}
21 | @include{002K}
22 | 


--------------------------------------------------------------------------------
/_nodes/002K.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Figures and figure shapes in the full subfibration
 3 | taxon: definition
 4 | ---
 5 | 
 6 | Let $E$ be a cartesian fibration and let $\FullSubfib{\bar{s}}\subseteq E$ be the full subfibration determined by a displayed object $\bar{s}\in E$ as in {%cref 0010%}. We now develop the following vocabulary:
 7 | 
 8 | 1. We will refer to each object of $\FullSubfib{\bar{s}}$ as a $\bar{s}$-figure
 9 | shape.
10 | 
11 | 2. A displayed morphism $\bar{z}\to \bar{x}$ is called a $\bar{s}$-figure
12 |    whenever $\bar{z}\in\FullSubfib{\bar{s}}$.
13 | 
14 | 


--------------------------------------------------------------------------------
/_nodes/002L.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Definable classes à la Bénabou
 3 | ---
 4 | 
 5 | A class of sets $\mathfrak{C}$ is sometimes said to be *formally
 6 | definable* when there is a formula $\prn{x \mid
 7 | \phi\Sub{\mathfrak{C}}\prn{x}}$ in the language of set theory such that a
 8 | set $S$ lies in $\mathfrak{C}$ if and only if
 9 | $\phi\Sub{\mathfrak{C}}\prn{S}$ holds. This concept is a bit sensitive, as
10 | it presupposes that we have a notion of "class" whose constituents are not
11 | all definable in this sense.
12 | 
13 | A better behaved notion of definability for sets than the formal one is
14 | given *model-theoretically*, i.e. relative to a model $\mathscr{M}$ of set
15 | theory in some ambient set theory {%cite benabou:1985%}. We will refer to
16 | an element of $\mathscr{M}$ as a set, and a subcollection of $\mathscr{M}$
17 | as a class.
18 | 
19 | @include{002U}
20 | @include{002V}
21 | 
22 | Bénabou then generalizes the definitions above to an arbitrary fibration,
23 | in such a way that the general fibered notion of definable class is
24 | equivalent in the fundamental self-indexing $\SelfIx{\mathscr{M}}$ to that of
25 | definable classes in the sense of {%cref 002V%}.
26 | 
27 | @include{002Q}
28 | @include{002S}
29 | 


--------------------------------------------------------------------------------
/_nodes/002M.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | taxon: construction
 3 | title: Closure of a class of sets under base change
 4 | ---
 5 | 
 6 | Let $\mathfrak{C}\subseteq\mathscr{M}$ be a class of sets; there exists a class $\bar{\mathfrak{C}}$ of *families*
 7 | of sets that contains $\prn{S\Sub{i}}\Sub{i\in I}$ exactly when each
 8 | $S\Sub{i}$ lies in $\mathfrak{C}$. We will refer to $\bar{\mathfrak{C}}$ as the *closure under base change* of $\mathfrak{C}$, a name motivated by the fact that when $\prn{S\Sub{i}}\Sub{i\in I}$ lies in $\bar{\mathfrak{C}}$,
 9 | then for any $u:J\to I$, the base change $\prn{S\Sub{uj}}\Sub{j\in J}$ also lies in $\bar{\mathfrak{C}}$.
10 | 


--------------------------------------------------------------------------------
/_nodes/002N.md:
--------------------------------------------------------------------------------
1 | ---
2 | taxon: construction
3 | title: From classes of families to classes of sets
4 | ---
5 | 
6 | Conversely to {%cref 002M%} we may take a class of families of sets $\mathfrak{F}$ to the the class of sets $\mathfrak{F}\Sub{1}\subseteq{\mathscr{M}}$ spanned by sets $S$ such that the singleton family $\brc{S}$ lies in $\mathfrak{C}$.
7 | 
8 | 


--------------------------------------------------------------------------------
/_nodes/002O.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | taxon: remark
 3 | ---
 4 | 
 5 | Let $\mathscr{M}$ be a model of {%nlab ETCS%}; and let $\mathfrak{C}$ be a
 6 | class of families of sets in $\mathscr{M}$. Then $\mathfrak{C}$ is a class of
 7 | objects in the fundamental self-indexing $\SelfIx{\mathscr{M}}$ over
 8 | $\mathscr{M}$.  Furthermore, $\mathscr{M}$ a definable class in
 9 | $\SelfIx{\mathscr{M}}$ in the sense of {%cref 002W%} if and only if it is a
10 | definable class of families of sets in the sense of {%cref 002P%}.
11 | 


--------------------------------------------------------------------------------
/_nodes/002P.md:
--------------------------------------------------------------------------------
1 | ---
2 | taxon: definition
3 | title: Definable class of families of sets
4 | ---
5 | 
6 | A class of families of sets $\mathfrak{F}$ is said to be *definable* when it is stable and moreover, for any family of sets $\prn{S\Sub{i}}\Sub{i\in I}$, there exists a subset $J\subseteq I$ such that the base change $\prn{S\Sub{j}}\Sub{j\in J}$ lies in $\mathfrak{F}$, and moreover, such that $u:K\to I$ factors through $J\subseteq I$ whenever the base change $\prn{S\Sub{uk}}\Sub{k\in K}$ lies in $\mathfrak{F}$.
7 | 


--------------------------------------------------------------------------------
/_nodes/002Q.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Set-theoretic intuition for Bénabou's definability
 3 | ---
 4 | 
 5 | To motivate Bénabou's general notion of definability, we will first develop an alternative
 6 | perspective on definability for classes of sets in terms of *families* of
 7 | sets.
 8 | 
 9 | @include{002M}
10 | @include{002N}
11 | @include{002P}
12 | @include{002R}
13 | @include{002T}
14 | 


--------------------------------------------------------------------------------
/_nodes/002R.md:
--------------------------------------------------------------------------------
1 | ---
2 | taxon: intuition
3 | ---
4 | 
5 | In other words, a stable class of families of sets is definable when any family of sets can be restricted to a "biggest" subfamily that lies in the class.
6 | 


--------------------------------------------------------------------------------
/_nodes/002S.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Bénabou's notion of definability
 3 | ---
 4 | 
 5 | We will now construe {%cref 002Q%} as the instantiation at the fundamental
 6 | self-indexing $\SelfIx{\mathscr{M}}$ of a more general notion of
 7 | definability for classes of objects in a cartesian fibration, defined
 8 | forthwith. The following definition of definability in a cartesian fibration is due to Bénabou, but is discussed in print by Borceux {%cite borceux:1994:vol2 -A%}, Jacobs {%cite jacobs:1999 -A%}, Streicher {%cite streicher:2018 -A%}.
 9 | 
10 | @include{002W}
11 | @include{002O}
12 | 


--------------------------------------------------------------------------------
/_nodes/002T.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | taxon: lemma
 3 | title: Characterization of definable classes of families
 4 | ---
 5 | 
 6 | Let $\mathfrak{C}\subseteq\mathscr{M}$ be a class of sets; then $\bar{\mathfrak{C}}$ is a definable class of families of sets if and only if $\mathfrak{C}$ is a definable class of sets.
 7 | 
 8 | 
 9 | **Proof.** Suppose that $\mathfrak{C}$ is a definable class of sets. To
10 | check that $\bar{\mathfrak{C}}$ is a definable class of families of sets,
11 | we fix a family $\prn{S\Sub{i}}\Sub{i\in I}$ not necessarily lying in $\bar{\mathfrak{C}}$. Because $\mathfrak{C}$ is definable, the intersection $\mathfrak{C}\cap \bigcup\Sub{i\in I}S\Sub{i}$ is represented by a set $U$. We therefore take the subset $J = \brc{i \in I\mid S_i\in U}\subseteq I$, and verify that the base change $\prn{S\Sub{j}}\Sub{j\in J}$ is the largest approximation of $\prn{S\Sub{i}}\Sub{i\in I}$ by a family lying in $\bar{\mathfrak{C}}$.
12 | 
13 | Conversely suppose that $\bar{\mathfrak{C}}$ is a definable class of families of sets. To see that $\mathfrak{C}$ is definable, we fix a class $\mathfrak{U}$ represented by a set $U\in\mathscr{M}$ to check that $\mathfrak{C}\cap\mathfrak{U}$ is representable. We consider the family of sets $\prn{u}\Sub{u\in U}$; because $\bar{\mathfrak{C}}$ is definable, there is a largest subset $V\subseteq U$ such that the change of base $\prn{v}\Sub{v\in V}$ lies in $\bar{\mathfrak{C}}$, i.e. such that each $v\in V$ lies in $\mathfrak{C}$. Therefore $\mathfrak{C}\cap\mathfrak{U}$ is represented by the set $V$.
14 | {%qed%}
15 | 


--------------------------------------------------------------------------------
/_nodes/002U.md:
--------------------------------------------------------------------------------
1 | ---
2 | taxon: definition
3 | title: Representable class of sets
4 | ---
5 | 
6 | A class of sets $\mathfrak{C}\subseteq\mathscr{M}$ is called *representable* when there is a set $C\in\mathscr{M}$ such that $U\in C$ if and only if $U\in\mathfrak{C}$.
7 | 
8 | 


--------------------------------------------------------------------------------
/_nodes/002V.md:
--------------------------------------------------------------------------------
1 | ---
2 | taxon: definition
3 | title: Definable class of sets
4 | ---
5 | 
6 | A class of sets $\mathfrak{C}\subseteq\mathscr{M}$ is called *definable* when for any representable class $\mathfrak{S}$, the class $\mathfrak{C}\cap\mathfrak{S}$ is representable.
7 | 
8 | 


--------------------------------------------------------------------------------
/_nodes/002W.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Definable class in a fibration
 3 | taxon: definition
 4 | ---
 5 | 
 6 | Let $E$ be a cartesian fibration over $B$. A *definable class*
 7 | $\mathfrak{F}$ in $E$ is a stable subcollection of the displayed objects of $E$
 8 | such that for any $\bar{u}\in E\Sub{u}$, there exists a cartesian map $\bar{v}\to \bar{u}$ lying over a monomorphism $v\rightarrowtail u$ such that $\bar{v}\in \mathfrak{F}$ and, moreover, any cartesian morphism $\bar{w}\to\bar{u}$ such that $\bar{w}\in\mathfrak{F}$ factors through $\bar{v}\to\bar{u}$.
 9 | {%tex macrolib: topos%}
10 |   \DiagramSquare{
11 |     height = 1.5cm,
12 |     west/style = lies over,
13 |     east/style = lies over,
14 |     nw/style = pullback,
15 |     south/style = {exists,>->},
16 |     north/style = {->,exists},
17 |     nw = \bar{v}\in\mathfrak{F},
18 |     ne = \bar{u},
19 |     se = u,
20 |     sw = v,
21 |   }
22 |   \qquad
23 |   \begin{tikzpicture}[diagram,baseline=(l/sw.base)]
24 |     \SpliceDiagramSquare<l/>{
25 |       height = 1.5cm,
26 |       width = 2.25cm,
27 |       west/style = lies over,
28 |       east/style = lies over,
29 |       ne/style = pullback,
30 |       nw = \bar{w}\in\mathfrak{F},
31 |       sw = w,
32 |       ne = \bar{v},
33 |       se = v,
34 |       south/style = {->,exists},
35 |       north/style = {->,exists},
36 |       south = \exists!,
37 |       north = \exists!,
38 |       north/node/style = upright desc,
39 |     }
40 |     \SpliceDiagramSquare<r/>{
41 |       height = 1.5cm,
42 |       glue = west,
43 |       glue target = l/,
44 |       east/style = lies over,
45 |       south/style = >->,
46 |       ne = \bar{u},
47 |       se = u,
48 |     }
49 |     \draw[->,bend left=30] (l/nw) to node [above] {cart.} (r/ne);
50 |   \end{tikzpicture}
51 | {%endtex%}
52 | 


--------------------------------------------------------------------------------
/_nodes/002X.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: "Example: Revisiting the fundamental self-indexing"
 3 | ---
 4 | 
 5 | Recall that the fundamental self-indexing $\SelfIx{B}$ {%pref 001X %} of a category $B$
 6 | is a displayed category with $\SelfIx{B}\Sub{x} = \Sl{B}{x}$.
 7 | As discussed in {%cref 001Y %}, $\SelfIx{B}$ is a cartesian fibration over $B$ if and
 8 | only if $B$ has pullbacks. However, $\SelfIx{B}$ is unconditionally a *cocartesian*
 9 | fibration.
10 | 
11 | @include{002Y}
12 | 


--------------------------------------------------------------------------------
/_nodes/002Y.md:
--------------------------------------------------------------------------------
1 | ---
2 | taxon: exercise
3 | ---
4 | 
5 | Prove that $\SelfIx{B}$ from {%ref 001X%} is a cocartesian fibration for any category $B$.
6 | 


--------------------------------------------------------------------------------
/_nodes/002Z.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | taxon: construction
 3 | ---
 4 | 
 5 | Let $B$ be a category. Define the displayed category $\overline{B}$ over $B$ as
 6 | follows:
 7 | 
 8 | 1. For $x\in B$, define $\overline{B}\Sub{x}$ as the collection of pairs
 9 |    $(\bar{x}\in B,p\Sub{x}:x\to\bar{x})$.
10 | 2. For $f : x\to y\in B$, define $\overline{B}\Sub{f}$ to be the collection of
11 |    commuting squares in the following configuration:
12 | {%tex macrolib: topos%}
13 |   \DiagramSquare{
14 |     height = 1.7cm,
15 |     sw = \bar{x},
16 |     se = \bar{y},
17 |     nw = x,
18 |     ne = y,
19 |     west = p\Sub{x},
20 |     east = p\Sub{y},
21 |     south = \bar{f},
22 |     north = f,
23 |     west/style = {->,exists},
24 |     east/style = {->,exists},
25 |     south/style = {->,exists},
26 |   }
27 | {%endtex%}
28 | 


--------------------------------------------------------------------------------
/_nodes/0030.md:
--------------------------------------------------------------------------------
1 | ---
2 | taxon: exercise
3 | ---
4 | 
5 | Prove that $\overline{B}$ is a cartesian fibration for any category $B$.
6 | 


--------------------------------------------------------------------------------
/_nodes/0031.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: The identity morphism of a topos
 3 | taxon: computation
 4 | ---
 5 | 
 6 | Let $X$ be a topos over a geometric universe $E$; we will compute the identity morphism $\Idn{X}:X\to X$ in $\TOP{E}$ as a diagram in the meta-2-category $\AlgGU$ of geometric universes and algebraic morphisms. We will let $\Sh{\Idn{X}} : \Sh{X}\to \Sh{X}$ be the identity map $\Idn{\Sh{X}}$. The coherence $\varphi\Sub{\Idn{X}}$ is given by the identity 2-cell, as we have the equation $X; \Sh{\Idn{X}} = X$ in $[E,\Sh{X}]$:
 7 | {%tex macrolib: topos%}
 8 | \begin{ZX}
 9 | \zxN{} &\dar[close]{X}\\
10 | \zxN{} &\zxX[pilr]{\varphi\Sub{\Idn{X}}=\Idn{X}}\dlar[swap,-o,close]{X}\drar[N,close,near end]{\Sh{\Idn{X}} = \Idn{\Sh{X}}}\\[3mm]
11 | \zxN{} & \zxN{} & \zxN{}
12 | \end{ZX}
13 | {%endtex%}
14 | {%qed%}
15 | 


--------------------------------------------------------------------------------
/_nodes/0032.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Composing morphisms of topoi
 3 | taxon: computation
 4 | ---
 5 | 
 6 | Let $X,Y,Z$ be topoi over a geometric universe $E$ and let $f:X\to Y$ and $g:Y\to Z$ be morphisms of topoi; we will construct the composite morphism $f;g:X\to Z$.
 7 | 
 8 | First of all, let us unfold assumptions into the language of $\AlgGU$. In particular, we have morphisms $\Sh{f}:\Sh{Y}\to \Sh{X}$ and $\Sh{g} : \Sh{Z}\to \Sh{Y}$, together with invertible 2-cells $\varphi_f,\varphi_g$ as depicted below:
 9 | {%tex macrolib: topos%}
10 | \begin{ZX}
11 | \zxN{} &\dar[close]{X}\\
12 | \zxN{} &\zxX[pilr]{\varphi_f}\dlar[swap,-o,close]{Y}\drar[N,close]{\Sh{f}}\\[3mm]
13 | \zxN{} & \zxN{} & \zxN{}
14 | \end{ZX}
15 | \qquad
16 | \begin{ZX}
17 | \zxN{} &\dar[close]{Y}\\
18 | \zxN{} &\zxX[pilr]{\varphi_g}\dlar[swap,-o,close]{Z}\drar[N,close]{\Sh{g}}\\[3mm]
19 | \zxN{} & \zxN{} & \zxN{}
20 | \end{ZX}
21 | {%endtex%}
22 | 
23 | We define $\Sh{f;g}$ to be the composite $\Sh{g};\Sh{f} : \Sh{Z}\to \Sh{Y}$; then we define the coherence isomorphism $\varphi\Sub{f;g}$ to be the 2-cell depicted below:
24 | {%tex macrolib: topos%}
25 | \begin{ZX}
26 | \zxN{} &\dar[close]{X}\\
27 | \zxN{} &\zxX[pilr]{\varphi\Sub{f;g}}\dlar[swap,-o,close]{Z}\drar[N,close]{\Sh{f;g}}\\[3mm]
28 | \zxN{} & \zxN{} & \zxN{}
29 | \end{ZX}
30 | :=
31 | \begin{ZX}
32 | \zxN{} & \zxN{} &\dar[close]{X}\\
33 | \zxN{} & \zxN{} &\zxX[pilr]{\varphi_f} \dlar[swap,-N,close]{Y} \drar[N,close,near end]{\Sh{f}}\\
34 | \zxN{} & \zxX[pilr]{\varphi_g}\dlar[swap,-o,close]{Z} \drar[N,close,near end]{\Sh{g}} & \zxN{} & \zxN{} \\[3mm]
35 | \zxN{} & \zxN{} & \zxN{} & \zxN{}
36 | \end{ZX}
37 | {%endtex%}
38 | {%qed%}
39 | 


--------------------------------------------------------------------------------
/_nodes/0033.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: The identity 2-cell of a morphism of topoi
 3 | taxon: computation
 4 | ---
 5 | 
 6 | Let $X,Y$ be a pair of topoi over a geometric universe $E$, and let $f : X \to Y$ be a morphism of $E$-topoi; we will construct the identity 2-cell $\Idn{f} : f \to f$ in the hom category $[X,Y]$. First of all, we unfold the data that we are given into the language of $\AlgGU$:
 7 | {%tex macrolib: topos%}
 8 | \begin{ZX}
 9 | \zxN{} &\dar[close]{X}\\
10 | \zxN{} &\zxX[pilr]{\varphi_f}\dlar[swap,-o,close]{Y}\drar[N,close]{\Sh{f}}\\[3mm]
11 | \zxN{} & \zxN{} & \zxN{}
12 | \end{ZX}
13 | {%endtex%}
14 | 
15 | We define $\Sh{\Idn{f}}$ to be the identity 2-cell $\Idn{\Sh{f}} : \Sh{f}\to \Sh{f}$; it remains to observe the following equation of 2-cells:
16 | {%tex macrolib: topos%}
17 | \begin{ZX}
18 |   \zxN{} &\dar[close]{X}\\
19 |    \zxN{} & \zxX[pilr]{\varphi_f} \arrow[-N', near end, close, swap]{ddl}{Y} \drar[N, close, pos=.6]{\Sh{f}}\\
20 |    \zxN{} & \zxN{} & \zxX[pilr]{\Sh{\Idn{f}}=\Idn{\Sh{f}}} \dar{\Sh{f}}\\
21 |    \zxN{} & \zxN{} & \zxN{}
22 | \end{ZX}
23 | =
24 | \begin{ZX}
25 |   \zxN{} &\dar[close]{X}\\
26 |   \zxN{} &\zxX[pilr]{\varphi_g}\dlar[swap,-o,close]{Y}\drar[N,close]{\Sh{f}}\\[3mm]
27 |   \zxN{} & \zxN{} & \zxN{}
28 | \end{ZX}
29 | {%endtex%}
30 | {%qed%}
31 | 


--------------------------------------------------------------------------------
/_nodes/0034.md:
--------------------------------------------------------------------------------
  1 | ---
  2 | title: Composition of 2-cells between morphisms of topoi
  3 | taxon: computation
  4 | ---
  5 | 
  6 | Let $X,Y$ be a pair of topoi over a geometric morphism $E$ and let $f,g,h : X\to Y$ be morphisms of topoi; given 2-cells $\alpha : f \to g$ and $\beta : g \to h$, we will define the composite 2-cell $\alpha;\beta : f \to h$.
  7 | 
  8 | First we unfold what we are given into the language of geometric universes; the morphisms of topoi unravel to the following morphisms and 2-cells of geometric universes:
  9 | {%tex macrolib: topos%}
 10 | \begin{ZX}
 11 | \zxN{} &\dar[close]{X}\\
 12 | \zxN{} &\zxX[pilr]{\varphi_f}\dlar[swap,-o,close]{Y}\drar[N,close]{\Sh{f}}\\[3mm]
 13 | \zxN{} & \zxN{} & \zxN{}
 14 | \end{ZX}
 15 | \qquad
 16 | \begin{ZX}
 17 | \zxN{} &\dar[close]{X}\\
 18 | \zxN{} &\zxX[pilr]{\varphi_g}\dlar[swap,-o,close]{Y}\drar[N,close]{\Sh{g}}\\[3mm]
 19 | \zxN{} & \zxN{} & \zxN{}
 20 | \end{ZX}
 21 | \qquad
 22 | \begin{ZX}
 23 | \zxN{} &\dar[close]{X}\\
 24 | \zxN{} &\zxX[pilr]{\varphi_h}\dlar[swap,-o,close]{Y}\drar[N,close]{\Sh{h}}\\[3mm]
 25 | \zxN{} & \zxN{} & \zxN{}
 26 | \end{ZX}
 27 | {%endtex%}
 28 | 
 29 | The given 2-cells of topoi $\alpha:f\to g$ and $\beta : g \to h$ unravel into 2-cells between morphisms of geometric universe $\Sh{\alpha} : \Sh{f}\to \Sh{g}$ and $\Sh{\beta}:\Sh{g}\to\Sh{f}$ satisfying the following equations:
 30 | {%tex macrolib: topos%}
 31 | \begin{aligned}
 32 | \begin{ZX}[ampersand replacement=\&]
 33 |   \zxN{} \&\dar[close]{X}\\
 34 |    \zxN{} \& \zxX[pilr]{\varphi_f} \arrow[-N', near end, close, swap]{ddl}{Y} \drar[N, close, pos=.6]{\Sh{f}}\\
 35 |    \zxN{} \& \zxN{} \& \zxX[pilr]{\Sh{\alpha}} \dar{\Sh{g}}\\
 36 |    \zxN{} \& \zxN{} \& \zxN{}
 37 | \end{ZX}
 38 | &=
 39 | \begin{ZX}[ampersand replacement=\&]
 40 |   \zxN{} \&\dar[close]{X}\\
 41 |   \zxN{} \&\zxX[pilr]{\varphi_g}\dlar[swap,-o,close]{Y}\drar[N,close]{\Sh{g}}\\[3mm]
 42 |   \zxN{} \& \zxN{} \& \zxN{}
 43 | \end{ZX}
 44 | \\
 45 | \begin{ZX}[ampersand replacement=\&]
 46 |   \zxN{} \&\dar[close]{X}\\
 47 |   \zxN{} \& \zxX[pilr]{\varphi_g} \arrow[-N', near end, close, swap]{ddl}{Y} \drar[N, close, pos=.6]{\Sh{g}}\\
 48 |   \zxN{} \& \zxN{} \& \zxX[pilr]{\Sh{\beta}} \dar{\Sh{h}}\\
 49 |   \zxN{} \& \zxN{} \& \zxN{}
 50 | \end{ZX}
 51 | &=
 52 | \begin{ZX}[ampersand replacement=\&]
 53 |   \zxN{} \&\dar[close]{X}\\
 54 |   \zxN{} \&\zxX[pilr]{\varphi_h}\dlar[swap,-o,close]{Y}\drar[N,close]{\Sh{h}}\\[3mm]
 55 |   \zxN{} \& \zxN{} \& \zxN{}
 56 | \end{ZX}
 57 | \end{aligned}
 58 | {%endtex%}
 59 | 
 60 | We choose $\Sh{\alpha;\beta}$ to be the composite 2-cell depicted below:
 61 | {%tex macrolib: topos%}
 62 | \begin{ZX}
 63 | \dar[close]{\Sh{f}}\\
 64 | \zxX[pilr]{\Sh{\alpha;\beta}}\dar[close]{\Sh{h}}\\
 65 | \zxN{}
 66 | \end{ZX}
 67 | :=
 68 | \begin{ZX}
 69 | \dar[close]{\Sh{f}}\\
 70 | \zxX[pilr]{\Sh{\alpha}} \dar[close]{\Sh{g}}\\
 71 | \zxX[pilr]{\Sh{\beta}} \dar[close]{\Sh{h}}\\
 72 | \zxN{}
 73 | \end{ZX}
 74 | {%endtex%}
 75 | 
 76 | We must check the following equation of 2-cells:
 77 | {%tex macrolib: topos%}
 78 | \begin{ZX}
 79 |   \zxN{} &\dar[close]{X}\\
 80 |    \zxN{} & \zxX[pilr]{\varphi_f} \arrow[-N', near end, close, swap]{ddl}{Y} \drar[N, close, pos=.6]{\Sh{f}}\\
 81 |    \zxN{} & \zxN{} & \zxX[pilr]{\Sh{\alpha;\beta}} \dar{\Sh{h}}\\
 82 |    \zxN{} & \zxN{} & \zxN{}
 83 | \end{ZX}
 84 | =
 85 | \begin{ZX}
 86 |   \zxN{} &\dar[close]{X}\\
 87 |   \zxN{} &\zxX[pilr]{\varphi_h}\dlar[swap,-o,close]{Y}\drar[N,close]{\Sh{h}}\\[3mm]
 88 |   \zxN{} & \zxN{} & \zxN{}
 89 | \end{ZX}
 90 | {%endtex%}
 91 | 
 92 | The above follows by unfolding the definition of $\Sh{\alpha;\beta}$ and using the assumed equations concerning $\Sh{\alpha}$ and $\Sh{\beta}$:
 93 | {%tex macrolib: topos%}
 94 | \begin{aligned}
 95 | \begin{ZX}[ampersand replacement = \&]
 96 |    \zxN{} \&\dar[close]{X}\\
 97 |    \zxN{} \& \zxX[pilr]{\varphi_f} \arrow[-N', near end, close, swap]{ddl}{Y} \drar[N, close, pos=.6]{\Sh{f}}\\
 98 |    \zxN{} \& \zxN{} \& \zxX[pilr]{\Sh{\alpha;\beta}} \dar{\Sh{h}}\\
 99 |    \zxN{} \& \zxN{} \& \zxN{}
100 | \end{ZX}
101 | &=
102 | \begin{ZX}[ampersand replacement = \&]
103 |    \zxN{} \&\dar[close]{X}\\
104 |    \zxN{} \& \zxX[pilr]{\varphi_f} \arrow[-N', near end, close, swap]{ddl}{Y} \drar[N, close, pos=.6]{\Sh{f}}\\
105 |    \zxN{} \& \zxN{} \& \zxX[pilr]{\Sh{\alpha}} \dar{\Sh{g}}\\
106 |    \zxN{} \& \zxN{} \& \zxX[pilr]{\Sh{\beta}} \dar{\Sh{h}}\\
107 |    \zxN{} \& \zxN{} \& \zxN{}
108 | \end{ZX}
109 | \\
110 | &=
111 | \begin{ZX}[ampersand replacement = \&]
112 |    \zxN{} \&\dar[close]{X}\\
113 |    \zxN{} \& \zxX[pilr]{\varphi_g} \arrow[-N', near end, close, swap]{ddl}{Y} \drar[N, close, pos=.6]{\Sh{g}}\\
114 |    \zxN{} \& \zxN{} \& \zxX[pilr]{\Sh{\beta}} \dar{\Sh{h}}\\
115 |    \zxN{} \& \zxN{} \& \zxN{}
116 | \end{ZX}
117 | \\
118 | &=
119 | \begin{ZX}[ampersand replacement = \&]
120 |    \zxN{} \&\dar[close]{X}\\
121 |    \zxN{} \& \zxX[pilr]{\varphi_h} \arrow[-N', near end, close, swap]{dl}{Y} \drar[N, close]{\Sh{h}}\\[3mm]
122 |    \zxN{} \& \zxN{} \& \zxN{}
123 | \end{ZX}
124 | \end{aligned}
125 | {%endtex%}
126 | {%qed%}
127 | 


--------------------------------------------------------------------------------
/_nodes/0035.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Explicit computations in the 2-category of topoi
 3 | ---
 4 | 
 5 | In what follows, we will give explicit computations of the 2-categorical structure of $\TOP{E}$ in the language of $\AlgGU$.
 6 | 
 7 | @include{0031}
 8 | @include{0032}
 9 | @include{0033}
10 | @include{0034}
11 | 


--------------------------------------------------------------------------------
/_nodes/0036.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Morphisms of geometric universes
 3 | taxon: definition
 4 | ---
 5 | 
 6 | We describe three different kinds of morphism between geometric universes.
 7 | 
 8 | 1. A *partial cartesian morphism* $f : E \to F$ is a pullback preserving functor $f:E\to F$.
 9 | 2. A *cartesian morphism* $f : E \to F$ is a partial cartesian morphism $f : E\to F$ that preserves the terminal object.
10 | 4. A *logical morphism* $f : E \to F$ is a cartesian morphism $f : E \to F$ that additionally preserves the subobject classifier.
11 | 3. An *algebraic morphism* $f : E \to F$ is a pair $f = (\InvImg{f}\dashv \DirImg{f})$ such that $\InvImg{f} : E \to F$ a cartesian morphism of geometric universes. The left adjoint $\InvImg{f}$ is called the *inverse image* and the right adjoint $\DirImg{f}$ is called the *direct image*.
12 | 


--------------------------------------------------------------------------------
/_nodes/0037.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: 2-categories of geometric universes
 3 | taxon: definition
 4 | ---
 5 | 
 6 | Each of the different notions of geometric universe above gives rise to a 2-category. We will fix our conventions below:
 7 | 
 8 | 1. $\PCartGU$ is the 2-category of geometric universes, partial cartesian morphisms, and natural transformations between them.
 9 | 2. $\CartGU$ is the 2-category of geometric universes, cartesian morphisms, and natural transformations between them.
10 | 3. $\LogGU$ is the locally groupoidal 2-category of geometric universes, logical morphisms, and natural isomorphisms between them.
11 | 4. $\AlgGU$ is the 2-category of geometric universes and algebraic morphisms $f = (\InvImg{f}\dashv\DirImg{f})$, such that a 2-morphism $f \to g$ is given by a natural transformation $\InvImg{f}\to\InvImg{g}$.
12 | 


--------------------------------------------------------------------------------
/_nodes/0038.md:
--------------------------------------------------------------------------------
1 | ---
2 | title: Finite colimits, natural numbers objects, and logical functors
3 | taxon: remark
4 | ---
5 | 
6 | Note that any geometric universe has finite colimits, which can be constructed using the subobject classifier. A logical morphism between geometric universes necessarily preserves finite colimits for this reasons. Using Freyd's characterization of natural numbers objects in terms of a coequalization property, it follows that logical morphisms also preserve the natural numbers object.
7 | 


--------------------------------------------------------------------------------
/_nodes/0039.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Fibrations of geometric universes
 3 | ---
 4 | 
 5 | Logical morphisms play an important role in the study of geometric universes; in particular, the most important notion of *fibered* geometric universe involves logical morphisms rather than algebraic morphisms.
 6 | 
 7 | @include{003A}
 8 | @include{003C}
 9 | @include{003D}
10 | @include{003E}
11 | 
12 | It will follow in the scenario of {%cref 003E%}, the total category $\TotCat{E}$ is a geometric universe and the projection $\TotCat{E}\to B$ is both a logical morphism of geometric universes and an algebraic morphism of geometric universes. Moreover, the Artin gluing theory of geometric universes will reveal that there exists a subterminal object $U\rightarrowtail \ObjTerm{\TotCat{E}}$ such that $\TotCat{E}\to B$ factors as  $\TotCat{E}\to \Sl{\TotCat{E}}{U} \simeq B$; indeed, $U$ is the initial object of the fiber category $E\Sub{\ObjTerm{B}}$. Then $F = E\Sub{\ObjTerm{B}}$ will be revealed to be equivalent to the full subcategory of $\TotCat{E}$ spanned by objects $X$ such that the projection $X\times U\to U$ is an isomorphism.
13 | 


--------------------------------------------------------------------------------
/_nodes/003A.md:
--------------------------------------------------------------------------------
1 | ---
2 | title: Logical fibrations of geometric universes
3 | taxon: definition
4 | ---
5 | 
6 | Let $B$ be a geometric universes; a *logical fibration of geometric universes* over $B$ is given by a fibered category $E$ over $B$ such that each that fiber category $E_u$ is a geometric universe and each restriction map $E_v\to E_u$ is a logical morphism.
7 | 


--------------------------------------------------------------------------------
/_nodes/003C.md:
--------------------------------------------------------------------------------
 1 | ---
 2 | title: Moens' Theorem
 3 | taxon: theorem
 4 | ---
 5 | 
 6 | Let $E$ be a logical fibration of geometric universes over a geometric universe $B$. Then the following are equivalent:
 7 | 
 8 | 1. $E$ has stable and disjoint sums over $B$.
 9 | 2. $E$ is the *gluing fibration* for a cartesian morphism $B\to F$ of geometric universes.
10 | 
11 | Indeed, in the situation described above we may define $F$ to be the fiber $E\Sub{\ObjTerm{B}}$ over the terminal object. The functor $B\to E\Sub{\ObjTerm{B}}$ sends each $b\in B$ to the sum $\coprod_{b}\ObjTerm{E}$.
12 | 


--------------------------------------------------------------------------------
/_nodes/003D.md:
--------------------------------------------------------------------------------
1 | ---
2 | title: Jibladze's Theorem
3 | taxon: theorem
4 | ---
5 | 
6 | Let $E$ be a logical fibration of geometric universes over $B$; if $E$ has small sums over $B$, then these sums are stable and disjoint.
7 | 


--------------------------------------------------------------------------------
/_nodes/003E.md:
--------------------------------------------------------------------------------
1 | ---
2 | taxon: corollary
3 | ---
4 | 
5 | Combining {%cref 003C%} and {%cref 003D%}, the following are equivalent for a logical fibration $E$ of geometric universes over $B$:
6 | 
7 | 1. $E$ has small sums over $B$.
8 | 2. $E$ is the gluing fibration for a cartesian morphism $B \to F$ of geometric universes.
9 | 


--------------------------------------------------------------------------------
/_plugins/changelog.rb:
--------------------------------------------------------------------------------
 1 | require 'open3'
 2 | require 'date'
 3 | 
 4 | module Changelog
 5 |   class Generator < Jekyll::Generator
 6 |     NODES_COLLECTIONS_NAMES = ["nodes", "lectures"]
 7 |     NODES_COLLECTION_PATHS = ["_lectures/", "_nodes/"]
 8 |     CHANGELOG_PAGE_PATH = "changelog.md"
 9 | 
10 |     def generate(site)
11 |       nodes = site.collections.values_at(*NODES_COLLECTIONS_NAMES).flat_map(&:docs)
12 |       changelog = site.pages.find { |page| page.relative_path == CHANGELOG_PAGE_PATH }
13 |       authors = site.config.fetch("changelog", {}).fetch("authors", {})
14 |       commits = parse_git_data(get_git_data, authors)
15 |       commits.each do |commit|
16 |         commit["diffs"].each do |diff|
17 |           diff["node"] = nodes.find do |node|
18 |             node.relative_path == diff["path"]
19 |           end
20 |         end
21 |       end
22 |       changelog.data['commits'] = commits
23 |     end
24 | 
25 |     def get_git_data
26 |       stdout, stderr, status = Open3.capture3(
27 |         [
28 |           "git log",
29 |           "--pretty='format:#{%w{%H %at %aN %aE %s}.join('%x00')}'",
30 |           # we're interested in ADM of files (not Rs or Cs)
31 |           "--name-status",
32 |           "--no-renames",
33 |           "--diff-filter=ADM",
34 |           # we match folders to get deleted objects too
35 |           "--",
36 |           *NODES_COLLECTION_PATHS,
37 |         ].join(" "),
38 |       )
39 |       stdout
40 |     end
41 | 
42 |     def parse_git_data(data, authors)
43 |       [].tap do |changelog|
44 |         data.split("\n\n").each do |block|
45 |           commit, *diff = block.split("\n")
46 |           hash, timestamp, name, mail, subject = commit.split("\u0000")
47 |           changelog << {
48 |             "hash" => hash,
49 |             "timestamp" => Time.at(timestamp.to_i),
50 |             "subject" => subject,
51 |             "diffs" => [],
52 |             "author" => {
53 |               "name" => name,
54 |               "mail" => mail,
55 |             }.merge(authors.fetch(mail, {})),
56 |           }
57 |           diff.each do |line|
58 |             status, path = line.split("\t")
59 |             changelog.last.fetch("diffs") << {
60 |               "status" => status,
61 |               "path" => path,
62 |             }
63 |           end
64 |         end
65 |       end
66 |     end
67 |   end
68 | end
69 | 


--------------------------------------------------------------------------------
/_plugins/display_title_filter.rb:
--------------------------------------------------------------------------------
 1 | module DisplayTitleFilter
 2 |   extend self
 3 | 
 4 |   def display_numbering(resource)
 5 |     info = NodeInfo.new resource
 6 |     info.display_numbering
 7 |   end
 8 | 
 9 |   def display_title(resource)
10 |     info = NodeInfo.new resource
11 |     info.display_title
12 |   end
13 | 
14 |   def display_title_parenthetical(resource)
15 |     info = NodeInfo.new resource
16 |     info.display_title_parenthetical
17 |   end
18 | 
19 | end
20 | 
21 | Liquid::Template.register_filter(DisplayTitleFilter)
22 | 
23 | 


--------------------------------------------------------------------------------
/_plugins/nlab_tag.rb:
--------------------------------------------------------------------------------
 1 | class NLabTag < Liquid::Tag
 2 |   def initialize(tag_name, title, tokens)
 3 |     super
 4 |     @title = title.strip
 5 |   end
 6 | 
 7 |   def render(context)
 8 |     page = @title.gsub(/\s/, '+')
 9 |     "<a href='http://ncatlab.org/nlab/show/#{page}'>#{@title}</a>"
10 |   end
11 | end
12 | 
13 | Liquid::Template.register_tag('nlab', NLabTag)
14 | 


--------------------------------------------------------------------------------
/_plugins/node_info.rb:
--------------------------------------------------------------------------------
 1 | # frozen_string_literal: true
 2 | require 'cgi'
 3 | 
 4 | class NodeInfo
 5 |   def initialize(resource)
 6 |     @node = resource
 7 |   end
 8 | 
 9 |   def title
10 |     @node["title"]
11 |   end
12 | 
13 |   def slug
14 |     @node["slug"]
15 |   end
16 | 
17 |   # This is a very strange routine, but at least the logic is factored out here.
18 |   def has_title?
19 |     title != slug.downcase
20 |   end
21 | 
22 |   def taxon
23 |     @node["taxon"]
24 |   end
25 | 
26 |   def numbering
27 |     ToNumberingFilter.to_numbering @node["clicks"]
28 |   end
29 | 
30 |   def genus
31 |     @node["genus"] || "Result"
32 |   end
33 | 
34 |   def display_numbering
35 |     @node["genus"].nil? ? "<span class='numbering'>#{numbering}</span>" : "<span class='numbering'>#{genus} #{numbering}</span>"
36 |   end
37 | 
38 |   def has_numbering?
39 |     @node["depth"] != 0 && !taxon.nil?
40 |   end
41 | 
42 |   def display_title
43 |     has_numbering? ? (display_numbering + (has_title? ? ". #{title}" : "")) : title
44 |   end
45 | 
46 |   def display_title_parenthetical
47 |     has_numbering? ? (display_numbering + (has_title? ? " (#{title})" : "")) : title
48 |   end
49 | 
50 |   def aria_label
51 |     label = has_numbering? ? ("#{genus} #{numbering}." + (has_title? ? " #{title}" : "")) : title
52 |     CGI.escapeHTML label
53 |   end
54 | end
55 | 


--------------------------------------------------------------------------------
/_plugins/patch-antex.rb:
--------------------------------------------------------------------------------
 1 | # frozen_string_literal: true
 2 | 
 3 | module Antex
 4 |   class Job
 5 |     include LiquidHelpers
 6 | 
 7 |     def prepare_code
 8 |       @code = liquid_render @options['template'],
 9 |                             'preamble' => @options['preamble'],
10 |                             'append' => @options['append'],
11 |                             'prepend' => @options['prepend'],
12 |                             'macrolib' => @options['macrolib'],
13 |                             'snippet' => @snippet
14 |     end
15 |   end
16 | end
17 | 


--------------------------------------------------------------------------------
/_plugins/qed_tag.rb:
--------------------------------------------------------------------------------
 1 | # frozen_string_literal: true
 2 | 
 3 | # A tag to generate a [XXXXX] reference link to a node
 4 | class QedTag < Liquid::Tag
 5 |   def render(context)
 6 |     "<span style='float:right'>∎</span>"
 7 |   end
 8 | end
 9 | 
10 | Liquid::Template.register_tag('qed', QedTag)
11 | 
12 | 


--------------------------------------------------------------------------------
/_plugins/ref_tag.rb:
--------------------------------------------------------------------------------
 1 | # frozen_string_literal: true
 2 | #
 3 | class RefTag < Liquid::Tag
 4 |   def initialize(tag_name, slug, tokens)
 5 |     super
 6 |     @slug = slug.strip
 7 |   end
 8 | 
 9 |   def render_node(node, info, url)
10 |     "<a href='#{url}' role='tooltip' aria-label='#{info.aria_label}' class='slug'>[#{@slug}]</a>"
11 |   end
12 | 
13 |   def render(context)
14 |     registers = context.registers
15 |     site = registers[:site]
16 |     node = site.collections['nodes'].docs.detect { |doc| doc.data['slug'] == @slug }
17 |     info = NodeInfo.new node.data
18 |     url = "#{site.baseurl}#{node.url}"
19 |     render_node(node, info, url)
20 |   end
21 | end
22 | 
23 | class CleverRefTag < RefTag
24 |   def render_node(node, info, url)
25 |     "<a href='#{url}' role='tooltip' aria-label='#{info.aria_label}' class='cref'>#{info.display_numbering} <span class='slug'>[#{@slug}]</span></a>"
26 |   end
27 | end
28 | 
29 | class ParenCleverRefTag < CleverRefTag
30 |   def render_node(node, info, url)
31 |     "(#{super})"
32 |   end
33 | end
34 | 
35 | 
36 | Liquid::Template.register_tag('ref', RefTag)
37 | Liquid::Template.register_tag('cref', CleverRefTag)
38 | Liquid::Template.register_tag('pref', ParenCleverRefTag)
39 | 


--------------------------------------------------------------------------------
/_plugins/to_numbering_filter.rb:
--------------------------------------------------------------------------------
 1 | module ToNumberingFilter
 2 |   extend self
 3 | 
 4 |   def to_numbering(clicks)
 5 |     label = ""
 6 |     # NOTE: we skip the first one because it's the lesson
 7 |     clicks[1..].each do |tick|
 8 |       label << case tick["clicker"]
 9 |       when "section" then ".#{tick["value"] + 1}"
10 |       when "result" then "·#{(tick["value"] + 97).chr}"
11 |       else ".#{tick["value"]}"
12 |       end
13 |     end
14 |     label[1..]
15 |   end
16 | end
17 | 
18 | Liquid::Template.register_filter(ToNumberingFilter)
19 | 


--------------------------------------------------------------------------------
/assets/macros/jon-tikz.sty:
--------------------------------------------------------------------------------
  1 | \RequirePackage{xparse}
  2 | \RequirePackage{tikz}
  3 | \RequirePackage{zx-calculus}
  4 | \RequirePackage{tikz-cd}
  5 | \usetikzlibrary{matrix,arrows}
  6 | \usetikzlibrary{fit,positioning,calc,shapes}
  7 | \usetikzlibrary{decorations.pathreplacing}
  8 | \usetikzlibrary{decorations.pathmorphing}
  9 | \usetikzlibrary{decorations.markings}
 10 | 
 11 | 
 12 | \renewcommand{\zxDefaultRowSep}{5mm}
 13 | \renewcommand{\zxDefaultColumnSep}{2mm}
 14 | 
 15 | \tikzset{
 16 |   /zx/styles/rounded style/.append style={
 17 |     zxShort/.append style={
 18 |       font={\fontsize{8}{10}\selectfont},
 19 |     },
 20 |     stylePhaseInLabel/.append style={
 21 |       font={\fontsize{8}{10}\selectfont},
 22 |     },
 23 |   },
 24 |   close/.style = {
 25 |     outer sep = -1.7pt
 26 |   }
 27 | }
 28 | 
 29 | 
 30 | 
 31 | \tikzset{
 32 |   desc/.style={sloped, fill=white,inner sep=2pt},
 33 |   upright desc/.style={fill=white,inner sep=2pt},
 34 |   pullback/.style = {
 35 |     append after command={
 36 |       \pgfextra{
 37 |         \draw ($(\tikzlastnode) + (.2cm,-.5cm)$) -- ++(0.3cm,0) -- ++(0,0.3cm);
 38 |       }
 39 |     }
 40 |   },
 41 |   pullback 45/.style = {
 42 |     append after command={
 43 |       \pgfextra{
 44 |         \draw[rotate = 45] ($(\tikzlastnode) + (.2cm,-.5cm)$) -- ++(0.3cm,0) -- ++(0,0.3cm);
 45 |       }
 46 |     }
 47 |   },
 48 |   ne pullback/.style = {
 49 |     append after command={
 50 |       \pgfextra{
 51 |         \draw ($(\tikzlastnode) + (-.2cm,-.5cm)$) -- ++(-0.3cm,0) -- ++(0,0.3cm);
 52 |       }
 53 |     }
 54 |   },
 55 |   sw pullback/.style = {
 56 |     append after command={
 57 |       \pgfextra{
 58 |         \draw ($(\tikzlastnode) + (.2cm,.5cm)$) -- ++(0.3cm,0) -- ++(0,-0.3cm);
 59 |       }
 60 |     }
 61 |   },
 62 |   dotted pullback/.style = {
 63 |     append after command={
 64 |       \pgfextra{
 65 |         \draw [densely dotted] ($(\tikzlastnode) + (.2cm,-.5cm)$) -- ++(0.3cm,0) -- ++(0,0.3cm);
 66 |       }
 67 |     }
 68 |   },
 69 |   muted pullback/.style = {
 70 |     append after command={
 71 |       \pgfextra{
 72 |         \draw [gray] ($(\tikzlastnode) + (.2cm,-.5cm)$) -- ++(0.3cm,0) -- ++(0,0.3cm);
 73 |       }
 74 |     }
 75 |   },
 76 |   pushout/.style = {
 77 |     append after command={
 78 |       \pgfextra{
 79 |         \draw ($(\tikzlastnode) + (-.2cm,.5cm)$) -- ++(-0.3cm,0) -- ++(0,-0.3cm);
 80 |       }
 81 |     }
 82 |   },
 83 |   between/.style args={#1 and #2}{
 84 |     at = ($(#1)!0.5!(#2)$)
 85 |   },
 86 |   diagram/.style = {
 87 |     on grid,
 88 |     node distance=2cm,
 89 |     commutative diagrams/every diagram,
 90 |     line width = .5pt,
 91 |     every node/.append style = {
 92 |       commutative diagrams/every cell,
 93 |     }
 94 |   },
 95 |   fibration/.style = {
 96 |     -{Triangle[open]}
 97 |   },
 98 |   etale/.style = {
 99 |     -{Triangle[open]}
100 |   },
101 |   etale cover/.style= {
102 |     >={Triangle[open]},->.>
103 |   },
104 |   opfibration/.style = {
105 |     -{Triangle}
106 |   },
107 |   lies over/.style = {
108 |     |-{Triangle[open]}
109 |   },
110 |   op lies over/.style = {
111 |     |-{Triangle}
112 |   },
113 |   embedding/.style = {
114 |     {right hook}->
115 |   },
116 |   open immersion/.style = {
117 |     {right hook}-{Triangle[open]}
118 |   },
119 |   closed immersion/.style = {
120 |     {right hook}-{Triangle}
121 |   },
122 |   closed immersion*/.style = {
123 |     {left hook}-{Triangle}
124 |   },
125 |   embedding*/.style = {
126 |     {left hook}->
127 |   },
128 |   open immersion*/.style = {
129 |     {left hook}-{Triangle[open]}
130 |   },
131 |   exists/.style = {
132 |     densely dashed
133 |   },
134 | }
135 | 
136 | \newlength{\dontworryaboutit}
137 | 
138 | \tikzset{
139 |   inline diagram/.style = {
140 |     commutative diagrams/every diagram,
141 |     commutative diagrams/cramped,
142 |     line width = .5pt,
143 |     every node/.append style = {
144 |       commutative diagrams/every cell,
145 |       anchor = base,
146 |       inner sep = 0pt
147 |     },
148 |     every path/.append style = {
149 |       outer xsep = 2pt
150 |     }
151 |   }
152 | }
153 | 
154 | \ExplSyntaxOn
155 | 
156 | \cs_new:Npn \jms_tikz_morphism:nnn #1 #2 #3 {
157 |   \tikz[inline~diagram, baseline = (Y.base)]{
158 |     \node (X) {$#2$};
159 |     \node (Y) [right = 3ex~of~X] {$#3$};
160 |     \path[#1] (X) edge (Y);
161 |   }
162 | }
163 | 
164 | \ExplSyntaxOff
165 | 
166 | 
167 | 
168 | 
169 | \tikzset{
170 |   square/nw/.style = {},
171 |   square/ne/.style = {},
172 |   square/se/.style = {},
173 |   square/sw/.style = {},
174 |   square/north/.style = {->},
175 |   square/south/.style = {->},
176 |   square/west/.style = {->},
177 |   square/east/.style = {->},
178 |   square/north/node/.style = {above},
179 |   square/south/node/.style = {below},
180 |   square/west/node/.style = {left},
181 |   square/east/node/.style = {right},
182 | }
183 | 
184 | \ExplSyntaxOn
185 | 
186 | \bool_new:N \l_jon_glue_west
187 | 
188 | \keys_define:nn { jon-tikz/diagram } {
189 |   nw .tl_set:N = \l_jon_tikz_diagram_nw,
190 |   sw .tl_set:N = \l_jon_tikz_diagram_sw,
191 |   ne .tl_set:N = \l_jon_tikz_diagram_ne,
192 |   se .tl_set:N = \l_jon_tikz_diagram_se,
193 | 
194 |   width .tl_set:N = \l_jon_tikz_diagram_width,
195 |   height .tl_set:N = \l_jon_tikz_diagram_height,
196 | 
197 |   north .tl_set:N = \l_jon_tikz_diagram_north,
198 |   south .tl_set:N = \l_jon_tikz_diagram_south,
199 |   west .tl_set:N = \l_jon_tikz_diagram_west,
200 |   east .tl_set:N = \l_jon_tikz_diagram_east,
201 | 
202 |   nw/style .code:n = {\tikzset{square/nw/.style = {#1}}},
203 |   sw/style .code:n = {\tikzset{square/sw/.style = {#1}}},
204 |   ne/style .code:n = {\tikzset{square/ne/.style = {#1}}},
205 |   se/style .code:n = {\tikzset{square/se/.style = {#1}}},
206 | 
207 |   glue .choice:,
208 |   glue / west .code:n = {\bool_set:Nn \l_jon_glue_west \c_true_bool},
209 | 
210 |   glue~target .tl_set:N = \l_jon_tikz_glue_target,
211 | 
212 |   north/style .code:n = {\tikzset{square/north/.style = {#1}}},
213 |   north/node/style .code:n = {\tikzset{square/north/node/.style = {#1}}},
214 |   south/style .code:n = {\tikzset{square/south/.style = {#1}}},
215 |   south/node/style .code:n = {\tikzset{square/south/node/.style = {#1}}},
216 |   west/style .code:n = {\tikzset{square/west/.style = {#1}}},
217 |   west/node/style .code:n = {\tikzset{square/west/node/.style = {#1}}},
218 |   east/style .code:n = {\tikzset{square/east/.style = {#1}}},
219 |   east/node/style .code:n = {\tikzset{square/east/node/.style = {#1}}},
220 | 
221 |   draft .meta:n = {
222 |     nw = {\__jon_tikz_diagram_fmt_placeholder:n {nw}},
223 |     sw = {\__jon_tikz_diagram_fmt_placeholder:n {sw}},
224 |     se = {\__jon_tikz_diagram_fmt_placeholder:n {se}},
225 |     ne = {\__jon_tikz_diagram_fmt_placeholder:n {ne}},
226 |     north = {\__jon_tikz_diagram_fmt_placeholder:n {north}},
227 |     south = {\__jon_tikz_diagram_fmt_placeholder:n {south}},
228 |     west = {\__jon_tikz_diagram_fmt_placeholder:n {west}},
229 |     east = {\__jon_tikz_diagram_fmt_placeholder:n {east}},
230 |   }
231 | }
232 | 
233 | \tl_set:Nn \l_jon_tikz_diagram_width { 2cm }
234 | \tl_set:Nn \l_jon_tikz_diagram_height { 2cm }
235 | 
236 | 
237 | \cs_new:Npn \__jon_tikz_diagram_fmt_placeholder:n #1 {
238 |   \texttt{\textcolor{red}{#1}}
239 | }
240 | 
241 | \keys_set:nn { jon-tikz/diagram } {
242 |   glue~target = {},
243 | }
244 | 
245 | 
246 | \cs_new:Nn \__jon_tikz_render_square:nn {
247 |   \group_begin:
248 |   \keys_set:nn {jon-tikz/diagram} {#2}
249 |   \bool_if:nTF \l_jon_glue_west {
250 |     \node (#1ne) [right = \l_jon_tikz_diagram_width~of~\l_jon_tikz_glue_target ne,square/ne] {$\l_jon_tikz_diagram_ne$};
251 |     \node (#1se) [below = \l_jon_tikz_diagram_height~of~#1ne,square/se] {$\l_jon_tikz_diagram_se$};
252 |     \draw[square/north] (\l_jon_tikz_glue_target ne) to node [square/north/node] {$\l_jon_tikz_diagram_north$} (#1ne);
253 |     \draw[square/east] (#1ne) to node [square/east/node] {$\l_jon_tikz_diagram_east$} (#1se);
254 |     \draw[square/south] (\l_jon_tikz_glue_target se) to node [square/south/node] {$\l_jon_tikz_diagram_south$} (#1se);
255 |   } {
256 |     \node (#1nw) [square/nw] {$\l_jon_tikz_diagram_nw$};
257 |     \node (#1sw) [below = \l_jon_tikz_diagram_height~of~#1nw,square/sw] {$\l_jon_tikz_diagram_sw$};
258 |     \draw[square/west] (#1nw) to node [square/west/node] {$\l_jon_tikz_diagram_west$} (#1sw);
259 | 
260 |     \node (#1ne) [right = \l_jon_tikz_diagram_width~of~#1nw,square/ne] {$\l_jon_tikz_diagram_ne$};
261 |     \node (#1se) [below = \l_jon_tikz_diagram_height~of~#1ne,square/se] {$\l_jon_tikz_diagram_se$};
262 |     \draw[square/north] (#1nw) to node [square/north/node] {$\l_jon_tikz_diagram_north$} (#1ne);
263 |     \draw[square/east] (#1ne) to node [square/east/node] {$\l_jon_tikz_diagram_east$} (#1se);
264 |     \draw[square/south] (#1sw) to node [square/south/node] {$\l_jon_tikz_diagram_south$} (#1se);
265 |   }
266 |   \group_end:
267 | }
268 | 
269 | \NewDocumentCommand\SpliceDiagramSquare{D<>{}m}{
270 |   \__jon_tikz_render_square:nn {#1} {#2}
271 | }
272 | 
273 | 
274 | \NewDocumentCommand\DiagramSquare{D<>{}O{}m}{
275 |   \begin{tikzpicture}[diagram,#2,baseline=(#1sw.base)]
276 |     \__jon_tikz_render_square:nn {#1} {#3}
277 |   \end{tikzpicture}
278 | }
279 | 
280 | 
281 | \ExplSyntaxOff
282 | 


--------------------------------------------------------------------------------
/assets/macros/topos.katex.json:
--------------------------------------------------------------------------------
1 | {"\\AlgGU":"\\mathbf{GU}_{\\textit{alg}}","\\BoldSymbol":"\\pmb{#1}","\\CandHom":"\\mathbf{H}_{#1}(#2,#3)","\\CartGU":"\\mathbf{GU}_{\\textit{cart}}","\\CoSl":"#2^{\\setminus #1}","\\Cod":"\\mathsf{cod}_{#1}","\\Con":"\\mathsf{#1}","\\DirImg":"#1_{*}","\\FAM":"\\pmb{\\mathcal{F}}_{#1}","\\FullSubfib":"\\mathbf{Full}(#1)","\\GL":"\\pmb{\\mathcal{G}}_{#1}","\\Gl":"\\mathsf{gl}_{#1}","\\Idn":"1_{#1}","\\InvImg":"#1^{*}","\\LogGU":"\\mathbf{GU}_{\\textit{log}}","\\ObjTerm":"\\mathbf{1}_{#1}","\\OpCat":"#1^{\\mathsf{o}}","\\OpGL":"\\overline{\\pmb{\\mathcal{G}}}_{#1}","\\OpGl":"\\overline{\\mathsf{gl}}_{#1}","\\PCartGU":"\\mathbf{GU}_{\\textit{pcart}}","\\SET":"\\mathbf{Set}","\\SelfIx":"\\underline{#1}","\\Sh":"\\pmb{\\mathcal{S}}_{#1}","\\Sl":"#1_{/#2}","\\Sub":"_{#1}","\\Sup":"^{#1}","\\TOP":"\\mathbf{Top}_{#1}","\\TotCat":"\\widetilde{#1}","\\TotOpCat":"#1^{\\tilde{\\mathsf{o}}}","\\brc":"\\{ #1 \\}","\\brk":"[#1]","\\gl":"\\langle #1 \\rangle","\\prn":"(#1)","\\tick":"'"}


--------------------------------------------------------------------------------
/assets/macros/topos.mathjax.json:
--------------------------------------------------------------------------------
1 | {"AlgGU":["\\mathbf{GU}_{\\textit{alg}}",0],"BoldSymbol":["\\boldsymbol{#1}",1],"CandHom":["\\mathbf{H}_{#1}(#2,#3)",3],"CartGU":["\\mathbf{GU}_{\\textit{cart}}",0],"CoSl":["#2^{\\setminus #1}",2],"Cod":["\\mathsf{cod}_{#1}",1],"Con":["\\mathsf{#1}",1],"DirImg":["#1_{*}",1],"FAM":["\\boldsymbol{\\mathcal{F}}_{#1}",1],"FullSubfib":["\\mathbf{Full}(#1)",1],"GL":["\\boldsymbol{\\mathcal{G}}_{#1}",1],"Gl":["\\mathsf{gl}_{#1}",1],"Idn":["1_{#1}",1],"InvImg":["#1^{*}",1],"LogGU":["\\mathbf{GU}_{\\textit{log}}",0],"ObjTerm":["\\mathbf{1}_{#1}",1],"OpCat":["#1^{\\mathsf{o}}",1],"OpGL":["\\overline{\\boldsymbol{\\mathcal{G}}}_{#1}",1],"OpGl":["\\overline{\\mathsf{gl}}_{#1}",1],"PCartGU":["\\mathbf{GU}_{\\textit{pcart}}",0],"SET":["\\mathbf{Set}",0],"SelfIx":["\\underline{#1}",1],"Sh":["\\boldsymbol{\\mathcal{S}}_{#1}",1],"Sl":["#1_{/#2}",2],"Sub":["_{#1}",1],"Sup":["^{#1}",1],"TOP":["\\mathbf{Top}_{#1}",1],"TotCat":["\\widetilde{#1}",1],"TotOpCat":["#1^{\\tilde{\\mathsf{o}}}",1],"brc":["\\{ #1 \\}",1],"brk":["[#1]",1],"gl":["\\langle #1 \\rangle",1],"prn":["(#1)",1],"tick":["'",0]}


--------------------------------------------------------------------------------
/assets/macros/topos.sty:
--------------------------------------------------------------------------------
 1 | \newcommand\AlgGU{\mathbf{GU}_{\textit{alg}}}
 2 | \newcommand\BoldSymbol[1]{\boldsymbol{#1}}
 3 | \newcommand\brc[1]{\{ #1 \}}
 4 | \newcommand\brk[1]{[#1]}
 5 | \newcommand\CandHom[3]{\mathbf{H}_{#1}(#2,#3)}
 6 | \newcommand\CartGU{\mathbf{GU}_{\textit{cart}}}
 7 | \newcommand\Cod[1]{\mathsf{cod}_{#1}}
 8 | \newcommand\Con[1]{\mathsf{#1}}
 9 | \newcommand\CoSl[2]{#2^{\setminus #1}}
10 | \newcommand\DirImg[1]{#1_{*}}
11 | \newcommand\FAM[1]{\boldsymbol{\mathcal{F}}_{#1}}
12 | \newcommand\FullSubfib[1]{\mathbf{Full}(#1)}
13 | \newcommand\Gl[1]{\mathsf{gl}_{#1}}
14 | \newcommand\gl[1]{\langle #1 \rangle}
15 | \newcommand\GL[1]{\boldsymbol{\mathcal{G}}_{#1}}
16 | \newcommand\Idn[1]{1_{#1}}
17 | \newcommand\InvImg[1]{#1^{*}}
18 | \newcommand\LogGU{\mathbf{GU}_{\textit{log}}}
19 | \newcommand\ObjTerm[1]{\mathbf{1}_{#1}}
20 | \newcommand\OpCat[1]{#1^{\mathsf{o}}}
21 | \newcommand\OpGL[1]{\overline{\boldsymbol{\mathcal{G}}}_{#1}}
22 | \newcommand\OpGl[1]{\overline{\mathsf{gl}}_{#1}}
23 | \newcommand\PCartGU{\mathbf{GU}_{\textit{pcart}}}
24 | \newcommand\prn[1]{(#1)}
25 | \newcommand\SelfIx[1]{\underline{#1}}
26 | \newcommand\SET{\mathbf{Set}}
27 | \newcommand\Sh[1]{\boldsymbol{\mathcal{S}}_{#1}}
28 | \newcommand\Sl[2]{#1_{/#2}}
29 | \newcommand\Sub[1]{_{#1}}
30 | \newcommand\Sup[1]{^{#1}}
31 | \newcommand\tick{'}
32 | \newcommand\TOP[1]{\mathbf{Top}_{#1}}
33 | \newcommand\TotCat[1]{\widetilde{#1}}
34 | \newcommand\TotOpCat[1]{#1^{\tilde{\mathsf{o}}}}
35 | 


--------------------------------------------------------------------------------
/assets/search.js:
--------------------------------------------------------------------------------
  1 | class Search {
  2 |   constructor() {
  3 |     this.NODE_TAG = "SECTION"; // MUST be uppercase to be both selector and nodeName
  4 |     this.FOLDED_CLASS = "ellipsis";
  5 |     this.MARKJS_EXCLUSIONS = [
  6 |       `.${this.FOLDED_CLASS} *`, // folded nodes
  7 |       "nav *", // toc, next/prev, etc.
  8 |     ];
  9 |     this.INPUT_NAME = "q";
 10 |     this.URL_PARAM = "q";
 11 |     this.DOC_REF_KEY = "slug";
 12 |     this.DOC_VAL_KEY = "text";
 13 | 
 14 |     this.form = document.getElementById("search-form");
 15 |     this.fieldset = this.form.querySelector("fieldset");
 16 |     this.input = document.getElementById("search-query");
 17 |     this.toggle = document.getElementById("search-toggle");
 18 |     this.bindEvents();
 19 | 
 20 |     this.tocItems = {};
 21 |     this.nodeDocuments = [];
 22 |     this.nodeElements = {};
 23 |     this.nodeVisibilities = {};
 24 |     this.initialized = false;
 25 | 
 26 |     this.checkURL();
 27 |   }
 28 | 
 29 |   //=[ Initialization ]=========================================================
 30 | 
 31 |   async lazyInit() {
 32 |     if (this.initialized) return;
 33 |     this.initBookkeeping();
 34 |     this.initIndex(this.nodeDocuments);
 35 |     this.initialized = true;
 36 |   }
 37 | 
 38 |   initBookkeeping() {
 39 |     document.querySelectorAll(this.NODE_TAG).forEach((element) => {
 40 |       this.tocItems[element.id] = document
 41 |         .querySelector(".toc")
 42 |         .querySelector(`a[href="#${element.id}"]`)
 43 |         ?.closest("li");
 44 |       this.nodeElements[element.id] = element;
 45 |       this.nodeVisibilities[element.id] = true;
 46 |       this.nodeDocuments.push({
 47 |         [this.DOC_REF_KEY]: element.id,
 48 |         [this.DOC_VAL_KEY]: this.extractSearchableText(element),
 49 |       });
 50 |     });
 51 |   }
 52 | 
 53 |   extractSearchableText(element) {
 54 |     return [...element.children]
 55 |       .filter((e) => e.nodeName != this.NODE_TAG)
 56 |       .map((e) => e.innerText)
 57 |       .join("\n");
 58 |   }
 59 | 
 60 |   //=[ URI ]====================================================================
 61 | 
 62 |   checkURL() {
 63 |     const q = this.getURLQuery();
 64 |     if (q) {
 65 |       this.handleToggle();
 66 |       this.input.value = q;
 67 |       this.run(q);
 68 |     }
 69 |   }
 70 | 
 71 |   getURLQuery() {
 72 |     var url = new URL(window.location.href);
 73 |     return url.searchParams.get(this.URL_PARAM);
 74 |   }
 75 | 
 76 |   setURLQuery(q) {
 77 |     if (!history.replaceState) return;
 78 |     var url = new URL(window.location.href);
 79 |     url.searchParams.set(this.URL_PARAM, q);
 80 |     window.history.replaceState(null, null, url);
 81 |   }
 82 | 
 83 |   deleteURLQuery() {
 84 |     if (!history.replaceState) return;
 85 |     var url = new URL(window.location.href);
 86 |     url.searchParams.delete(this.URL_PARAM);
 87 |     window.history.replaceState(null, null, url);
 88 |   }
 89 | 
 90 |   //=[ Operation ]==============================================================
 91 | 
 92 |   async run(q) {
 93 |     this.lazyInit();
 94 | 
 95 |     await this.setFormLock(true);
 96 | 
 97 |     const results = this.searchIndex(q);
 98 | 
 99 |     this.setVisibilities(false);
100 |     this.setVisibilitiesFromResults(results);
101 |     this.applyVisibilities();
102 | 
103 |     await this.promiseToUnmarkTree();
104 |     await this.promiseToMarkTree(results);
105 | 
106 |     await this.setFormLock(false);
107 | 
108 |     this.setURLQuery(q);
109 |   }
110 | 
111 |   async clear() {
112 |     await this.setFormLock(true);
113 | 
114 |     await this.promiseToUnmarkTree();
115 | 
116 |     this.setVisibilities(true);
117 |     this.applyVisibilities();
118 | 
119 |     await this.setFormLock(false);
120 | 
121 |     this.deleteURLQuery();
122 |   }
123 | 
124 |   //=[ Indexing ]===============================================================
125 | 
126 |   initIndex(documents) {
127 |     const that = this; // sigh.
128 |     this.index = lunr(function () {
129 |       this.ref(that.DOC_REF_KEY);
130 |       this.field(that.DOC_VAL_KEY);
131 |       documents.forEach(this.add.bind(this));
132 |     });
133 |   }
134 | 
135 |   searchIndex(query) {
136 |     return this.index.search(query);
137 |   }
138 | 
139 |   //=[ (Un)folding ]============================================================
140 | 
141 |   setVisibilities(status) {
142 |     Object.keys(this.nodeVisibilities).forEach(
143 |       this.setVisibility.bind(this, status)
144 |     );
145 |   }
146 | 
147 |   setVisibility(status, slug) {
148 |     this.nodeVisibilities[slug] = status;
149 |   }
150 | 
151 |   showSectionAndAncestors(nodeEl) {
152 |     if (nodeEl.nodeName != this.NODE_TAG) return;
153 |     if (this.nodeVisibilities[nodeEl.id]) return;
154 |     this.nodeVisibilities[nodeEl.id] = true;
155 |     this.showSectionAndAncestors(nodeEl.parentNode);
156 |   }
157 | 
158 |   setVisibilitiesFromResults(results) {
159 |     results.map(({ ref }) =>
160 |       this.showSectionAndAncestors(this.nodeElements[ref])
161 |     );
162 |   }
163 | 
164 |   applyVisibilities() {
165 |     Object.keys(this.nodeVisibilities).forEach(this.applyVisibility.bind(this));
166 |   }
167 | 
168 |   applyVisibility(slug) {
169 |     if (this.nodeVisibilities[slug]) {
170 |       this.tocItems[slug]?.classList?.remove(this.FOLDED_CLASS);
171 |       this.nodeElements[slug].classList.remove(this.FOLDED_CLASS);
172 |     } else {
173 |       this.tocItems[slug]?.classList?.add(this.FOLDED_CLASS);
174 |       this.nodeElements[slug].classList.add(this.FOLDED_CLASS);
175 |     }
176 |   }
177 | 
178 |   //=[ Highlighting ]===========================================================
179 | 
180 |   promiseToMarkNode(slug, terms) {
181 |     return new Promise((resolve, reject) => {
182 |       new Mark(this.nodeElements[slug]).mark(terms, {
183 |         exclude: this.MARKJS_EXCLUSIONS,
184 |         accuracy: "complementary",
185 |         done: resolve,
186 |       });
187 |     });
188 |   }
189 | 
190 |   promiseToMarkTree(results) {
191 |     const promisesToMarkNodes = results.map(
192 |       ({ ref, matchData: { metadata } }) => {
193 |         this.promiseToMarkNode(ref, Object.keys(metadata));
194 |       }
195 |     );
196 |     return Promise.allSettled(promisesToMarkNodes);
197 |   }
198 | 
199 |   promiseToUnmarkTree() {
200 |     return new Promise((resolve, reject) => {
201 |       new Mark(document).unmark({ done: resolve });
202 |     });
203 |   }
204 | 
205 |   //=[ (Un)locking ]============================================================
206 | 
207 |   async setFormLock(locked) {
208 |     this.fieldset.disabled = locked;
209 |     // we need to give slow machines some time to redraw
210 |     await this.sleep(1);
211 |   }
212 | 
213 |   sleep(ms) {
214 |     return new Promise((resolve) => setTimeout(resolve, ms));
215 |   }
216 | 
217 |   //=[ Events ]=================================================================
218 | 
219 |   async handleReset(event) {
220 |     await this.clear();
221 |   }
222 | 
223 |   async handleSubmit(event) {
224 |     event.preventDefault();
225 |     var formData = new FormData(event.target);
226 |     var query = formData.get(this.INPUT_NAME);
227 |     await this.run(query);
228 |   }
229 | 
230 |   handleClick(event) {
231 |     if (!this.initialized) return;
232 |     const section = event.target.closest(this.NODE_TAG);
233 |     if (!section) return;
234 |     const slug = section.id;
235 |     if (this.nodeVisibilities[slug]) return;
236 |     this.setVisibility(true, slug);
237 |     this.applyVisibility(slug);
238 |   }
239 | 
240 |   handleToggle() {
241 |     this.form.hidden = !this.form.hidden;
242 |     if (!this.form.hidden) this.input.focus();
243 |   }
244 | 
245 |   bindEvents() {
246 |     this.toggle.addEventListener("click", this.handleToggle.bind(this), false);
247 |     document.addEventListener("click", this.handleClick.bind(this), false);
248 |     this.form.addEventListener("submit", this.handleSubmit.bind(this), false);
249 |     this.form.addEventListener("reset", this.handleReset.bind(this), false);
250 |   }
251 | }
252 | 
253 | document.addEventListener("DOMContentLoaded", function () {
254 |   window.search = new Search();
255 | });
256 | 


--------------------------------------------------------------------------------
/assets/simple.css:
--------------------------------------------------------------------------------
  1 | @import url('https://fonts.googleapis.com/css2?family=Inria+Sans:ital,wght@0,300;0,400;0,700;1,300;1,400;1,700&display=swap');
  2 | 
  3 | /* Global variables. */
  4 | :root {
  5 |   /* Set sans-serif & mono fonts */
  6 |   --sans-font: "Inria Sans", -apple-system, BlinkMacSystemFont, "Avenir Next", Avenir,
  7 |     "Nimbus Sans L", Roboto, "Noto Sans", "Segoe UI", Arial, Helvetica,
  8 |     "Helvetica Neue", sans-serif;
  9 |   --mono-font: Consolas, Menlo, Monaco, "Andale Mono", "Ubuntu Mono", monospace;
 10 | 
 11 |   /* Default (light) theme */
 12 |   --bg: #fff;
 13 |   --accent-bg: #f5f7ff;
 14 |   --text: #212121;
 15 |   --text-light: #585858;
 16 |   --border: #d8dae1;
 17 |   --accent: #1756a9;
 18 |   --code: #d81b60;
 19 |   --preformatted: #444;
 20 |   --marked: #ffdd33;
 21 |   --disabled: #efefef;
 22 | }
 23 | 
 24 | /* Reset box-sizing */
 25 | *, *::before, *::after {
 26 |   box-sizing: border-box;
 27 | }
 28 | 
 29 | /* Reset default appearance */
 30 | 
 31 | html {
 32 |   /* Set the font globally */
 33 |   font-family: var(--sans-font);
 34 |   scroll-behavior: smooth;
 35 | }
 36 | 
 37 | /* Make the body a nice central block */
 38 | body {
 39 |   color: var(--text);
 40 |   background: var(--bg);
 41 |   font-size: 1rem;
 42 |   line-height: 1.5;
 43 |   display: grid;
 44 |   grid-template-columns: 1fr min(42rem, 90%) 1fr;
 45 |   margin: 0;
 46 | }
 47 | 
 48 | body > * {
 49 |   grid-column: 2;
 50 | }
 51 | 
 52 | body > p {
 53 |   margin-top: .25rem;
 54 |   margin-bottom: .25rem;
 55 | }
 56 | 
 57 | /* Add a little padding to ensure spacing is correct between content and header > nav */
 58 | main {
 59 |   padding-top: 1.5rem;
 60 | }
 61 | 
 62 | body > footer {
 63 |   margin-top: 4rem;
 64 |   padding: 2rem 1rem 1.5rem 1rem;
 65 |   color: var(--text-light);
 66 |   font-size: 0.9rem;
 67 |   text-align: center;
 68 | }
 69 | 
 70 | /* Format headers */
 71 | h1 {
 72 |   font-size: 3rem;
 73 |   margin-bottom: 1rem;
 74 | }
 75 | 
 76 | h2 {
 77 |   font-size: 2.6rem;
 78 |   margin-top: 3rem;
 79 | }
 80 | 
 81 | h3 {
 82 |   font-size: 2rem;
 83 |   margin-top: 3rem;
 84 | }
 85 | 
 86 | h4 {
 87 |   font-size: 1.44rem;
 88 | }
 89 | 
 90 | h5 {
 91 |   font-size: 1.15rem;
 92 | }
 93 | 
 94 | h6 {
 95 |   font-size: 0.96rem;
 96 | }
 97 | 
 98 | /* Prevent long strings from overflowing container */
 99 | p, h1, h2, h3, h4, h5, h6 {
100 |   overflow-wrap: break-word;
101 | }
102 | 
103 | /* Fix line height when title wraps */
104 | h1,
105 | h2,
106 | h3 {
107 |   line-height: 1.1;
108 | }
109 | 
110 | /* Reduce header size on mobile */
111 | @media only screen and (max-width: 720px) {
112 |   h1 {
113 |     font-size: 2.5rem;
114 |   }
115 | 
116 |   h2 {
117 |     font-size: 2.1rem;
118 |   }
119 | 
120 |   h3 {
121 |     font-size: 1.75rem;
122 |   }
123 | 
124 |   h4 {
125 |     font-size: 1.25rem;
126 |   }
127 | }
128 | 
129 | /* Format links & buttons */
130 | 
131 | a {
132 |   text-decoration: none;
133 |   color: var(--accent);
134 | }
135 | 
136 | a:hover {
137 |   text-decoration: underline;
138 | }
139 | 
140 | 
141 | 
142 | /* Set the cursor to '?' while hovering over an abbreviation */
143 | abbr {
144 |   cursor: help;
145 | }
146 | /* Format navigation */
147 | header > nav {
148 |   font-size: 1rem;
149 |   line-height: 1.25;
150 |   padding: 1rem 0 0 0;
151 |   display: flex;
152 |   justify-content: space-between;
153 | }
154 | 
155 | header > nav h2 {
156 |   margin: 0 0 0 0 ;
157 |   padding: 0.1rem 1rem;
158 |   font-family: var(--sans-font);
159 |   font-size: 1rem;
160 | }
161 | 
162 | /* Use flexbox to allow items to wrap, as needed */
163 | header > nav ul,
164 | header > nav ol {
165 |   align-items: center;
166 |   list-style-type: none;
167 |   display: flex;
168 |   margin: 0;
169 |   padding: 0;
170 | }
171 | 
172 | /* List items are inline elements, make them behave more like blocks */
173 | header > nav ul li,
174 | header > nav ol li {
175 |   padding-left: 1rem;
176 |   display: inline-block;
177 | }
178 | 
179 | body > header {
180 |   margin-bottom: .6rem;
181 |   padding-bottom: .6em;
182 |   border-bottom: 1px solid var(--border);
183 | }
184 | 
185 | .page-link {
186 |   color: black;
187 | }
188 | 
189 | aside {
190 |   font-size: 1rem;
191 |   width: 30%;
192 |   padding: 0 15px;
193 |   margin-left: 15px;
194 |   float: right;
195 | }
196 | 
197 | /* Make aside full-width on mobile */
198 | @media only screen and (max-width: 720px) {
199 |   aside {
200 |     width: 100%;
201 |     float: none;
202 |     margin-left: 0;
203 |   }
204 | }
205 | 
206 | article, fieldset {
207 |   margin-bottom: 1rem;
208 | }
209 | 
210 | article h2:first-child,
211 | section h2:first-child {
212 |   margin-top: 1rem;
213 | }
214 | 
215 | section {
216 |   margin-bottom: 1.5rem;
217 | }
218 | 
219 | summary {
220 |   cursor: pointer;
221 |   font-weight: bold;
222 |   padding: 0.6rem 1rem;
223 | }
224 | 
225 | 
226 | /* Format tables */
227 | table {
228 |   border-collapse: collapse;
229 |   display: block;
230 |   margin: 1.5rem 0;
231 |   overflow: auto;
232 |   width: 100%;
233 | }
234 | 
235 | td,
236 | th {
237 |   border: 1px solid var(--border);
238 |   text-align: left;
239 |   padding: 0.5rem;
240 | }
241 | 
242 | table caption {
243 |   font-weight: bold;
244 |   margin-bottom: 0.5rem;
245 | }
246 | /* Misc body elements */
247 | hr {
248 |   color: var(--border);
249 |   border-top: 1px;
250 |   margin: 1rem auto;
251 | }
252 | 
253 | mark {
254 |   padding: 2px 5px;
255 |   border-radius: 4px;
256 |   background: var(--marked);
257 | }
258 | 
259 | img,
260 | video {
261 |   max-width: 100%;
262 |   height: auto;
263 |   border-radius: 5px;
264 | }
265 | 
266 | figure {
267 |   margin: 0;
268 |   text-align: center;
269 | }
270 | 
271 | figcaption {
272 |   font-size: 0.9rem;
273 |   color: var(--text-light);
274 |   margin-bottom: 1rem;
275 | }
276 | 
277 | blockquote {
278 |   margin: 2rem 0 2rem 2rem;
279 |   padding: 0.4rem 0.8rem;
280 |   border-left: 0.35rem solid var(--accent);
281 |   color: var(--text-light);
282 |   font-style: italic;
283 | }
284 | 
285 | cite {
286 |   font-size: 0.9rem;
287 |   color: var(--text-light);
288 |   font-style: normal;
289 | }
290 | /* Use mono font for code elements */
291 | code,
292 | pre,
293 | pre span,
294 | kbd,
295 | samp {
296 |   font-family: var(--mono-font);
297 |   color: var(--code);
298 | }
299 | 
300 | kbd {
301 |   color: var(--preformatted);
302 |   border: 1px solid var(--preformatted);
303 |   border-bottom: 3px solid var(--preformatted);
304 |   border-radius: 5px;
305 |   padding: 0.1rem 0.4rem;
306 | }
307 | 
308 | pre {
309 |   padding: 1rem 1.4rem;
310 |   max-width: 100%;
311 |   overflow: auto;
312 |   color: var(--preformatted);
313 | }
314 | 
315 | /* Fix embedded code within pre */
316 | pre code {
317 |   color: var(--preformatted);
318 |   background: none;
319 |   margin: 0;
320 |   padding: 0;
321 | }
322 | 
323 | /* Progress bars */
324 | /* Declarations are repeated because you */
325 | /* cannot combine vendor-specific selectors */
326 | progress {
327 |   width: 100%;
328 | }
329 | 
330 | progress:indeterminate {
331 |   background-color: var(--accent-bg);
332 | }
333 | 
334 | progress::-webkit-progress-bar {
335 |   border-radius: 5px;
336 |   background-color: var(--accent-bg);
337 | }
338 | 
339 | progress::-webkit-progress-value {
340 |   border-radius: 5px;
341 |   background-color: var(--accent);
342 | }
343 | 
344 | progress::-moz-progress-bar {
345 |   border-radius: 5px;
346 |   background-color: var(--accent);
347 |   transition-property: width;
348 |   transition-duration: 0.3s;
349 | }
350 | 
351 | progress:indeterminate::-moz-progress-bar {
352 |   background-color: var(--accent-bg);
353 | }
354 | 


--------------------------------------------------------------------------------
/assets/style.css:
--------------------------------------------------------------------------------
  1 | @import "simple.css";
  2 | 
  3 | h1,
  4 | h2,
  5 | h3,
  6 | h4,
  7 | h5,
  8 | h6 {
  9 |   font-family: "Palatino", "Inria Sans" !important;
 10 |   font-weight: bold !important;
 11 | }
 12 | 
 13 | h1 {
 14 |   font-size: 2em !important;
 15 | }
 16 | 
 17 | h2 {
 18 |   font-size: 1.5em !important;
 19 |   margin-block-end: .4em;
 20 |   margin-top: 1.5em;
 21 | }
 22 | 
 23 | h3 {
 24 |   font-size: 1.2em !important;
 25 |   margin-block-end: .4em;
 26 |   margin-top: .3em;
 27 | }
 28 | 
 29 | h4 {
 30 |   font-size: 1.1em !important;
 31 |   margin-top: .3em;
 32 |   margin-block-end: .4em;
 33 | }
 34 | 
 35 | h5 {
 36 |   font-size: 1em !important;
 37 | }
 38 | 
 39 | h6 {
 40 |   font-size: 1em !important;
 41 |   font-style: italic;
 42 | }
 43 | 
 44 | .trans-note {
 45 |   color: gray;
 46 | }
 47 | 
 48 | 
 49 | .site-title {
 50 |   color: black;
 51 |   font-weight: bold;
 52 | }
 53 | 
 54 | .slug {
 55 |   font-family: "Inria Sans";
 56 |   font-weight: 200;
 57 |   color: gray !important;
 58 | }
 59 | 
 60 | ul.toc {
 61 |   list-style-type: none;
 62 |   padding: 0;
 63 |   margin-left: 10pt;
 64 | }
 65 | 
 66 | div.bibtex-code {
 67 |   margin-top: 1em;
 68 | }
 69 | 
 70 | header.post-header {
 71 |   margin-bottom: 1em !important;
 72 | }
 73 | 
 74 | header.inline {
 75 |   display: inline;
 76 |   margin-right: .5ex;
 77 | }
 78 | 
 79 | header.inline+p {
 80 |   display: inline;
 81 | }
 82 | 
 83 | header.inline+p+* {
 84 |   margin-top: 10px;
 85 | }
 86 | 
 87 | header.inline .numbering {
 88 |   font-weight: bold;
 89 | }
 90 | 
 91 | .katex {
 92 |   line-height: 1 !important;
 93 | }
 94 | 
 95 | .antex.display {
 96 |   display: flex;
 97 |   padding: 1em;
 98 | }
 99 | 
100 | .antex.display>svg,
101 | .antex.display>img {
102 |   margin: 0 auto !important;
103 | }
104 | 
105 | ul.toc {
106 |   margin-top: 0;
107 | }
108 | 


--------------------------------------------------------------------------------
/changelog.md:
--------------------------------------------------------------------------------
  1 | ---
  2 | layout: page
  3 | title: Recent changes
  4 | limit: 50
  5 | ---
  6 | 
  7 | <style>
  8 | 
  9 |   details > summary {
 10 |     margin-left: 30px;
 11 |     margin-bottom: 15px;
 12 |     padding-inline-start: 1ch;
 13 |   }
 14 | 
 15 |   details[open] > summary {
 16 |     display: none;
 17 |   }
 18 | 
 19 |   summary { list-style: none; }
 20 |   summary::-webkit-details-marker { display: none; }
 21 | 
 22 |   details summary::before {
 23 |     position: absolute;
 24 |     transform: translateX(-100%);
 25 |     padding-right: 1ch;
 26 |     content: '⧆';
 27 |     color: hsl(0, 0%, 40%);
 28 |   }
 29 | 
 30 |   summary {
 31 |     cursor: pointer;
 32 |     margin-left: 30px;
 33 |     padding-inline-start: 1ch;
 34 |   }
 35 | 
 36 |   li.diff-A, li.diff-M, li.diff-D {
 37 |     padding-inline-start: 1ch;
 38 |   }
 39 |   li.diff-A { list-style-type: '⊞'; }
 40 |   li.diff-M { list-style-type: '⊡'; } /* ⧇ */
 41 |   li.diff-D { list-style-type: '⊟'; }
 42 |   li.diff-A::marker { color: hsl(120, 100%, 40%); }
 43 |   li.diff-M::marker { color: hsl(240, 100%, 40%); }
 44 |   li.diff-D::marker { color: hsl(  0, 100%, 40%); }
 45 | 
 46 |   .author {
 47 |     float: right;
 48 |     font-style: italic;
 49 |   }
 50 | 
 51 | </style>
 52 | 
 53 | {% if page.commits.size > page.limit %}
 54 | These are the last {{ page.limit }} of the {{ page.commits.size }} commits involving nodes.
 55 | {% endif %}
 56 | 
 57 | <dl>
 58 | 
 59 | {% for commit in page.commits limit: page.limit %}
 60 | 
 61 | <dt>
 62 | 
 63 | <strong>{{ commit.timestamp | date_to_string }}</strong>
 64 | 
 65 | <a href="https://github.com/jonsterling/math/commit/{{ commit.hash }}">
 66 |   <code>{{ commit.hash | truncate: 7, "" }}</code>
 67 | </a>
 68 | 
 69 | {{ commit.subject }}
 70 | 
 71 | {% unless commit.author.hide -%}
 72 |   <span class="author">by
 73 |     {% if commit.author.link -%}
 74 |       <a href="{{ commit.author.link }}">{{ commit.author.name }}</a>
 75 |     {%- else -%}
 76 |       {{ commit.author.name }}
 77 |     {%- endif -%}
 78 |   </span>
 79 | {%- endunless %}
 80 | 
 81 | </dt>
 82 | 
 83 | <dd>
 84 | 
 85 | {% capture commit_list %}
 86 | 
 87 | <ul>
 88 |   {% for diff in commit.diffs %}
 89 |   <li class="diff-{{ diff.status }}">
 90 | 
 91 |     {% if diff.node %}
 92 |       <a href="{{diff.node.url | relative_url}}">
 93 |         {{diff.node | display_title}}
 94 |         {% if diff.node.collection == "nodes" %}
 95 |           <span class="slug">[{{diff.node.slug}}]</span>
 96 |         {% endif %}
 97 |       </a>
 98 |     {% else %}
 99 |       <del>{{ diff.path | split: "/" | last | remove: ".md" }}</del>
100 |     {% endif%}
101 | 
102 |   </li>
103 |   {% endfor %}
104 | </ul>
105 | 
106 | {% endcapture %}
107 | 
108 | <details {% if commit.diffs.size < 7 %}open{% endif %}>
109 |   <summary markdown='span'>Click to (un)fold the list of all {{ commit.diffs.size }} affected nodes.</summary>
110 |   {{ commit_list }}
111 | </details>
112 | 
113 | </dd>
114 | 
115 | {% endfor %}
116 | 
117 | </dl>
118 | 


--------------------------------------------------------------------------------
/deploy.sh:
--------------------------------------------------------------------------------
 1 | #!/bin/bash
 2 | 
 3 | bundle exec jekyll build
 4 | ./setup-worktree.sh
 5 | 
 6 | rm -rf _deploy/*
 7 | cp -rf _site/* _deploy/
 8 | 
 9 | cd _deploy
10 | git add .
11 | git commit -am "Deploy"
12 | git push -u origin gh-pages
13 | cd ..
14 | 
15 | 


--------------------------------------------------------------------------------
/index.md:
--------------------------------------------------------------------------------
1 | ---
2 | layout: default
3 | title: Lectures
4 | ---
5 | 
6 | {% for lecture in site.lectures %}
7 |   <a href="{{ lecture.url | relative_url }}">{{lecture.title}}</a>
8 | {% endfor %}
9 | 


--------------------------------------------------------------------------------
/macros/macro-kit.rkt:
--------------------------------------------------------------------------------
  1 | #lang at-exp racket
  2 | 
  3 | (require json)
  4 | 
  5 | (provide raw define-global define-local publish-macro-library target)
  6 | (struct macro-repr (name arity inst) #:transparent)
  7 | 
  8 | (define target (make-parameter null))
  9 | 
 10 | (define (macro-reprs->mathjax-jsexpr reprs)
 11 |   (parameterize ([target 'mathjax])
 12 |     (make-hash
 13 |      (for/list ([repr reprs])
 14 |        (cons
 15 |         (string->symbol (macro-repr-name repr))
 16 |         (list
 17 |          ((macro-repr-inst repr))
 18 |          (macro-repr-arity repr)))))))
 19 | 
 20 | (define (macro-reprs->katex-jsexpr reprs)
 21 |   (parameterize ([target 'katex])
 22 |     (make-hash
 23 |      (for/list ([repr reprs])
 24 |        (cons
 25 |         (string->symbol (string-append "\\" (macro-repr-name repr)))
 26 |         ((macro-repr-inst repr)))))))
 27 | 
 28 | (define (write-macro-repr-latex repr port)
 29 |   (parameterize ([target 'latex])
 30 |     (display "\\newcommand\\" port)
 31 |     (display (macro-repr-name repr) port)
 32 |     (when (> (macro-repr-arity repr) 0)
 33 |       (display "[" port)
 34 |       (display (macro-repr-arity repr) port)
 35 |       (display "]" port))
 36 |     (display "{" port)
 37 |     (display ((macro-repr-inst repr)) port)
 38 |     (displayln "}" port)))
 39 | 
 40 | (define macro-set (make-parameter (mutable-set)))
 41 | 
 42 | (define (macro-list)
 43 |   (define (lt? repr1 repr2)
 44 |     (string-ci<? (macro-repr-name repr1) (macro-repr-name repr2)))
 45 |   
 46 |   (sort (set->list (macro-set)) lt?))
 47 | 
 48 | (define (render-macros-to-tex filename)
 49 |   (call-with-output-file filename #:exists 'replace
 50 |     (λ (port)
 51 |       (for ([repr (in-list (macro-list))])
 52 |         (write-macro-repr-latex repr port)))))
 53 | 
 54 | (define (render-macros-to-mathjax filename)
 55 |   (define macros (macro-list))
 56 |   (call-with-output-file filename #:exists 'replace
 57 |     (λ (port)
 58 |       (write-json (macro-reprs->mathjax-jsexpr macros) port))))
 59 | 
 60 | (define (render-macros-to-katex filename)
 61 |   (define macros (macro-list))
 62 |   (call-with-output-file filename #:exists 'replace
 63 |     (λ (port)
 64 |       (write-json (macro-reprs->katex-jsexpr macros) port))))
 65 | 
 66 | (define (publish-macro-library sym)
 67 |   (define (filename ext)
 68 |     (string-append "../assets/macros/" (symbol->string sym) "." ext))
 69 |   (render-macros-to-tex (filename "sty"))
 70 |   (render-macros-to-mathjax (filename "mathjax.json"))
 71 |   (render-macros-to-katex (filename "katex.json")))
 72 | 
 73 | (define (index a b)
 74 |   (let [(tail (member a (reverse b)))]
 75 |     (and tail (length (cdr tail)))))
 76 |  
 77 | (define-syntax id->string
 78 |   (lambda (stx)
 79 |     (syntax-case stx ()
 80 |       ((_ id)
 81 |        (identifier? #'id)
 82 |        (datum->syntax #'id (symbol->string (syntax->datum #'id)))))))
 83 | 
 84 | 
 85 | (define (name-arg arg args)
 86 |   (string-append
 87 |    "#"
 88 |    (number->string (+ 1 (index arg args)))))
 89 | 
 90 | (define (raw . args)
 91 |   (string-append* args))
 92 | 
 93 | (define-syntax-rule (define-local (id arg ...) bdy ...)
 94 |   (begin
 95 |     (define (id arg ...) (raw bdy ...))))
 96 |  
 97 | (define-syntax-rule (define-global (id arg ...) bdy ...)
 98 |   (begin
 99 |     (define (id arg ...) (raw bdy ...))
100 |     (let*
101 |         ([name (id->string id)]
102 |          [args (list (syntax->datum #'arg) ...)]
103 |          [arity (length args)]       
104 |          [inst
105 |           (let ([arg (name-arg (syntax->datum #'arg) args)] ...)
106 |             (thunk (raw bdy ...)))])
107 |       (set-add! (macro-set) (macro-repr name arity inst)))))
108 | 


--------------------------------------------------------------------------------
/macros/topos.rkt:
--------------------------------------------------------------------------------
  1 | #lang at-exp racket
  2 | 
  3 | (require "macro-kit.rkt")
  4 | 
  5 | (define-global (Sub x)
  6 |   "_{" x "}")
  7 | (define-global (Sup x)
  8 |   "^{" x "}")
  9 | 
 10 | (define-global (BoldSymbol x)
 11 |   (match (target)
 12 |     ['katex @raw{\pmb{@x}}]
 13 |     [_ @raw{\boldsymbol{@x}}]))
 14 | 
 15 | (define-local (ShSymbol)
 16 |   (BoldSymbol @raw{\mathcal{S}}))
 17 | 
 18 | (define-local (GlSymbol)
 19 |   (BoldSymbol @raw{\mathcal{G}}))
 20 | 
 21 | (define-local (FamSymbol)
 22 |   (BoldSymbol @raw{\mathcal{F}}))
 23 | 
 24 | (define-global (Con x)
 25 |   @raw{\mathsf{@x}})
 26 | 
 27 | (define-global (Sh X)
 28 |   (ShSymbol)
 29 |   (Sub X))
 30 | 
 31 | (define-global (GL X)
 32 |   (GlSymbol)
 33 |   (Sub X))
 34 | 
 35 | (define-global (Gl x)
 36 |   (Con "gl")
 37 |   (Sub x))
 38 | 
 39 | (define-local (overline x)
 40 |   @raw{\overline{@x}})
 41 | 
 42 | (define-global (OpGL x)
 43 |   (overline (GlSymbol))
 44 |   (Sub x))
 45 | 
 46 | (define-global (OpGl x)
 47 |   (overline (Con "gl"))
 48 |   (Sub x))
 49 | 
 50 | (define-global (OpCat C)
 51 |   C (Sup (Con "o")))
 52 | 
 53 | (define-global (TotOpCat C)
 54 |   C (Sup @raw{\tilde{@(Con "o")}}))
 55 | 
 56 | (define-global (Cod C)
 57 |   (Con "cod")
 58 |   (Sub C))
 59 | 
 60 | (define-local (bold kwd)
 61 |   @raw{\mathbf{@kwd}})
 62 | 
 63 | (define-global (PCartGU)
 64 |   (bold "GU")
 65 |   (Sub @raw{\textit{pcart}}))
 66 | 
 67 | (define-global (CartGU)
 68 |   (bold "GU")
 69 |   (Sub @raw{\textit{cart}}))
 70 | 
 71 | (define-global (AlgGU)
 72 |   (bold "GU")
 73 |   (Sub @raw{\textit{alg}}))
 74 | 
 75 | (define-global (LogGU)
 76 |   (bold "GU")
 77 |   (Sub @raw{\textit{log}}))
 78 | 
 79 | 
 80 | 
 81 | (define-global (TOP E)
 82 |   (bold "Top")
 83 |   (Sub E))
 84 | 
 85 | (define-global (Idn x)
 86 |   "1" (Sub x))
 87 | 
 88 | (define-global (TotCat E)
 89 |   @raw{\widetilde{@E}})
 90 | 
 91 | (define-global (SelfIx B)
 92 |   @raw{\underline{@B}})
 93 | 
 94 | (define-global (Sl E e)
 95 |   E (Sub @raw{/@e}))
 96 | 
 97 | (define-global (CoSl e E)
 98 |   E (Sup @raw{\setminus @e}))
 99 | 
100 | 
101 | (define-global (InvImg f)
102 |   f (Sup @raw{*}))
103 | (define-global (DirImg f)
104 |   f (Sub @raw{*}))
105 | 
106 | (define-global (SET)
107 |   (bold "Set"))
108 | 
109 | (define-global (FAM cat)
110 |   (FamSymbol) (Sub cat))
111 | 
112 | (define-global (CandHom i u v)
113 |   (bold "H") (Sub i) "(" u "," v ")")
114 | 
115 | (define-global (brc x)
116 |   @raw{\{ @x \}})
117 | 
118 | (define-global (gl x)
119 |   @raw{\langle @x \rangle})
120 | 
121 | (define-global (brk x)
122 |   @raw{[@x]})
123 | 
124 | (define-global (prn x)
125 |   @raw{(@x)})
126 | 
127 | (define-global (FullSubfib u)
128 |   (bold "Full") (prn u))
129 | 
130 | (define-global (tick)
131 |   @raw{'})
132 | 
133 | (define-global (ObjTerm E)
134 |   (bold "1") (Sub E))
135 | 
136 | (publish-macro-library 'topos)
137 | 


--------------------------------------------------------------------------------
/mint-node.rkt:
--------------------------------------------------------------------------------
 1 | #! /usr/bin/env racket
 2 | #lang racket
 3 | 
 4 | (require racket/runtime-path)
 5 | (require left-pad)
 6 | 
 7 | (define-runtime-path NODE_DIR "_nodes/")
 8 | 
 9 | (define (path-remove-extension path)
10 |   (path-replace-extension path #""))
11 | 
12 | (define (node-path->uid-string path)
13 |   (path->string (path-remove-extension path)))
14 | 
15 | (define DIGIT_TABLE
16 |   (string->list "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ"))
17 | 
18 | (define BASE
19 |   (length DIGIT_TABLE))
20 | 
21 | (define (uid-string->int uid)
22 |   (define digits (string->list uid))
23 |   (for/fold ([sum 0]
24 |              [place 1]
25 |              #:result sum)
26 |             ([x (in-list (reverse digits))])
27 |     (values
28 |      (+ sum (* place (index-of DIGIT_TABLE x)))
29 |      (* BASE place))))
30 | 
31 | (define (int->uid-string ix)
32 |   (do ([digits
33 |         (list)
34 |         (cons (list-ref DIGIT_TABLE (modulo n BASE)) digits)]
35 |        [n ix (quotient n BASE)])
36 |     ((<= n 0)
37 |      (left-pad (list->string digits) 4 "0"))))
38 | 
39 | (define (node-path->int path)
40 |   (uid-string->int (node-path->uid-string path)))
41 | 
42 | (define (uid-committed? uid-string)
43 |   (not (string-prefix? uid-string "_")))
44 | 
45 | (define all-committed-uids
46 |   (filter
47 |    uid-committed?
48 |    (map
49 |     node-path->uid-string
50 |     (directory-list NODE_DIR))))
51 | 
52 | (define next-uid-string
53 |   (int->uid-string
54 |    (+ 1
55 |       (apply max (map uid-string->int all-committed-uids)))))
56 | 
57 | (printf "Next node: ~a\n" next-uid-string)
58 | 


--------------------------------------------------------------------------------
/setup-worktree.sh:
--------------------------------------------------------------------------------
1 | #!/bin/bash
2 | 
3 | rm -rf _deploy
4 | git worktree prune
5 | git worktree add -B gh-pages _deploy origin/gh-pages
6 | 


--------------------------------------------------------------------------------