├── .gitignore ├── README.md ├── exam.pdf └── exam.tex /.gitignore: -------------------------------------------------------------------------------- 1 | ## Core latex/pdflatex auxiliary files: 2 | *.aux 3 | *.lof 4 | *.log 5 | *.lot 6 | *.fls 7 | *.out 8 | *.toc 9 | *.fmt 10 | *.fot 11 | *.cb 12 | *.cb2 13 | .*.lb 14 | 15 | ## Intermediate documents: 16 | *.dvi 17 | *.xdv 18 | *-converted-to.* 19 | # these rules might exclude image files for figures etc. 20 | # *.ps 21 | # *.eps 22 | # *.pdf 23 | 24 | ## Generated if empty string is given at "Please type another file name for output:" 25 | .pdf 26 | 27 | ## Bibliography auxiliary files (bibtex/biblatex/biber): 28 | *.bbl 29 | *.bcf 30 | *.blg 31 | *-blx.aux 32 | *-blx.bib 33 | *.run.xml 34 | 35 | ## Build tool auxiliary files: 36 | *.fdb_latexmk 37 | *.synctex 38 | *.synctex(busy) 39 | *.synctex.gz 40 | *.synctex.gz(busy) 41 | *.pdfsync 42 | 43 | ## Auxiliary and intermediate files from other packages: 44 | # algorithms 45 | *.alg 46 | *.loa 47 | 48 | # achemso 49 | acs-*.bib 50 | 51 | # amsthm 52 | *.thm 53 | 54 | # beamer 55 | *.nav 56 | *.pre 57 | *.snm 58 | *.vrb 59 | 60 | # changes 61 | *.soc 62 | 63 | # cprotect 64 | *.cpt 65 | 66 | # elsarticle (documentclass of Elsevier journals) 67 | *.spl 68 | 69 | # endnotes 70 | *.ent 71 | 72 | # fixme 73 | *.lox 74 | 75 | # feynmf/feynmp 76 | *.mf 77 | *.mp 78 | *.t[1-9] 79 | *.t[1-9][0-9] 80 | *.tfm 81 | 82 | #(r)(e)ledmac/(r)(e)ledpar 83 | *.end 84 | *.?end 85 | *.[1-9] 86 | *.[1-9][0-9] 87 | *.[1-9][0-9][0-9] 88 | *.[1-9]R 89 | *.[1-9][0-9]R 90 | *.[1-9][0-9][0-9]R 91 | *.eledsec[1-9] 92 | *.eledsec[1-9]R 93 | *.eledsec[1-9][0-9] 94 | *.eledsec[1-9][0-9]R 95 | *.eledsec[1-9][0-9][0-9] 96 | *.eledsec[1-9][0-9][0-9]R 97 | 98 | # glossaries 99 | *.acn 100 | *.acr 101 | *.glg 102 | *.glo 103 | *.gls 104 | *.glsdefs 105 | 106 | # gnuplottex 107 | *-gnuplottex-* 108 | 109 | # gregoriotex 110 | *.gaux 111 | *.gtex 112 | 113 | # htlatex 114 | *.4ct 115 | *.4tc 116 | *.idv 117 | *.lg 118 | *.trc 119 | *.xref 120 | 121 | # hyperref 122 | *.brf 123 | 124 | # knitr 125 | *-concordance.tex 126 | # TODO Comment the next line if you want to keep your tikz graphics files 127 | *.tikz 128 | *-tikzDictionary 129 | 130 | # listings 131 | *.lol 132 | 133 | # makeidx 134 | *.idx 135 | *.ilg 136 | *.ind 137 | *.ist 138 | 139 | # minitoc 140 | *.maf 141 | *.mlf 142 | *.mlt 143 | *.mtc[0-9]* 144 | *.slf[0-9]* 145 | *.slt[0-9]* 146 | *.stc[0-9]* 147 | 148 | # minted 149 | _minted* 150 | *.pyg 151 | 152 | # morewrites 153 | *.mw 154 | 155 | # nomencl 156 | *.nlg 157 | *.nlo 158 | *.nls 159 | 160 | # pax 161 | *.pax 162 | 163 | # pdfpcnotes 164 | *.pdfpc 165 | 166 | # sagetex 167 | *.sagetex.sage 168 | *.sagetex.py 169 | *.sagetex.scmd 170 | 171 | # scrwfile 172 | *.wrt 173 | 174 | # sympy 175 | *.sout 176 | *.sympy 177 | sympy-plots-for-*.tex/ 178 | 179 | # pdfcomment 180 | *.upa 181 | *.upb 182 | 183 | # pythontex 184 | *.pytxcode 185 | pythontex-files-*/ 186 | 187 | # thmtools 188 | *.loe 189 | 190 | # TikZ & PGF 191 | *.dpth 192 | *.md5 193 | *.auxlock 194 | 195 | # todonotes 196 | *.tdo 197 | 198 | # easy-todo 199 | *.lod 200 | 201 | # xmpincl 202 | *.xmpi 203 | 204 | # xindy 205 | *.xdy 206 | 207 | # xypic precompiled matrices 208 | *.xyc 209 | 210 | # endfloat 211 | *.ttt 212 | *.fff 213 | 214 | # Latexian 215 | TSWLatexianTemp* 216 | 217 | ## Editors: 218 | # WinEdt 219 | *.bak 220 | *.sav 221 | 222 | # Texpad 223 | .texpadtmp 224 | 225 | # Kile 226 | *.backup 227 | 228 | # KBibTeX 229 | *~[0-9]* 230 | 231 | # auto folder when using emacs and auctex 232 | ./auto/* 233 | *.el 234 | 235 | # expex forward references with \gathertags 236 | *-tags.tex 237 | 238 | # standalone packages 239 | *.sta 240 | 241 | # generated if using elsarticle.cls 242 | *.spl 243 | notes/*.pdf 244 | /exam-hott/exam-hott.pdf 245 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # Homotopy (type) theory 2 | 3 | A doctoral course on homotopy theory and homotopy type theory given by 4 | [Andrej Bauer](http://www.andrej.com) and [Jaka Smrekar](https://www.fmf.uni-lj.si/~smrekar/) 5 | at the [Faculty of mathematics and Physics](http://www.fmf.uni-lj.si/), University of Ljubljana, 6 | in the Spring of 2019. 7 | 8 | In this course we first overview the basics of classical homotopy theory. Starting with 9 | the notion of locally trivial bundles, we motivate the classical definitions of 10 | fibrations, from which we proceed to identify the abstract strucure of Quillen model 11 | categories. We outline the basics of abstract homotopy theory in a Quillen model. 12 | 13 | In the second part we introduce homotopy type theory from the point of view of classical 14 | homotopy theory, deliberately avoiding the connections between homotopy type theory and 15 | computer science. We show how type theory can be used to carry out homotopy-theoretic 16 | arguments abstractly and "synthetically". The fact that any construction expressed in type 17 | theory is homotopy invariant is both a blessing and a curse: a blessing because it never 18 | lets us step outside of the realm of homotopy theory, and a curse because we never 19 | step outside of the realm of homotopy theory. 20 | 21 | ## Course administration 22 | 23 | We meet weekly on Tuesdays from 14:00 to 16:00 in lecture room 3.06 at the Mathematics 24 | department. 25 | 26 | The take-home exam is now available in [`exam.pdf`](./exam.pdf). 27 | 28 | 29 | ## Course materials 30 | 31 | ### Homotopy theory 32 | 33 | [Anja Petković](https://anjapetkovic.com) has kindly provided her [course 34 | notes](http://www.andrej.com/zapiski/HoTT-2019/course-notes-part-I.pdf) for the first part 35 | of the course, with the caveat that they very likely contain mistakes. Thank you, Anja! 36 | 37 | ### Homotopy type theory 38 | 39 | All lectures are recorded on video and can be watched in the [HoTT-2019 video 40 | channel](https://www.youtube.com/playlist?list=PL-47DDuiZOMCRDiXDZ1fI0TFLQgQqOdDa). The lecture notes are also available 41 | [here](http://www.andrej.com/zapiski/HoTT-2019/). 42 | 43 | 44 | ## External resources 45 | 46 | ### Homotopy theory 47 | 48 | There is a wealth of resources available on the topic of homotopy theory. The following literature is recommended as reading material: 49 | 50 | * J. Davis & P. Kirk, [Lecture Notes in Algebraic Topology](http://www.indiana.edu/~jfdavis/teaching/m623/book.pdf) (Chapter 4) 51 | * R. Piccinini, [Lectures on Homotopy Theory](https://www.sciencedirect.com/bookseries/north-holland-mathematics-studies/vol/171) (Chapter 2) 52 | * M. Hovey, [Model categories](https://web.math.rochester.edu/people/faculty/doug/otherpapers/hovey-model-cats.pdf) (Chapters 1-3) 53 | * A. Strom, [The homotopy category is a homotopy category](https://mathscinet.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=strom&s5=the%20homotopy%20category&s6=&s7=&s8=All&sort=Newest&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq) 54 | * P. G. Goerss & J. F. Jardine, [Simplicial homotopy theory](https://www.springer.com/us/book/9783034601887) (Chapters I and II) 55 | * Daniel G. Quillen, [Homotopical algebra](https://www.springer.com/gp/book/9783540039143) 56 | 57 | ### Homotopy type theory 58 | 59 | Being a new topic, homotopy type theory is still developing. Consequently, reading 60 | material and resources are a bit more fluid and scattered. A central resource is the "HoTT book", although it is hard-going for the unexperienced: 61 | 62 | * Univalent foundations program, [Homotopy type theory: Univalent foundations of mathematics](https://homotopytypetheory.org/book/) 63 | 64 | The following introductory notes are targeted at teaching homotopy type theory: 65 | 66 | * Egbert Rijke, [Introduction to Homotopy Type Theory](http://www.andrew.cmu.edu/user/erijke/hott/), a graduate course taught at Carnegie Mellon University. The accompanying [lecture notes](http://www.andrew.cmu.edu/user/erijke/hott/hott_intro.pdf) is recommended as reading material. 67 | 68 | * [Martín Escardó](https://www.cs.bham.ac.uk/~mhe/), [Introduction to Univalent Foundations of Mathematics with Agda](https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/HoTT-UF-Agda.html) is a set of lecture notes written in literal Agda (text interspersed with [Agda code](http://wiki.portal.chalmers.se/agda)). It explains the basics very thoroughly and very well. You may safely ignore the fact that it is all formalized and computer-checked, altought you may also be interested in learning how to use Agda, in which case you can consult the [lecture notes GitHub repository](https://github.com/martinescardo/HoTT-UF-Agda-Lecture-Notes). 69 | 70 | Here are some additional resources: 71 | 72 | * [Daniel Grayson](https://faculty.math.illinois.edu/~dan/): [An introduction to univalent foundations for mathematicians](https://arxiv.org/pdf/1711.01477.pdf), targeted at a general mathematical audience, 73 | * [Michael Shulman](https://home.sandiego.edu/~shulman/): [Homotopy type theory: the logic of space](https://arxiv.org/abs/1703.03007), 74 | worth reading if you would like to expand your understanding of "space", 75 | * [Steve Awodey](http://www.andrew.cmu.edu/user/awodey/) and [Michael Warren](http://mawarren.net): [Homotopy theoretic models of identity types](https://arxiv.org/abs/0709.0248), the original paper introducing the interpreation of identity types in Quillen model categories. 76 | 77 | ## Course outline 78 | 79 | ### Homotopy theory 80 | 81 | #### Background & bundles 82 | 83 | ##### Background 84 | 85 | * What does (a part of) math deal with? 86 | * Distinguishing between objects: relax, distinguish, stiffen 87 | * Homotopy theory as a natural domain of algebraic invariants 88 | * Homotopy theory as means of rephrasing a geometric problem 89 | 90 | #### Bundles 91 | 92 | * Bundles are omnipresent 93 | * Vector bundle and its frame bundle 94 | * Bundles with structure group 95 | * Principal bundles and classification 96 | * Lifting properties 97 | * Homotopization of bundles 98 | 99 | #### Fibrations of topological spaces, and their classification 100 | 101 | * Hurewicz fibrations; definition in terms of the right lifting property with respect to inclusions `Z → Z × [0,1]` 102 | * Pullbacks and retracts of fibrations are fibrations 103 | * Any map decomposes functorially as a composite of a SDR inclusion followed by fibration 104 | * The concept of a lifting function 105 | * The fibres of a fibration are homotopy equivalent 106 | * Homotopy fibre 107 | * Puppe sequence 108 | 109 | #### Cofibrations & model structure 110 | 111 | * Homotopy extension property, cofibration, Eckmann-Hilton duality 112 | * Pushouts and retracts of cofibrations are cofibrations 113 | * Any map decomposes functorially as a composite of a cofibration followed by a fibration 114 | * Quillen closed model category 115 | * Model category on `Top` with homotopy equivalences, Hurewicz fibrations, and Hurewicz cofibrations 116 | 117 | #### Homotopy and the homotopy category 118 | 119 | * Cylinder object, left homotopy 120 | * Path object, right homotopy 121 | * Homotopy classes of maps `X → Y` where `X` is cofibrant and `Y` is fibrant 122 | * Abstract localization and the homotopy category 123 | * Cylinder object, path objects, and correspondence in the category of pointed topological spaces 124 | 125 | #### Homotopies and suspensions in model categories 126 | 127 | * Left homotopy of left homotopies in a model category, motivated by `Top` 128 | * Correspondence between left and right homotopies 129 | * The category `Hom(X,Y)` with objects morphisms `X → Y` and morphisms equivalence classes 130 | of left homotopies; interpretation in `Top` 131 | * Suspension in a pointed model category and the suspension functor on its homotopy 132 | category 133 | 134 | ### Homotopy type theory 135 | 136 | #### Type theory (motivated by simplicial sets): `Π`, `Σ` 137 | 138 | [Video](https://youtu.be/FBjz8mFXH7M) and [notes](http://www.andrej.com/zapiski/HoTT-2019/HoTT-2019-04-16/HoTT-2019-04-16%20Type%20theory.pdf). 139 | 140 | * Type theory as a theory of constructions 141 | * The notion of a dependent type 142 | * Types as sets, dependent types as families of sets 143 | * Types as spaces, dependent types as fibrations 144 | * Basic type-theoretic constructions: 145 | * dependent products 146 | * dependent sums 147 | * basic types `0`, `1`, `N` 148 | 149 | #### Identity types as path objects 150 | 151 | [Video](https://youtu.be/ZMF_2Hr16M0) and [notes](http://www.andrej.com/zapiski/HoTT-2019/HoTT-2019-04-23/HoTT-2019-04-23%20Identity%20types.pdf). 152 | 153 | * Identity types as path objects 154 | * Type-theoretic rules for identity types 155 | * Basic homotopy-theoretic constructions in terms of identity types: 156 | * the groupoid of paths 157 | * Iterated identity types 158 | 159 | #### Homotopy levels 160 | 161 | [Video](https://youtu.be/Yy_U6_cvyLM) and [notes](http://www.andrej.com/zapiski/HoTT-2019/HoTT-2019-05-06/HoTT-2019-05-06%20Homotopy%20levels.pdf). 162 | 163 | * Contractible spaces 164 | * Propositions, sets, and h-levels 165 | * Truncation as a type-theoretic construction 166 | * The embedding of logic into type theory (using propositional truncation) 167 | * Basic homotopy-theoretic constructions in terms of truncation: 168 | * loop space vs. fundamental group 169 | * path-connectedness vs. contractibility 170 | * surjectivity vs. split epimorphism 171 | 172 | #### Equivalences 173 | 174 | [Video](https://youtu.be/BuI62-o3Ds8) and [notes](http://www.andrej.com/zapiski/HoTT-2019/HoTT-2019-05-14/HoTT-2019-05-14%20Equivalences.pdf). 175 | 176 | * Structure vs. property in mathematics 177 | * Having an inverse is not a property 178 | * Homotopy equivalences 179 | * Example: universal property stated as an equivalence 180 | 181 | #### Higher-inductive types 182 | 183 | [Video](https://youtu.be/6K80dcmz7nI) and [notes](http://www.andrej.com/zapiski/HoTT-2019/HoTT-2019-05-21/HoTT-2019-05-21%20Higher%20inductive%20types.pdf). 184 | 185 | * Inductive types, examples 186 | * Higher-inductive types (HITs) 187 | * Examples: circle, interval, suspension 188 | * Truncations as HITs 189 | 190 | #### Univalence and `π₁(S¹) = Z` 191 | 192 | [Video](https://youtu.be/GX3igiADQVI) and [notes](http://www.andrej.com/zapiski/HoTT-2019/HoTT-2019-05-28/HoTT-2019-05-28%20Univalence%20and%20Π₁(S¹)%20=%20Z.pdf). 193 | 194 | * The idea that equivalent structures are equal 195 | * Univalence axiom 196 | * Some consequences of the univalence axiom 197 | * Example: `π₁(S¹) = Z` 198 | 199 | 200 | -------------------------------------------------------------------------------- /exam.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/andrejbauer/homotopy-type-theory-course/ceaa72ed6844477e8fcb7b7de7e120a2180ce867/exam.pdf -------------------------------------------------------------------------------- /exam.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage[T1]{fontenc} 3 | \usepackage[utf8]{inputenc} 4 | 5 | \usepackage{a4wide} 6 | \usepackage{amsmath,amsthm,amssymb} 7 | \usepackage{color} 8 | \usepackage{times} 9 | \usepackage{enumerate} 10 | \usepackage{xypic} 11 | 12 | \newcommand\RR{\mathbb{R}} 13 | \newcommand\ZZ{\mathbb{Z}} 14 | \newcommand\EEE{\mathcal{E}} 15 | \newcommand\CCC{\mathcal{C}} 16 | \newcommand\DDD{\mathcal{D}} 17 | \newcommand{\booktitle}[1]{\textit{#1}} 18 | \newcommand{\setof}[1]{\left\{#1\right\}} 19 | \newcommand\lqs{\leqslant} 20 | \newcommand\gqs{\geqslant} 21 | \newcommand\id{\mathop{\mathrm{id}}\nolimits} 22 | \newcommand\pr{\mathop{\mathrm{pr}}\nolimits} 23 | \newcommand\bdsc{{\mathrm{bdSeq}}({\mathcal C})} 24 | \newcommand\ctop{\underline{\mathrm{Top}}} 25 | \newcommand\colim{\qopname\relax n{colim}} 26 | 27 | \newcommand{\dsum}[1]{\Sigma (#1) \,.\,} 28 | \newcommand{\dprod}[1]{\Pi (#1) \,.\,} 29 | \newcommand{\univ}{\mathcal{U}} 30 | \newcommand{\susp}[1]{\mathsf{Susp}(#1)} 31 | \newcommand{\two}{\mathsf{2}} 32 | \newcommand{\eqv}{\simeq} 33 | 34 | {\theoremstyle{definition} 35 | \newtheorem{problem}{Problem}} 36 | 37 | \begin{document} 38 | \title{Homotopy (Type) Theory take-home exam} 39 | \date{May 31, 2019} 40 | \author{} 41 | \maketitle 42 | 43 | For full credit solve \emph{at least 51 points} worth of problems. 44 | % 45 | As you are training to become a researcher, you are free to refer to constructions and 46 | proofs in existing literature, namely peer-reviewed papers and monographs. References to 47 | blog posts and other non-standard sources are allowed, but in those cases you need to 48 | verify the veracity of the claims yourself. It is probably a good idea to verify your 49 | sources even when they are of a reputable origin. In the end, you are responsible for your 50 | solutions. 51 | 52 | \section*{Part I: homotopy theory} 53 | 54 | \begin{problem}[7 points] 55 | Let $n\gqs 1$, $1\lqs k\lqs n-1$, and let $G_k(\RR^n)$ denote the set of $k$-planes in 56 | $\RR^n$. Also, let $V_k(\RR^n)$ denote the set of (ordered) $k$-tuples of orthonormal 57 | vectors in $\RR^n$. Topologize the latter by viewing it as a subset of $\RR^{n\times k}$ 58 | in the obvious way. 59 | % 60 | \begin{enumerate}[(a)] 61 | \item Topologize $G_k(\RR^n)$ as a quotient space of $V_k(\RR^n)$. 62 | \item Show that 63 | $E_k^n=\setof{(\Lambda,x)\,\vert\,\Lambda\in 64 | G_k(\RR^n),\,x\in\Lambda}\subset G_k(\RR^n)\times\RR^n$, 65 | together with the obvious projection map, is a vector bundle of 66 | rank $k$ over $G_k(\RR^n)$. 67 | \item Let $S^2$ be the $2$-sphere and let $f\colon S^2\to G_2(\RR^3)$ assign to $\zeta\in S^2$ 68 | the plane perpendicular to $\zeta$. Show that $f$ is continuous and identify the pullback 69 | bundle $f^*(E_2^3)$. You may want to consult Davis-Kirk \cite{d-k} for the latter. 70 | \end{enumerate} 71 | \end{problem} 72 | 73 | \begin{problem}[7 points] 74 | Suppose given $p_0\colon E_0\to B$ and $p_1\colon E_1\to B$ in the category of 75 | topological spaces over $B$. A map $f\colon E_0\to E_1$ over $B$ is called a {\it fibre 76 | homotopy equivalence} if there exist a map $g\colon E_1\to E_0$ over $B$ and 77 | homotopies $gf\simeq\id_{E_0}$ and $fg\simeq\id_{E_1}$ over $B$. Here, $E_i\times[0,1]$ 78 | is a space over $B$ by virtue of $P_i=p_i\circ\pr_{E_i}$. Let $p\colon E\to B$ be a 79 | fibration and let $h\colon X\times[0,1]\to B$ be a homotopy from $h_0$ to $h_1$. Using a 80 | lifting function for $p$, construct an explicit fibre homotopy equivalence of pullbacks 81 | $h_0^*(E)$ and $h_1^*(E)$ as spaces over $X$. 82 | \end{problem} 83 | 84 | 85 | \begin{problem}[7 points] 86 | Look up the definition of a {\it diagram} in $\CCC$ with a given {\it shape} $\DDD$ and its colimit in Dwyer-Spalinski \cite{d-s}. 87 | \begin{enumerate}[(a)] 88 | \item Make sense of the colimit functor $\colim\CCC^\DDD\to\CCC$ for a finite (small) category $\CCC$ 89 | with finite (small) colimits and a finite (small) shape $\DDD$. (Define it and prove that it is a functor.) 90 | \item Consider the diagrams $\DDD$ and $\EEE$, 91 | % 92 | \begin{equation*} 93 | \DDD: 94 | % 95 | \vcenter{\xymatrix{ 96 | {\bullet} \ar[rr] 97 | \ar[dd] & & 98 | {\bullet} \ar[dd] \\ 99 | & & \\ 100 | {\bullet} \ar[rr] & & 101 | {\bullet} 102 | }} 103 | \qquad\qquad 104 | \EEE: 105 | \vcenter{\xymatrix{ 106 | {\bullet} \ar[rr] 107 | \ar[dd] & & 108 | {\bullet} \ar[dd] \ar[ld] \\ 109 | & {\bullet} \ar[rd] & \\ 110 | {\bullet} \ar[rr] \ar[ur] & & 111 | {\bullet} 112 | }} 113 | \end{equation*} 114 | % 115 | Employing the pushout, define a suitable map $\CCC^\DDD\to\CCC^\EEE$ and study its properties. 116 | \end{enumerate} 117 | \end{problem} 118 | 119 | \begin{problem}[7 points] 120 | Let $\CCC$ be a category. A bounded direct sequence in $\CCC$ is a diagram of objects and morphisms of $\CCC$ of the form 121 | % 122 | \begin{equation*} 123 | \dots \xrightarrow{\xi_{-2}} X_{-1} 124 | \xrightarrow{\xi_{-1}} X_0 125 | \xrightarrow{\xi_0} X_1 126 | \xrightarrow{\xi_1} X_2 127 | \xrightarrow{\xi_2} X_3 128 | \xrightarrow{\xi_3} 129 | \dots 130 | \end{equation*} 131 | % 132 | where for all small enough $i\in\ZZ$, the $\xi_i$ are identity morphisms. We denote such 133 | a direct sequence simply by $\setof{(X_i,\xi_i)}$. A morphism 134 | $f\colon\setof{(X_i,\xi_i)}\to\setof{(Y_i,\eta_i)}$ is a collection of morphisms 135 | $f_i\colon X_i\to Y_i$ in $\CCC$ satisfying $f_{i+1}\xi_i=\eta_if_i$ for all $i$, such 136 | that $f_i=f_{i-1}$ for all small enough $i$ (i.e. for all $i\lqs b$ where $b\in\ZZ$ 137 | depends on $f$). This defines a category of bounded direct sequences in $\CCC$, which we 138 | denote $\bdsc$. 139 | 140 | Suppose that $\CCC$ is a model category. We call $f\colon\setof{(X_i,\xi_i)}\to\setof{(Y_i,\eta_i)}$ a weak equivalence 141 | (respectively a fibration) if all $f_i$ are weak equivalences (respectively fibrations) in $\CCC$. Next, we call $f$ a cofibration 142 | if for all $i$, the natural morphism $Y_i\sqcup_{X_i}X_{i+1}\xrightarrow{\eta_i+f_{i+1}}Y_{i+1}$ is a cofibration in $\CCC$, and, moreover, $f_i$ is 143 | a cofibration in $\CCC$ for all small enough $i$. 144 | \begin{enumerate}[(a)] 145 | \item Prove that $\bdsc$ is a model category. 146 | \item Identify the fibrant and cofibrant objects in $\bdsc$. 147 | \item Suppose that $\CCC$ has small colimits. Define a colimit functor $\colim\bdsc\to\CCC$ and prove that 148 | it preserves cofibrations and trivial cofibrations. {\bf Hint.} Use adjoint functors. 149 | \end{enumerate} 150 | \end{problem} 151 | 152 | \begin{problem}[7 points] 153 | Let $\CCC$ be a pointed model category. For a cofibrant $X$, we defined an association 154 | $[\Sigma X,Y]\to\pi_1^l(X,Y)=\pi_1^l(X,Y;0,0)$ which is a natural equivalence of 155 | functors $[\Sigma X,\_]$ and $\pi_1^l(X,\_)$ on the category $\CCC_f$ (the full 156 | subcategory of $\CCC$ of fibrant objects). See Theorem 2 of Quillen \cite{quillen} for a 157 | proof. State the dual of the former and prove it. \textbf{Warning.} Mind the notation. 158 | \end{problem} 159 | 160 | \section*{Part II: homotopy type theory} 161 | 162 | \begin{problem}[5 points] 163 | Prove that the coproducts have the expected universal property: 164 | % 165 | \begin{equation*} 166 | (A + B \to C) \eqv (A \to C) \times (B \to C). 167 | \end{equation*} 168 | \end{problem} 169 | 170 | \begin{problem}[5 points] 171 | Let $A$ be a type and $a : A$ a point. Prove that $\dsum{x : A} a =_A x$ is 172 | contractible. 173 | \end{problem} 174 | 175 | \begin{problem}[5 points] 176 | Prove that $\mathbb{N}$ is a set. 177 | \end{problem} 178 | 179 | \begin{problem}[5 points] 180 | Show that $(\two \eqv \two) \eqv \two$. 181 | \end{problem} 182 | 183 | \begin{problem}[5 points] 184 | Show that $S^1 \eqv \susp{\two}$, where $S^1$ is the circle and $\susp{\two}$ the 185 | suspension of~$\two$. 186 | \end{problem} 187 | 188 | \begin{problem}[5 points] 189 | Construct the \emph{double cover} of the circle as a dependent type, i.e., 190 | a dependent type $D : S^1 \to \univ$ such that $D(\mathsf{base}) \eqv \two$ and 191 | $(\dsum{x : S^1} D(x)) \eqv S^1$. 192 | \end{problem} 193 | 194 | \begin{problem}[5 points] 195 | How would you define the \emph{Möbius band} as a type? 196 | \end{problem} 197 | 198 | \begin{thebibliography}{00} 199 | \bibitem{d-k} J.~F.~Davis, P.~Kirk, \booktitle{Lecture notes in algebraic topology.} Graduate Studies in Mathematics, 35. American Mathematical Society, Providence, RI, 2001. 200 | \bibitem{d-s} W.~G.~Dwyer, J.~Spalinski, \booktitle{Homotopy theories and model categories.} Handbook of algebraic topology, 73\--126, 201 | North-Holland, Amsterdam, 1995. 202 | \bibitem{quillen} D.~G.~Quillen, \booktitle{Homotopical algebra.} Lecture Notes in Mathematics, No. 43. Springer-Verlag, Berlin-New York, 1967. 203 | \end{thebibliography} 204 | 205 | \end{document} --------------------------------------------------------------------------------