├── Bohr.tex ├── Cohen.log ├── Cohen.tex ├── Gro.tex ├── NSA.log ├── NSA.tex ├── README.md ├── SDG.tex ├── Topos.bib ├── app.tex ├── cat-appendix.tex ├── cat.tex ├── cla.tex ├── cohesion.log ├── cohesion.tex ├── colimit-appendix.tex ├── colorboxes-printing.tex ├── colorboxes.tex ├── condensed.log ├── condensed.tex ├── eff.tex ├── figure-sieve.tex ├── foreword.tex ├── glossaries.tex ├── higher.log ├── higher.tex ├── infty-bun.tex ├── infty-cat-appendix.log ├── infty-cat-appendix.pdf ├── infty-cat-appendix.synctex.gz ├── infty-cat-appendix.tex ├── infty-cat.tex ├── infty-coh.tex ├── infty-cohesion.tex ├── infty-foreword.tex ├── infty-lan.log ├── infty-lan.tex ├── infty-topos.aux ├── infty-topos.bbl ├── infty-topos.bcf ├── infty-topos.bib ├── infty-topos.blg ├── infty-topos.log ├── infty-topos.maf ├── infty-topos.mtc ├── infty-topos.mtc0 ├── infty-topos.mtc1 ├── infty-topos.mtc2 ├── infty-topos.mtc3 ├── infty-topos.mtc4 ├── infty-topos.mtc5 ├── infty-topos.mtc6 ├── infty-topos.mtc7 ├── infty-topos.out ├── infty-topos.pdf ├── infty-topos.run.xml ├── infty-topos.synctex.gz ├── infty-topos.tex ├── infty-topos.toc ├── lan.tex ├── loc.tex ├── logic-appendix.tex ├── logic.tex ├── main (省墨版).pdf ├── main-2024-2-27.pdf ├── main-2024-6-10.pdf ├── main.aux ├── main.bbl ├── main.bcf ├── main.blg ├── main.log ├── main.maf ├── main.mtc ├── main.mtc0 ├── main.mtc1 ├── main.mtc10 ├── main.mtc11 ├── main.mtc2 ├── main.mtc3 ├── main.mtc4 ├── main.mtc5 ├── main.mtc6 ├── main.mtc7 ├── main.mtc8 ├── main.mtc9 ├── main.out ├── main.pdf ├── main.run.xml ├── main.synctex.gz ├── main.tex ├── main.toc ├── rel.log ├── rel.tex ├── topos.aux ├── topos.bbl ├── topos.bcf ├── topos.blg ├── topos.log ├── topos.maf ├── topos.mtc ├── topos.mtc0 ├── topos.mtc1 ├── topos.mtc10 ├── topos.mtc11 ├── topos.mtc2 ├── topos.mtc3 ├── topos.mtc4 ├── topos.mtc5 ├── topos.mtc6 ├── topos.mtc7 ├── topos.mtc8 ├── topos.mtc9 ├── topos.out ├── topos.pdf ├── topos.run.xml ├── topos.synctex.gz ├── topos.tex ├── topos.toc ├── toposCommands.tex ├── toposes-in-Mondovi.aux ├── toposes-in-Mondovi.log ├── toposes-in-Mondovi.out ├── toposes-in-Mondovi.pdf ├── toposes-in-Mondovi.synctex.gz └── toposes-in-Mondovi.tex /Bohr.tex: -------------------------------------------------------------------------------- 1 | \section{量子理论与 Bohr 意象} 2 | 3 | \philoquote{A description of physical reality is made in terms of two sets of objects: observables and states.}{Ludwig Faddeev,\\ \emph{Elementary Introduction to Quantum Field Theory }} 4 | 5 | 一个物理系统最核心的对象是其中的\emph{状态}与\emph{可观测量}. 可观测量的\emph{代数}与状态的\emph{空间}互为对偶: 例如经典力学中, 可观测量是状态空间上的函数; 反过来, 状态空间上的点可视为可观测量代数到 $\mathbb{R}$ 的代数同态. 6 | %完全类似地, 在量子力学中, 给定语境 $\mathcal A$, 对应的 "状态空间" $\Sigma(\mathcal A)$ 中的点就是 $\mathcal A$ 到 $\mathbb{C}$ 的代数同态, 而 $\mathcal A$ 中的可观测量则可视为状态空间 $\Sigma(\mathcal A)$ 上的函数. 7 | 8 | 量子理论有多种不同的公理化, 我们考虑的公理化使用 $C^*$-代数 $\mathcal A$ 表示量子系统, \emph{可观测量}是 $\mathcal A$ 中的自伴元素, 而\emph{状态}是线性映射 $\rho\colon \mathcal A \to \mathbb{C}.$ 此外人们常常以一个 Hilbert 空间 $H$ 表示系统中的纯态 (pure states), 可观测量通过一个表示 $\pi\colon \mathcal A \to \operatorname{End}(H)$ 对应到 $H$ 上的自伴算子, 而纯态 $\psi\in H$ 对应一个映射 9 | $ 10 | \rho\colon A\mapsto \langle\psi | A | \psi \rangle := \langle\psi,\pi(A)\psi\rangle, 11 | $ 12 | 它给出状态 $\psi$ 下可观测量 $A$ 的 ``期望值''. 13 | 在这种公理化的量子理论中, 每个量子系统都对应一个\topos{}, 称为 \emph{Bohr \topos{}}, 使得 $\mathcal A$ 在这个\topos{}的内语言中成为\emph{交换} $C^*$-代数, 且系统中的状态与可观测量在这个\topos{}的内语言中可理解为一个\emph{经典力学系统}的状态和可观测量. 14 | 15 | \subsection{$C^*$-代数, 经典语境与 Bohr 景} 16 | 17 | \begin{definition} 18 | {($C^*$-代数, $*$-子代数)} 19 | \emph{$C^*$-代数}是 $\mathbb{C}$ 上的 Banach 代数\footnotemark{} $\big(\mathcal A,\|{-}\|\big)$, 20 | 带有 ``伴随'' 运算 $(-)^*\colon \mathcal A \to \mathcal A$, 21 | 满足对任意 $x\in \mathcal A$, 22 | \begin{multicols} 23 | {2} 24 | \begin{itemize} 25 | \item $(a^*)^*=a$, 26 | \item $(ab)^*=b^*a^*$, 27 | \item $(\lambda a)^*=\bar\lambda a^*\,(\lambda\in\mathbb{C})$, 28 | \item $\|a^* a\|=\|a\|\cdot \|a^*\|=\|a\|^2$. 29 | \end{itemize} 30 | \end{multicols} 31 | $C^*$-代数的 \emph{$*$-子代数}是指关于 $(-)^*$ 封闭的子代数. 32 | \end{definition} 33 | \footnotetext{Banach 代数是配有乘法的完备赋范线性空间, 满足 $\|ab\|\leq\|a\|\cdot\|b\|$. 后面我们将要在\topos{}内部使用 $C^*$ 代数的概念, 这需要谨慎地定义\emph{实数}, 但我们忽略这一问题.} 34 | 35 | \begin{definition} 36 | {(量子力学系统, 可观测量)} 37 | \begin{itemize} 38 | \item 一个\emph{量子力学系统} (quantum mechanical system) 是一个 $C^*$-代数 $\mathcal A$; 39 | \item 系统中的\emph{可观测量} (observable) 是 $\mathcal A$ 中的自伴元素, 即满足 $a^*=a$ 的元素; 40 | \item 系统中的\emph{状态} (state) 是线性函数 $\rho\colon \mathcal A\to \mathbb{C}$, 满足 41 | \begin{itemize} 42 | \item (正性) $\rho(aa^*)\geq 0$; 43 | \item (归一性) $\rho(1)=1$. 44 | \end{itemize} 45 | \end{itemize} 46 | \end{definition} 47 | 48 | \begin{example} 49 | {(Hilbert 空间上的有界线性算子的代数)} 50 | 对于 Hilbert 空间 $H$, $H$ 上的有界线性算子的代数 $\mathcal B(H)$ 是 $C^*$-代数, 其中 $a^*$ 是 $a$ 的伴随算子. 事实上, 每个 $C^*$-代数都同构于某个形如 $\mathcal B(H)$ 的代数的 $*$-子代数, 因此后者也可作为 $C^*$-代数的一种具体定义. 51 | 量子力学最初就是使用 Hilbert 空间上的自伴算子叙述的. 52 | \end{example} 53 | 54 | 我们给出经典力学系统的一种定义. 注意经典与量子系统的相似性. 55 | 56 | \begin{definition} 57 | {(Poisson 代数)} 58 | \emph{Poisson 代数}是 $\mathbb{R}$ 上的含幺交换结合代数 $\mathcal A$ 配备一个运算 $\{-,-\}\colon \mathcal A\otimes \mathcal A\to \mathcal A$, 称为 \emph{Poisson 括号}, 满足 59 | \begin{itemize} 60 | \item $(\mathcal A,\{-,-\})$ 是 Lie 代数; 61 | \item 对任意 $a\in \mathcal A$, $\{a,-\}\colon \mathcal A\to \mathcal A$ 是导子, 也即 $\{a,xy\}=\{a,x\}y+x\{a,y\}$. 62 | \end{itemize} 63 | \end{definition} 64 | 65 | \begin{example} 66 | {(辛流形上的光滑函数代数)} 67 | 在经典力学中, \emph{相空间} (phase space) 是一个辛流形 $(X,\omega)$, 其上的光滑函数代数 $C^\infty (X)$ 有自然的 Poisson 代数结构: 对 $f\in C^\infty (X)$ 定义向量场 $v_f$ 满足 $\omega(v_f,-) = df$, 则 $\{f,g\}:=\omega(v_f,v_g)$ 给出 $C^\infty (X)$ 上的 Poisson 代数结构. 68 | \end{example} 69 | 70 | \begin{definition} 71 | [label={classical-mechanical-system}] 72 | {(经典力学系统)} 73 | \begin{itemize} 74 | \item 一个\emph{经典力学系统} (classical mechanical system) 是一个 Poisson 代数 $(\mathcal A,\{-,-\})$; 75 | \item 系统中的\emph{可观测量} (observable) 是 $\mathcal A$ 中的元素; 76 | \item 系统中的\emph{状态} (state) 是\emph{线性函数} $\rho\colon \mathcal A\to \mathbb{R}$, 满足 77 | \begin{itemize} 78 | \item (正性) $\rho(a^2)\geq 0$; 79 | \item (归一性) $\rho(1)=1$. 80 | \end{itemize} 81 | \item 系统中的\emph{纯态} (pure state) 是满足上面条件的\emph{代数同态} $\mathcal A\to\mathbb{R}$. 82 | \end{itemize} 83 | \end{definition} 84 | 85 | \begin{remark} 86 | {(相空间上的点对应纯态)} 87 | 由定义 \ref{classical-mechanical-system}, 对于辛流形 $(X,\omega)$, $X$ 上的一个点 $p$ 对应一个纯态 $C^\infty (X)\to\mathbb{R},f\mapsto f(p)$. 88 | 在 $X$ 为紧流形的情形, 可以证明纯态 $C^\infty \to\mathbb{R}$ 一定形如 $f\mapsto f(p)$. 89 | \end{remark} 90 | 91 | 量子力学中的 Heisenberg 不确定性原理表明, 不交换的可观测量不可同时确定, 而一族相交换的可观测量可以同时确定. 因此我们格外关注那些交换的子代数. 因为一个状态对应的函数 $\mathcal A\to \mathbb{C}$ 只有在交换的子代数上\emph{局部地}谈论才有意义, 我们自然应当视之为交换子代数范畴上的一个层. 92 | 93 | \begin{definition} 94 | {(经典语境)} 95 | 对于量子力学系统 $\mathcal A$, 称 $\mathcal A$ 的一个交换 $*$-子代数为一个\emph{经典语境} (classical context). 96 | 记 $\mathcal C(\mathcal A)$ 为经典语境在包含关系下构成的偏序集. 97 | \end{definition} 98 | 99 | \begin{remark} 100 | {} 101 | 语境这个名字的含义是, 一个可观测量只在某些特定的语境下才有确定的值. 在一个固定的交换 $*$-子代数中, 可观测量的表现无异于一个经典系统, 故称之为经典语境. 102 | \end{remark} 103 | 104 | 这里我们稍微偏题, 介绍偏序集上的层. 105 | 106 | \subsubsection{偏序集上的层} 107 | 108 | \begin{definition} 109 | [label={Alexandroff-space}] 110 | {(Alexandroff 空间)} 111 | 若一个拓扑空间中开集的任意交仍是开集, 则称其为 \emph{Alexandroff 空间}. 记 Alexandroff 空间构成的 $\mathsf {Top}$ 的全子范畴为 $\mathsf {AlexSp}$. 112 | \end{definition} 113 | 114 | \begin{definition} 115 | [label={Alex-space-on-poset}] 116 | {(偏序集上的 Alexandroff 空间)} 117 | 设 $P$ 为偏序集. 称子集 $Q\subset P$ 为\emph{向上封闭集}是指对任意 $x\in Q,y\in P$, 若 $x\leq y$, 则 $y\in Q$. 定义 \emph{$P$ 上的 Alexandroff 空间} $\operatorname{Alex} P$ 是以 $P$ 为底层集合, 以\emph{向上封闭集}为开集的拓扑空间; 它满足定义 \ref{Alexandroff-space} 的条件. 记 $\upward{x} = \{y\in P\mid x\leq y\}$; 那么所有 $\upward{x}$ 构成 $\operatorname{Alex} P$ 的开集基, 且 $\upward{x}$ 是包含 $x$ 的所有开集的交. 由偏序集给出 Alexandroff 空间的构造是一个函子 118 | \[ 119 | \operatorname{Alex}\colon \mathsf {Poset} \to \mathsf {AlexSp}. 120 | \] 121 | % 以向下封闭集为闭集 122 | \end{definition} 123 | 124 | \begin{prop} 125 | {(偏序集等价于 T0 Alexandroff 空间)} 126 | 记 $\text{T0}\mathsf {AlexSp}\hookrightarrow\mathsf {AlexSp}$ 为满足 T0 条件 (对任意两个不同的点, 存在开集包含其中一个而不包含另一个) 的 Alexandroff 空间的全子范畴, 则定义 \ref{Alex-space-on-poset} 给出了范畴等价 $$\operatorname{Alex}\colon \mathsf {Poset}\simeq \text{T0}\mathsf {AlexSp},$$ 其逆定义如下: 127 | 对一个 T0 Alexandroff 空间, 定义其底层集合上的关系 $\leq$ 使得 $x\leq y$ 当且仅当所有包含 $x$ 的开集都包含 $y$, 这给出了一个偏序集. 128 | \end{prop} 129 | 130 | \begin{prop} 131 | [label={presheaf-on-poset-equivalent-sheaf-alexsp}] 132 | {} 133 | 对任意偏序集 $P$ 有范畴等价 134 | \[ 135 | \operatorname{Presh}(P^{\op})\simeq \mathsf {Fun}(P,\mathsf {Set}) \simeq \operatorname{Sh}(\operatorname{Alex} P). 136 | \] 137 | \end{prop} 138 | \begin{proof} 139 | 设 $F$ 为 $\operatorname{Alex}P$ 上的层, 它限制在子范畴 $\{\upward{x}\mid x\in P\}^{\op} \simeq P$ 上即给出函子 $P\to\mathsf {Set}$. 140 | 另一方面, 对于函子 $G\colon P\to\mathsf {Set}$, 141 | 定义 $\operatorname{Alex} P$ 上的预层 142 | $$ 143 | F\colon\operatorname{Open}(\operatorname{Alex}P)\to\mathsf {Set},\ U\mapsto \operatorname{lim}_{x\in U}G(x), 144 | $$ 145 | % 那么其在 $x\in P$ 上的茎为 146 | % $$ 147 | % F_x :=\operatorname{colim}_{U\in\operatorname{Open}(\operatorname{Alex}P), x\in U}F(U) \simeq F(\upward{x}). 148 | % $$ 149 | % 对于 $w\leq x$, $F_x \to F_w$. 150 | 设 $U=\bigcup_{i\in I} U_i$ 为开覆盖, 对任意一族相容的元素 $(s_i\in F(U_i))_{i\in I}$, 151 | 设 $s_i = (t_x\in G(x))_{x\in U_i}$, 152 | 那么 $s:=(t_x)_{x\in U}\in F(U)$ 是满足 $s|_{U_i} = s_i$ 的唯一元素. 这说明 $F$ 是层. 153 | 容易验证以上两个构造互逆. 154 | \end{proof} 155 | 156 | % sheaf vs cosheaf 157 | % 158 | %\begin{definition} 159 | % {(Bohr 景)} 160 | % 范畴, 也称 \emph{Bohr 景}. Bohr 景上的意象将是我们主要的研究对象. 161 | %\end{definition} 162 | 163 | \subsection{Bohr 意象} 164 | 165 | 166 | \begin{definition} 167 | {(Bohr 景, Bohr \topos{})} 168 | 对于量子力学系统 $\mathcal A$, 定义其 \emph{Bohr 景}为拓扑空间 $\operatorname{Alex}\mathcal C(\mathcal A)$, \emph{Bohr \topos{}}为 $\operatorname{Sh}(\operatorname{Alex}\mathcal C(\mathcal A))$; 由命题 \ref{presheaf-on-poset-equivalent-sheaf-alexsp}, Bohr \topos{}也等价于函子范畴 $\mathsf {Fun}(\mathcal C(\mathcal A),\mathsf {Set})$. 169 | 进一步, Bohr \topos{}还配备如下环对象 (这样的结构称为\emph{环化\topos{}}, ringed topos), 170 | $$ 171 | \underline{\mathcal A} \colon \mathcal C(\mathcal A)\to \mathsf {Set} ,\ C\mapsto C. 172 | $$ 173 | 它是 Bohr \topos{}中的 \emph{交换 $C^*$-代数}. 174 | \end{definition} 175 | 176 | \subsubsection{Gelfand 对偶} 177 | 178 | \begin{definition} 179 | {(Gelfand 谱)} 180 | 对于交换 $C^*$-代数 $\mathcal A$, 定义其 \emph{Gelfand 谱} 181 | $$ 182 | \Sigma(\mathcal A) := \{C^*\text{-代数同态}\,\lambda\colon \mathcal A \to\mathbb{C}\}, 183 | $$ 184 | 其拓扑为使得所有映射 $\Sigma(\mathcal A)\to\mathbb{C}, \lambda \mapsto \lambda (x)$ 都连续的最弱拓扑. 由 Gelfand--Mazur 定理, Gelfand 谱 $\Sigma(\mathcal A)$ 也是 $\mathcal A$ 的极大理想的集合. 185 | \end{definition} 186 | 187 | $\Sigma(\mathcal A)$ 上拓扑的定义旨在保证每个元素 $x\in A$ 都对应 $\Sigma(\mathcal A)$ 上的一个复值连续函数. 如下定理表明这个对应实际上是一个同构; 这是代数--几何对偶的一例. 188 | 189 | %--Naimark 190 | \begin{prop} 191 | {(Gelfand 对偶)} 192 | 记 $\mathsf {CC}^*$ 为交换 $C^*$-代数的范畴, $\mathsf{CHaus}$ 为紧 Hausdorff 空间的范畴, 193 | 那么 Gelfand 谱给出反变函子 $\Sigma\colon \big(\mathsf {CC}^*\big)^{\op} \to \mathsf {CHaus}$, 且有范畴等价 194 | \[\begin{tikzcd}[ampersand replacement=\&] 195 | {\big(\mathsf {CC}^*\big)^{\op}} \& {\mathsf {CHaus},} 196 | \arrow["\Sigma", shift left, from=1-1, to=1-2] 197 | \arrow["{C({-},\mathbb{C})}", shift left, from=1-2, to=1-1] 198 | \end{tikzcd}\] 199 | 其中 $C(X,\mathbb{C})$ 是空间 $X$ 上复值连续函数的 $C^*$-代数. 200 | \end{prop} 201 | 202 | \begin{remark} 203 | {(Gelfand 对偶的适用范围)} 204 | Gelfand 对偶的证明需要选择公理 (命题 \ref{axiom-of-choice}), 从而不能在一般的\topos{}中使用. 选择公理此处用于构造空间的\emph{点}; 若将紧 Hausdorff 空间推广为\emph{无点拓扑学}中的相应概念------\emph{紧完全正则位象} (compact completely regular locale), 则可得到 Gelfand 对偶的构造性证明 (见 \cite{CGDC}), 从而可以将其推广到任何\topos{}. 205 | \end{remark} 206 | 207 | 对非交换的 $C^*$-代数, 我们也可赋予类似于 Gelfand 谱的一个 ``空间'', 只不过这个空间是以 Bohr \topos{}中的内位象的形式出现. 208 | 209 | \begin{definition} 210 | {(谱预层)} 211 | 对于量子力学系统 $\mathcal A$ 的两个经典语境 $A_1 \subset A_2$, 有限制映射 $\Sigma(A_2)\to\Sigma(A_1)$. 这定义了 $\mathcal C(\mathcal A)$ 上的预层 $\Sigma$. 212 | \end{definition} 213 | 214 | \begin{remark} 215 | {} 216 | 预层 $\Sigma$ 整合了所有经典语境的几何信息. 217 | 218 | 一般而言, 一个可观测量只能给出预层 $\Sigma$ 的局部截面, 而无法给出整体截面. 219 | \end{remark} 220 | 221 | Bohr 意象中对象 $\Sigma$ 的构造可视为将 Gelfand 谱的构造由交换代数推广到非交换代数, 成为与交换子代数相对偶的空间的系统. 它实际上是 Bohr 意象中的内蕴位象 (internal locale). 而交换子代数的全体构成 Bohr 意象中的一个\emph{内蕴代数}. 由此, Bohr 意象的内语言允许我们像谈论经典态一样谈论量子态. 222 | 223 | \subsection{Bohr 意象中的命题} 224 | 225 | 226 | 227 | 在一个经典系统中, 命题是状态空间的子集, 表示这个命题在何种状态下成立. 228 | 类似地, 量子系统中的命题是预层意象中 $\Sigma$ 的子对象, 或称子函子. -------------------------------------------------------------------------------- /NSA.tex: -------------------------------------------------------------------------------- 1 | \section{非标准分析} 2 | 3 | % , 滤商与超滤范畴 4 | 5 | \emph{非标准分析}起源于对无穷小与极限等概念的重新审视. 不同于 Cauchy--Weierstrass 的 $\varepsilon$-$\delta$ 方法, 它将无穷小量视为扩充实数集中实实在在的对象. 一种称作\emph{传达原理}的工具提供了经典分析与非标准分析之间的桥梁. 6 | 7 | \subsection{基本概念} 8 | 9 | \begin{definition} 10 | {(超滤)} 11 | Boole　代数　$B$ 上的\emph{超滤}是 Boole 代数同态 $B\to \{\bot,\top\}$ 下 $\top$ 的原像. 超滤 $\mathcal F$ 也可由如下等价的条件之一定义: 12 | \begin{itemize} 13 | \item $\mathcal F$ 是极大的真滤子; 14 | \item $\mathcal F$ 是真滤子, 且对任意 $a\in B$, 要么 $a\in\mathcal F$, 要么 $\neg a\in \mathcal F$. 15 | \end{itemize} 16 | 集合 $S$ 上的超滤是指 Boole 代数 $2^S$ 上的超滤. 17 | \end{definition} 18 | 19 | \begin{remark} 20 | {} 21 | 一个集合上超滤构成的空间是其子集 Boole 代数的 Stone 空间, 这是代数--几何对偶的一例. 参见定义 \ref{points-of-locale} 后的注, 以及 \ref{locales-and-logic} 节. 22 | \end{remark} 23 | 24 | \subsection{滤商} 25 | 26 | \subsection{超滤范畴} 27 | 28 | \todo{} -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # Topos 理论讲义 2 | 3 | 一本计划中的 Topos 理论 (我希望能跟随学长将它译为 "意象") 的中文讲义. 4 | 5 | 点[这里](topos.pdf)看 pdf. 6 | -------------------------------------------------------------------------------- /Topos.bib: -------------------------------------------------------------------------------- 1 | @book{SGL, 2 | title = {Sheaves in Geometry and Logic}, 3 | author = {Saunders Mac Lane and Ieke Moerdijk}, 4 | publisher = {Springer New York}, 5 | year = {1994} 6 | } 7 | 8 | @book{OSL, 9 | title = {Objects, Structures, and Logics}, 10 | author = {Gianluigi Oliveri and Claudio Ternullo and Stefano Boscolo}, 11 | publisher = {Springer Cham}, 12 | year = {2022} 13 | } 14 | 15 | @book{CWM, 16 | title = {Categories for the Working Mathematician}, 17 | author = {Saunders Mac Lane}, 18 | note = {第二版}, 19 | publisher = {Springer New York}, 20 | year = {1978} 21 | } 22 | 23 | @book{Elephant, 24 | title = {Sketches of an Elephant}, 25 | author = {Peter T. Johnstone}, 26 | publisher = {Oxford University Press}, 27 | year = {2002} 28 | } 29 | 30 | @book{MQF, 31 | title = {Towards the Mathematics of Quantum Field Theory}, 32 | author = {Frédéric Paugam}, 33 | year = {2014}, 34 | publisher = {Springer Cham} 35 | } 36 | 37 | @book{HCA1, 38 | title = {Handbook of Categorical Algebra 1}, 39 | author = {Francis Borceux}, 40 | publisher = {Cambridge Univrsity Press}, 41 | year = {1994} 42 | } 43 | 44 | @book{HCA2, 45 | title = {Handbook of Categorical Algebra 2}, 46 | author = {Francis Borceux}, 47 | publisher = {Cambridge Univrsity Press}, 48 | year = {1994} 49 | } 50 | 51 | @book{HCA3, 52 | title = {Handbook of Categorical Algebra 3}, 53 | author = {Francis Borceux}, 54 | publisher = {Cambridge Univrsity Press}, 55 | year = {1994} 56 | } 57 | 58 | @book{TST, 59 | title = {Theories, Sites, Toposes}, 60 | author = {Olivia Caramello}, 61 | publisher = {Oxford University Press}, 62 | year = {2018} 63 | } 64 | 65 | @book{lww2, 66 | author = {李文威}, 67 | title = {代数学方法: 卷二}, 68 | publisher = {高等教育出版社 (尚未出版)}, 69 | year = {2023}, 70 | url = {https://www.wwli.asia/downloads/books/Al-jabr-2.pdf} 71 | } 72 | 73 | @BOOK{SGA4, 74 | AUTHOR = "Artin, Michael and Grothendieck, Alexander and Verdier, Jean-Louis", 75 | TITLE = "Theorie de Topos et Cohomologie Etale des Schemas {I}, {II}, {III}", 76 | PUBLISHER = "Springer", 77 | YEAR = "1971", 78 | SERIES = "Lecture Notes in Mathematics", 79 | VOLUME = "269, 270, 305" 80 | } 81 | 82 | @book{SDG-Lavendhomme, 83 | author = {Ren\'e Lavendhomme}, 84 | title = {Basic Concepts of Synthetic Differential Geometry}, 85 | publisher = {Springer New York, NY}, 86 | year = {1996} 87 | } 88 | 89 | @book{CLTT, 90 | author = {Bart Jacobs}, 91 | title = {Categorical Logic and Type Theory}, 92 | publisher = {Elsevier}, 93 | year = {1999} 94 | } 95 | 96 | @book{MSIA, 97 | author = {Ieke Moerdijk and Gonzalo E. Reyes}, 98 | title = {Models for Smooth Infinitesimal Analysis}, 99 | publisher= {Springer New York, NY}, 100 | year = {1990} 101 | } 102 | 103 | @book{HTT, 104 | author = {Jacob Lurie}, 105 | title = {Higher Topos Theory}, 106 | publisher= {Princeton University Press}, 107 | year = {2009} 108 | } 109 | 110 | @book{DCCT, 111 | author = {Urs Schreiber}, 112 | title = {Diifferential Cohomology in a Cohesive Topos}, 113 | publisher = {(尚未出版)}, 114 | url = {https://ncatlab.org/schreiber/files/dcct170811.pdf} 115 | } 116 | 117 | @book{QFT, 118 | title = {Quantum Fields and Strings: A Course for Mathematicians}, 119 | editor = {Pierre Deligne and Pavel Etingof and Dan Freed and Lisa Jeffrey and David Kazhdan and John Morgan and David R. Morrison and Edward Witten}, 120 | year = {1999}, 121 | publisher = {American Mathematical Society}, 122 | url = {http://www.math.ias.edu/qft} 123 | } 124 | 125 | @book{LPAC, 126 | title = {Locally Presentable and Accessible Categories}, 127 | author = {Jiří Adámek and Jiří Rosický}, 128 | publisher = {Cambridge University Press}, 129 | year = {1994}, 130 | doi = {https://doi.org/10.1017/CBO9780511600579} 131 | } 132 | 133 | @inproceedings{Butz1998RegularCA, 134 | title={Regular Categories and Regular Logic}, 135 | author={Carsten Butz}, 136 | year={1998}, 137 | url={https://api.semanticscholar.org/CorpusID:115242756} 138 | } 139 | 140 | % 文章 141 | 142 | @misc{FSAG, 143 | author = {Felix Cherubini and Thierry Coquand and Matthias Hutzler}, 144 | title = {A Foundation for Synthetic Algebraic Geometry}, 145 | year = {2023}, 146 | url = {https://arxiv.org/abs/2307.00073}, 147 | eprint={2307.00073}, 148 | archivePrefix={arXiv}, 149 | primaryClass={math.AG} 150 | } 151 | 152 | @misc{ILAG, 153 | title={Using the internal language of toposes in algebraic geometry}, 154 | author={Ingo Blechschmidt}, 155 | year={2021}, 156 | eprint={2111.03685}, 157 | archivePrefix={arXiv}, 158 | primaryClass={math.AG} 159 | } 160 | 161 | @misc{HTTP, 162 | author={Urs Schreiber}, 163 | title={Higher Topos Theory in Physics}, 164 | book={Encyclopedia of Mathematical Physics 2nd ed (尚未出版)}, 165 | year={2023}, 166 | url={https://ncatlab.org/schreiber/show/Higher+Topos+Theory+in+Physics} 167 | } 168 | 169 | @article{Johnstone-OTT, 170 | title = {On a Topological Topos}, 171 | author = {Peter T. Johnstone}, 172 | year = {1979}, 173 | journal = {Proc. London Math. Soc.}, 174 | doi = {https://doi.org/10.1112/plms/s3-38.2.237} 175 | } 176 | 177 | @article{EGG, 178 | title = {An extension of the Galois theory of Grothendieck}, 179 | author = {Andr\'e Joyal and Myles Tierney}, 180 | journal = {Mem. Amer. Math. Soc.}, 181 | year = {1984}, 182 | volume = {51}, 183 | number = {309} 184 | } 185 | 186 | @article{FSCM, 187 | title = {Five stages of accepting constructive mathematics}, 188 | author = {Andrej Bauer}, 189 | journal = {Bull. Amer. Math. Soc.}, 190 | year = {2017}, 191 | doi = {http://dx.doi.org/10.1090/bull/1556}, 192 | url = {https://www.youtube.com/watch?v=21qPOReu4FI} 193 | } 194 | 195 | @article{RTTG, 196 | title = {Representing topoi by topological groupoids}, 197 | journal = {Journal of Pure and Applied Algebra}, 198 | volume = {130}, 199 | number = {3}, 200 | pages = {223-235}, 201 | year = {1998}, 202 | issn = {0022-4049}, 203 | doi = {https://doi.org/10.1016/S0022-4049(97)00107-2}, 204 | url = {https://www.sciencedirect.com/science/article/pii/S0022404997001072}, 205 | author = {Carsten Butz and Ieke Moerdijk} 206 | } 207 | 208 | @article{CGDC, 209 | title={Constructive Gelfand duality for C*-algebras}, 210 | url={https://arxiv.org/abs/0808.1518}, 211 | author={Thierry Coquand and Bas Spitters}, 212 | year={2008} 213 | } 214 | 215 | @article{FSAT, 216 | title = {Functorial Semantics of Algebraic Theories}, 217 | author = {William Lawvere}, 218 | journal = {Proceedings of the National Academy of Sciences of the United States of America}, 219 | year = {1963} 220 | } 221 | 222 | @article{SIMPSON20121642, 223 | title = {Measure, randomness and sublocales}, 224 | journal = {Annals of Pure and Applied Logic}, 225 | volume = {163}, 226 | number = {11}, 227 | pages = {1642-1659}, 228 | year = {2012}, 229 | note = {Kurt Goedel Research Prize Fellowships 2010}, 230 | issn = {0168-0072}, 231 | doi = {https://doi.org/10.1016/j.apal.2011.12.014}, 232 | url = {https://www.sciencedirect.com/science/article/pii/S0168007211001874}, 233 | author = {Alex Simpson}, 234 | keywords = {Locale theory, Foundations of measure theory, Foundations of probability theory}, 235 | abstract = {This paper investigates aspects of measure and randomness in the context of locale theory (point-free topology). We prove that every measure (σ-continuous valuation) μ, on the σ-frame of opens of a fitted σ-locale X, extends to a measure on the lattice of all σ-sublocales of X (Theorem 1). Furthermore, when μ is a finite measure with μ(X)=M, the σ-locale X has a smallest σ-sublocale of measure M (Theorem 2). In particular, when μ is a probability measure, X has a smallest σ-sublocale of measure 1. All σ prefixes can be dropped from these statements whenever X is a strongly Lindelöf locale, as is the case in the following applications. When μ is the Lebesgue measure on the Euclidean space Rn, Theorem 1 produces an isometry-invariant measure that, via the inclusion of the powerset P(Rn) in the lattice of sublocales, assigns a weight to every subset of Rn. (Contradiction is avoided because disjoint subsets need not be disjoint as sublocales.) When μ is the uniform probability measure on Cantor space {0,1}ω, the smallest measure-1 sublocale, given by Theorem 2, provides a canonical locale of random sequences, where randomness means that all probabilistic laws (measure-1 properties) are satisfied.} 236 | } 237 | 238 | % 网页 239 | 240 | @MISC {177894, 241 | TITLE = {What is a Lawvere--Tierney topology?}, 242 | AUTHOR = {Zhen Lin}, 243 | HOWPUBLISHED = {Mathematics Stack Exchange}, 244 | URL = {https://math.stackexchange.com/q/177894} 245 | } 246 | 247 | @misc{nlab:topos, 248 | author = {{nLab authors}}, 249 | title = {topos}, 250 | howpublished = {\url{https://ncatlab.org/nlab/show/topos}}, 251 | note = {\href{https://ncatlab.org/nlab/revision/topos/114}{Revision 114}}, 252 | month = jul, 253 | year = 2023 254 | } 255 | 256 | @misc{stacks-project, 257 | author = {The {Stacks project authors}}, 258 | title = {The Stacks project}, 259 | howpublished = {\url{https://stacks.math.columbia.edu}}, 260 | year = {2023}, 261 | } 262 | 263 | @online{Joyal-Crash-Course, 264 | title = {A crash course in topos theory : the big picture}, 265 | date = {2015}, 266 | organization = {IHES}, 267 | author = {Andr\'e Joyal}, 268 | url = {https://www.youtube.com/watch?v=Ro8KoFFdtS4}, 269 | } 270 | 271 | @misc{Lurie-Categorical-Logic, 272 | title = {Categorical Logic (278x)}, 273 | author = {Jacob Lurie}, 274 | url = {https://www.math.ias.edu/~lurie/278x.html}, 275 | year = {2018} 276 | } 277 | 278 | @misc{Trebor-History, 279 | author = {Trebor Huang}, 280 | title = {类型论简史}, 281 | year = {2023}, 282 | publisher = {GitHub}, 283 | journal = {GitHub repository}, 284 | howpublished = {\url{https://github.com/Trebor-Huang/history}} 285 | } 286 | 287 | @misc{nlab:sheafification, 288 | author = {{nLab authors}}, 289 | title = {sheafification}, 290 | howpublished = {\url{https://ncatlab.org/nlab/show/sheafification}}, 291 | note = {\href{https://ncatlab.org/nlab/revision/sheafification/35}{Revision 35}}, 292 | month = feb, 293 | year = 2024 294 | } -------------------------------------------------------------------------------- /app.tex: -------------------------------------------------------------------------------- 1 | \chapter{\topos{}理论的应用} 2 | 3 | 4 | \input{NSA} 5 | 6 | \input{eff} 7 | 8 | \input{SDG} 9 | 10 | \input{Bohr} 11 | 12 | \input{Cohen} 13 | 14 | \input{condensed} -------------------------------------------------------------------------------- /cohesion.tex: -------------------------------------------------------------------------------- 1 | \chapter{凝聚\topos{}} 2 | 3 | \minitoc 4 | 5 | \section{凝聚的动机, 基本概念} 6 | 7 | \label{cohesion-basics} 8 | 9 | 拓扑空间范畴 $\mathsf {Top}$ 与集合范畴 $\mathsf {Set}$ 之间存在如下的伴随四元组, 10 | \[\begin{tikzcd}[ampersand replacement=\&] 11 | {\mathsf{Top}} \&\& {\mathsf {Set}} 12 | \arrow[""{name=0, anchor=center, inner sep=0}, "\Gamma"{description, pos=0.7}, shift right=2, from=1-1, to=1-3] 13 | \arrow[""{name=1, anchor=center, inner sep=0}, "{\operatorname{disc}}"{description, pos=0.7}, shift right=2, from=1-3, to=1-1] 14 | \arrow[""{name=2, anchor=center, inner sep=0}, "{\Pi_0}"{description, pos=0.7}, shift left=6, from=1-1, to=1-3] 15 | \arrow[""{name=3, anchor=center, inner sep=0}, "{\operatorname{codisc}}"{description, pos=0.7}, shift left=6, from=1-3, to=1-1] 16 | \arrow["\dashv"{anchor=center, rotate=-90}, draw=none, from=2, to=1] 17 | \arrow["\dashv"{anchor=center, rotate=-90}, draw=none, from=1, to=0] 18 | \arrow["\dashv"{anchor=center, rotate=-90}, draw=none, from=0, to=3] 19 | \end{tikzcd}\] 20 | 其中 21 | \begin{itemize} 22 | \item $\Pi_0$ 给出拓扑空间的\emph{连通分支的集合}; 23 | \item $\operatorname{disc}$ 将集合对应到\emph{离散空间}; 24 | \item $\Gamma$ 将拓扑空间遗忘为其\emph{底层集合}; 25 | \item $\operatorname{codisc}$ 将集合对应到\emph{余离散空间} (即只有空集和全集两个开集的拓扑空间). 26 | \end{itemize} 27 | 28 | 29 | 30 | \begin{definition} 31 | {(凝聚\topos{})} 32 | \emph{凝聚\topos{}} (cohesive topos) 是指一个\topos{} $\mathcal E$ 带有如下伴随四元组, 33 | % https://q.uiver.app/#q=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 34 | \[\begin{tikzcd}[ampersand replacement=\&] 35 | {\mathcal E} \&\& {\mathsf {Set}} 36 | \arrow[""{name=0, anchor=center, inner sep=0}, "\Gamma"{description, pos=0.7}, shift right=2, from=1-1, to=1-3] 37 | \arrow[""{name=1, anchor=center, inner sep=0}, "{\operatorname{disc}}"{description, pos=0.7}, shift right=2, from=1-3, to=1-1] 38 | \arrow[""{name=2, anchor=center, inner sep=0}, "{\Pi_0}"{description, pos=0.7}, shift left=6, from=1-1, to=1-3] 39 | \arrow[""{name=3, anchor=center, inner sep=0}, "{\operatorname{codisc}}"{description, pos=0.7}, shift left=6, from=1-3, to=1-1] 40 | \arrow["\dashv"{anchor=center, rotate=-90}, draw=none, from=2, to=1] 41 | \arrow["\dashv"{anchor=center, rotate=-90}, draw=none, from=1, to=0] 42 | \arrow["\dashv"{anchor=center, rotate=-90}, draw=none, from=0, to=3] 43 | \end{tikzcd}\] 44 | 使得 $\Pi_0$ 保持有限乘积. 45 | \end{definition} 46 | 47 | \begin{example} 48 | [label={cohesion-family-of-sets}] 49 | {(集合族)} 50 | 考虑集合族范畴 $\mathsf {Fam}$ (例 \ref{family-of-sets-fibration}), 又称变集范畴 (例 \ref{varying-set-topos}), Sierpi\'nski \topos{} (定义 \ref{Sierpinski-space}). 这里, 我们将一个集合族 $W\to X$ 想象为一个大集合 $W$ 分成了 $X$ 那么多组, 于是有凝聚的直观. $\mathsf {Fam}$ 是一个凝聚\topos{}, 其中 51 | \begin{itemize} 52 | \item $\Pi_0\colon \mathsf {Fam}\to\mathsf {Set}$, $(W\to X)\mapsto X$; 53 | \item $\operatorname{disc}\colon\mathsf {Set}\to\mathsf {Fam}, X\mapsto (\operatorname{id}\colon X\to X)$, 一个集合 $X$ 可以完全拆散分成 $X$ 那么多组; 54 | \item $\Gamma \colon \mathsf {Fam}\to\mathsf {Set}$, $(W\to X)\mapsto W$; 55 | \item $\operatorname{codisc}\colon \mathsf {Set}\to \mathsf {Fam}$, $X\mapsto (X\to \{*\})$, 一个集合 $X$ 可以完全不拆, 分成 $1$ 组. 56 | \end{itemize} 57 | \end{example} 58 | 59 | \begin{example} 60 | {(单纯集)} 61 | 单纯集范畴 $\mathsf {sSet}$ 是一个凝聚\topos{}, 其中 62 | \begin{itemize} 63 | \item $\Pi_0\colon \mathsf {sSet}\to\mathsf {Set}$, $X\mapsto \operatorname{coeq}(X_1\rightrightarrows X_0)$, 即 $X$ 的连通分支的集合; 64 | \item $\operatorname{disc}\colon\mathsf {Set}\to\mathsf {sSet}$, 将集合 $X$ 对应到常值单纯集 (也就是离散单纯集) $X$; 65 | \item $\Gamma \colon \mathsf {sSet}\to\mathsf {Set}$, $X\mapsto X_0 = \operatorname{Hom}(\Delta^0,X)$; 66 | \item $\operatorname{codisc}\colon \mathsf {Set}\to \mathsf {sSet}$, $\operatorname{codisc}(X)_n := X^{n+1}$. 67 | \end{itemize} 68 | \end{example} 69 | 70 | \begin{example} 71 | {(光滑空间)} 72 | 光滑空间范畴 $\operatorname{Sh}(\mathsf {CartSp})$ (例 \ref{cartsp-site}) 是一个凝聚\topos{}, 其中 73 | \begin{itemize} 74 | \item $\Pi_0\colon \mathsf {sSet}\to\mathsf {Set}$, $X\mapsto \operatorname{coeq}(X_1\rightrightarrows X_0)$, 即 $X$ 的连通分支的集合; 75 | \item $\operatorname{disc}\colon\mathsf {Set}\to\mathsf {sSet}$, 将集合 $X$ 对应到常值单纯集 (也就是离散单纯集) $X$; 76 | \item $\Gamma \colon \mathsf {sSet}\to\mathsf {Set}$, $X\mapsto X_0 = \operatorname{Hom}(\Delta^0,X)$; 77 | \item $\operatorname{codisc}\colon \mathsf {Set}\to \mathsf {sSet}$, $\operatorname{codisc}(X)_n := X^{n+1}$. 78 | \end{itemize} 79 | \end{example} -------------------------------------------------------------------------------- /colimit-appendix.tex: -------------------------------------------------------------------------------- 1 | \section{\topos{}中余极限的构造} 2 | 3 | \label{colimit-appendix} 4 | 5 | 本节从\topos{}的基础定义出发, 证明余极限的存在性. 6 | 7 | \subsection{始对象} 8 | 9 | 始对象 $0$ 是空的余极限. 作为热身, 我们先用一个相对简单的方法构造始对象. 10 | 11 | 若集合 $Z$ 只有一个子集, 那么 $Z$ 是空集. 类似地有如下命题. 12 | \begin{prop} 13 | {} 14 | 在\topos{}中, 若对象 $Z$ 只有一个子对象, 也即 $Z$ 到 $\Omega$ 有唯一的态射, 那么 $Z$ 是始对象. 15 | \end{prop} 16 | 17 | \begin{proof} 18 | 设 $X$ 是任意对象. 19 | 首先注意到, 单元集映射 $\{-\}_X\colon X \to \Omega^X$ (例 \ref{singleton-map}) 是单射. 20 | 21 | \end{proof} 22 | 23 | 假设 $0$ 存在, 那么态射 $0\to 1$ 对应 ``假'' $\bot \colon 1 \to \Omega$. 24 | 25 | % https://q.uiver.app/#q=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 26 | \[\begin{tikzcd}[ampersand replacement=\&] 27 | {\widetilde 0} \&\& 1 \\ 28 | \& X \&\& 1 \\ 29 | 1 \&\& {\Omega^\Omega} \\ 30 | \& 1 \&\& \Omega 31 | \arrow["{\operatorname{id}_\Omega}"'{pos=0.7}, from=3-1, to=3-3] 32 | \arrow["\top"{pos=0.7}, from=1-3, to=3-3] 33 | \arrow[from=1-1, to=3-1] 34 | \arrow[from=1-1, to=1-3] 35 | \arrow["{\operatorname{ev}_X}"{description}, from=3-3, to=4-4] 36 | \arrow["X"', from=4-2, to=4-4] 37 | \arrow[Rightarrow, no head, from=3-1, to=4-2] 38 | \arrow["\top", from=2-4, to=4-4] 39 | \arrow[Rightarrow, no head, from=1-3, to=2-4] 40 | \arrow["{\exists !}"{description}, dashed, from=1-1, to=2-2] 41 | \arrow[from=2-2, to=4-2] 42 | \arrow[from=2-2, to=2-4] 43 | \end{tikzcd}\] 44 | 45 | 46 | 47 | \subsection{幂对象函子} 48 | 49 | \begin{prop} 50 | {(幂对象函子是自身的伴随)} 51 | 在一个\topos{} $\mathsf C$ 中, 幂对象函子 (见定义\ref{power-object-functor}) $P \colon \mathsf C^{\op} \to \mathsf C$ 是其对偶函子 $P^{\op} \colon \mathsf C \to \mathsf C^{\op}$ 的右伴随. 52 | \end{prop} 53 | 54 | \begin{proof} 55 | 这是由于自然同构 56 | \begin{equation} 57 | \begin{aligned} 58 | \operatorname{Hom}_{\mathsf C}(X,PY) 59 | &\simeq \operatorname{Hom}_{\mathsf C}(X\times Y,\Omega)\\ 60 | &\simeq \operatorname{Hom}_{\mathsf C}(Y,PX) \simeq \operatorname{Hom}_{\mathsf C^{\op}}(PX,Y). 61 | \end{aligned} 62 | \label{power-object-functor-adjoint} 63 | \end{equation} 64 | \end{proof} 65 | 66 | 在 (\ref{power-object-functor-adjoint}) 中令 $Y=PX$, 考察 $\operatorname{id}_{PX}$ 对应的 $\operatorname{Hom}_{\mathsf C}(X,PPX)$ 的元素, 67 | 我们发现成员关系 (定义\ref{membership-relation}) $\in_Y \hookrightarrow PX \times X$ 给出了这个伴随的\emph{单位} $\eta \colon \operatorname{id}_{\mathsf C} \to PP^{\op}$. 68 | 69 | \todo{幂对象函子的单子性} -------------------------------------------------------------------------------- /colorboxes-printing.tex: -------------------------------------------------------------------------------- 1 | 2 | 3 | % 彩色方框, 参数的使用有点意思 4 | % 这是打印版的参数 (用于省墨) 5 | 6 | \usepackage{tcolorbox} 7 | 8 | \tcbuselibrary{breakable} 9 | 10 | \newcommand{\framecolor}{gray!50!white} 11 | 12 | \newtcolorbox[ 13 | auto counter, 14 | number within=section, 15 | ]{remark}[2][]{ 16 | colback=white, 17 | colbacktitle=white, 18 | colframe=\framecolor, 19 | coltitle=black, 20 | breakable, 21 | title={\textsf{注~\thetcbcounter} #2}, 22 | #1 23 | } 24 | 25 | \def\examplecolor{white} 26 | \newtcolorbox[ 27 | use counter from=remark, 28 | %number within=chapter, 29 | ]{example}[2][]{ 30 | colback=white, 31 | colbacktitle=white, 32 | colframe=\framecolor, 33 | coltitle=black, 34 | breakable, 35 | title={\textsf{例~\thetcbcounter} #2}, 36 | #1 37 | } 38 | \newtcolorbox[ 39 | use counter from=remark, 40 | %number within=chapter, 41 | ]{definition}[2][]{ 42 | colback=white, 43 | colbacktitle=white, 44 | colframe=\framecolor, 45 | coltitle=black, 46 | breakable, 47 | title={\textsf{定义~\thetcbcounter} #2}, 48 | #1 49 | } 50 | 51 | \def\propcolor{white} 52 | \newtcolorbox[ 53 | use counter from=remark, 54 | ]{prop}[2][]{ 55 | colback=white, 56 | colbacktitle=white, 57 | colframe=\framecolor, 58 | coltitle=black, 59 | breakable, 60 | title={\textsf{命题~\thetcbcounter} #2}, 61 | #1 62 | } 63 | \newtcolorbox[ 64 | use counter from=remark, 65 | ]{propdef}[2][]{ 66 | colback=white, 67 | colbacktitle=white, 68 | colframe=\framecolor, 69 | coltitle=black, 70 | breakable, 71 | title={\textsf{命题-定义~\thetcbcounter} #2}, 72 | #1 73 | } 74 | 75 | \newtcolorbox[ 76 | auto counter, 77 | number within=chapter 78 | ]{exercise}[2][]{ 79 | colback=white, 80 | colbacktitle=white, 81 | colframe=\framecolor, 82 | coltitle=black, 83 | title={\textsf{习题~\alph{\thetcbcounter}} #2}, 84 | #1 85 | } 86 | \newtcolorbox[ 87 | use counter from=remark, 88 | ]{axiom}[2][]{ 89 | colback=white, 90 | colbacktitle=white, 91 | colframe=\framecolor, 92 | coltitle=black, 93 | title={\textsf{公理~\thetcbcounter} #2}, 94 | #1 95 | } -------------------------------------------------------------------------------- /colorboxes.tex: -------------------------------------------------------------------------------- 1 | 2 | 3 | % 彩色方框, 参数的使用有点意思 4 | 5 | \usepackage{tcolorbox} 6 | 7 | \tcbuselibrary{breakable} 8 | 9 | \def\remarkcolor{blue!10!white} 10 | \newtcolorbox[ 11 | auto counter, 12 | number within=section, 13 | ]{remark}[2][]{ 14 | colback=\remarkcolor, 15 | colframe=\remarkcolor, 16 | coltitle=black, 17 | breakable, 18 | title={\textsf{注~\thetcbcounter} #2}, 19 | #1 20 | } 21 | 22 | \def\examplecolor{pink!25!white} 23 | \newtcolorbox[ 24 | use counter from=remark, 25 | %number within=chapter, 26 | ]{example}[2][]{ 27 | colback=\examplecolor, 28 | colframe=\examplecolor, 29 | coltitle=black, 30 | breakable, 31 | title={\textsf{例~\thetcbcounter} #2}, 32 | #1 33 | } 34 | \newtcolorbox[ 35 | use counter from=remark, 36 | %number within=chapter, 37 | ]{definition}[2][]{ 38 | colback=red!10!white, 39 | colframe=red!10!white, 40 | coltitle=black, 41 | breakable, 42 | title={\textsf{定义~\thetcbcounter} #2}, 43 | #1 44 | } 45 | 46 | \def\propcolor{green!20!white} 47 | \newtcolorbox[ 48 | use counter from=remark, 49 | ]{prop}[2][]{ 50 | colback=\propcolor, 51 | colframe=\propcolor, 52 | coltitle=black, 53 | breakable, 54 | title={\textsf{命题~\thetcbcounter} #2}, 55 | #1 56 | } 57 | \newtcolorbox[ 58 | use counter from=remark, 59 | ]{propdef}[2][]{ 60 | colback=orange!20!white, 61 | colframe=orange!20!white, 62 | coltitle=black, 63 | breakable, 64 | title={\textsf{命题-定义~\thetcbcounter} #2}, 65 | #1 66 | } 67 | 68 | \newtcolorbox[ 69 | auto counter, 70 | number within=chapter 71 | ]{exercise}[2][]{ 72 | colback=gray!10!white, 73 | colframe=gray!10!white, 74 | coltitle=black, 75 | title={\textsf{习题~\alph{\thetcbcounter}} #2}, 76 | #1 77 | } 78 | \newtcolorbox[ 79 | use counter from=remark, 80 | ]{axiom}[2][]{ 81 | colback=cyan!30!white, 82 | colframe=cyan!30!white, 83 | coltitle=black, 84 | title={\textsf{公理~\thetcbcounter} #2}, 85 | #1 86 | } -------------------------------------------------------------------------------- /condensed.tex: -------------------------------------------------------------------------------- 1 | \section{凝聚态数学} 2 | 3 | % 【潜水】岩豚鼠: 我们要解决拓扑abel群范畴不是abel范畴的问题, 首先要解决拓扑空间范畴中连续双射不可逆的问题. 注意到紧Hausdorff空间没有这个问题, 我们就尝试将一般的拓扑空间换成紧Hausdorff空间范畴上的层, 而这个景有一族基叫做投射有限集. 这里的拓扑是取连续满射为覆盖. 每个紧Hausdorff空间都被它自己的集合作为离散空间覆盖, 注意到紧Hausdorff空间到一般拓扑空间范畴的嵌入有个左伴随叫Stone—Cech紧化, 我们就可以把基取为离散空间的SC紧化. 4 | 5 | 作为原理的介绍, 本节忽视集合论问题. 6 | 7 | \begin{definition} 8 | {(投射有限集景)} 9 | \emph{投射有限集景} $\mathsf{ProFin}$ 是投射有限集的范畴 (例 \ref{pro-finite-set}), 配备有限联合满射族生成的覆盖. 该覆盖对应 $\mathsf {ProFin}$ 上的典范 Grothendieck 拓扑 (定义 \ref{canonical-topology}). 10 | \end{definition} 11 | 12 | % 为什么典范? 13 | % 紧空间上联合有效满射族必然有有限的子族构成联合满射. 14 | 15 | 16 | \begin{definition} 17 | {(凝聚态集合)} 18 | \emph{凝聚态集合} (condensed set) 是投射有限集景 $\mathsf{ProFin}$ 上的层. 记凝聚态集合的范畴为 $\mathsf{Cond}$. 19 | 定义凝聚态群 (Abel 群, 环, ...) 为 $\mathsf{Cond}$ 中的群 (Abel 群, 环, ...). 20 | \end{definition} 21 | 22 | 任何一个 Grothendieck \topos{}中的 Abel 群构成 Abel 范畴. 凝聚态 Abel 群也构成一个 Abel 范畴. 相比之下, 拓扑 Abel 群范畴没有这样好的性质: 考虑 Abel 群 $\mathbb{R}$ 带有离散拓扑 $\mathbb{R}_{\text{散}}$ 和通常拓扑 $\mathbb{R}_{\text{常}}$. 恒等映射 $\mathbb{R}_{\text{散}}\to \mathbb{R}_{\text{常}}$ 既单又满, 却不是同构. 23 | 24 | \begin{example} 25 | [label={top-space-as-cond-set}] 26 | {(拓扑空间视为凝聚态集合)} 27 | 对于拓扑空间 $X$, 定义凝聚态集合 $\underline{X}\colon S\mapsto \operatorname{Hom}_{\mathsf {Top}}(S,X)$. 28 | 当然, 对于拓扑群 (Abel 群, 环, ...), 也有相应的凝聚态群 (Abel 群, 环, ...). 29 | 30 | 凝聚态集合 $\underline{X}$ 包含了 $X$ 的许多重要的拓扑信息. 例如, 考虑投射有限集 $\mathbb{N}\cup\infty$ (例 \ref{N-cup-infty}), $\underline{X}(\mathbb{N}\cup\infty)$ 的元素等同于 $X$ 中的\emph{收敛序列}. 31 | 32 | 对于好的空间 $X$ (所谓\emph{紧生成 Hausdorff 空间}), $\underline{X}$ 包含的信息足够还原 $X$ 的拓扑, 这就是说这类拓扑空间的范畴全忠实地嵌入凝聚态集合范畴. 33 | \end{example} 34 | 35 | \begin{remark} 36 | [label={remark-topological-topos}] 37 | {(凝聚态集合作为拓扑空间的推广, Johnstone 拓扑\topos{})} 38 | 由例 \ref{top-space-as-cond-set}, 凝聚态集合可视为拓扑空间的推广: 对于凝聚态集合 $X$ 与投射有限集 $S$, $X(S)$ 的元素可视为 $S$ 到 $X$ 的假想的 ``连续映射''; 39 | 特别地, $X(\mathbb{N}\cup\infty)$ 的元素可视为 $X$ 中假想的 ``收敛序列''. 40 | 41 | 称拓扑空间 $X$ 为\emph{序列空间} (sequential space) 是指: 对任意拓扑空间 $Y$, 集合映射 $f\colon X\to Y$ 连续当且仅当 $f$ 将 $X$ 中的收敛序列映射到 $Y$ 中的收敛序列. 很多常见的拓扑空间 (如 CW 复形) 都是序列空间. 42 | 考虑单点 $\text{pt}$ 和 $\mathbb{N}\cup\infty$ 构成的 $\mathsf {Top}$ 的全子范畴, 配备典范 Grothendieck 拓扑成为一个景, 这个景上的层范畴即是 Johnstone 的\emph{拓扑\topos{}} $\mathcal T$. 那么序列空间的范畴全忠实地嵌入 $\mathcal T$, 正如紧生成 Hausdorff 空间全忠实地嵌入凝聚态集合范畴一样. 43 | \end{remark} -------------------------------------------------------------------------------- /eff.tex: -------------------------------------------------------------------------------- 1 | \section{可计算性理论与有效意象} 2 | 3 | 通常数学中可定义的函数不一定能在计算机上编程计算, 其中最著名的是停机问题: 不存在一个程序能够判断任何程序是否停机. 研究类似问题的学科称作\emph{可计算性理论}. 有效\topos{} (effective topos) $\mathsf {Eff}$ 是一个可用内语言研究可计算性理论的\topos{}, 或用一种诗意的表达, 是 ``可计算数学的世界'' (相对于 $\mathsf {Set}$ 是 ``通常数学的世界''). 4 | 5 | \subsection{基础知识} 6 | 7 | 首先我们需要用形式化的语言描述\emph{计算}的概念. 我们知道, 通用的\emph{计算机}是这样工作的: 8 | \begin{itemize} 9 | \item 计算机可以执行一些 (有限个) 基本\emph{指令} (instructions); \emph{程序} (program) 是有限个指令的序列 (特别地, 这意味着只有可数个程序); 10 | \item 计算机可以存储, 输入或输出\emph{数据} (data)\footnote{我们假设储存空间是无限的.}, 程序也是一种数据; 11 | \item 计算机一次只执行一条指令\footnote{并行计算机可一次执行多条指令, 但它不改变计算的能力, 只是增加计算的效率.}. 12 | \end{itemize} 13 | 14 | 数据是多种多样的, 例如一个程序可以作为数据输入另一个程序, 两个数据可以放在一起成为一个数据. 为了将所有数据表示为同一种东西, 我们需要\emph{编码} (code). 一种常用的编码是 \emph{G\"odel 数}, 它将任何一个数据对应到一个确定的自然数; 我们不需要其细节, 而只要知道如下性质: 15 | \begin{itemize} 16 | \item 17 | \item 对于两个自然数 $n,m$, 数对 $(n,m)$ 可编码为一个自然数, 记为 $\langle n,m \rangle$; 18 | \end{itemize} 19 | 20 | 若以 G\"odel 数来编码所有数据, 则我们考虑的程序是 $\mathbb{N}$ 到 $\mathbb{N}$ 的某种函数; 它在可计算性理论中称为\emph{部分递归函数} (partial recursive function). 21 | 22 | \begin{definition} 23 | {(部分递归函数)} 24 | 对于集合 $X,Y$, \emph{部分函数} $f\colon X \to Y$ 是指定义在 $X$ 的一个子集上取值于 $Y$ 的函数. 对于 $x\in X$, 以符号 $f(x)\downarrow$ 表示 $f(x)$ 有定义. 25 | 在部分函数 $\mathbb{N}^k \to \mathbb{N} (k\geq 0)$ 中, 归纳地定义\emph{基础递归函数} (primitive recursive function): 26 | \begin{itemize} 27 | \item 常值函数 $\bar n\colon (x_1,\cdots,x_k)\mapsto n$ 是基础递归函数; 28 | \item 后继 $\mathsf {succ} \colon n\mapsto n+1$ 是基础递归函数; 29 | \item 投影 $(x_1,\cdots,x_k)\mapsto x_i$ 是基础递归函数; 30 | \item 基础递归函数的复合是基础递归函数; 31 | \item 给定基础递归函数 $g\colon \mathbb{N}^k\to \mathbb{N}$ 以及 $h\colon \mathbb{N}^{k+2}\to\mathbb{N}$, 如下定义的函数 $f$ 也是基础递归函数, 这个过程称作\emph{基础递归} (primitive recursion): 32 | $$ 33 | \begin{aligned} 34 | f(0,x_1,\cdots,x_k) &= g(x_1,\cdots,x_k), 35 | \\ 36 | f(n+1,x_1,\cdots,x_k) &= h(n,f(n,x_1,\cdots,x_k),x_1,\cdots,x_k). 37 | \end{aligned} 38 | $$ 39 | %由 ``$\mathtt{for}$ 循环'' 定义的函数是基础递归函数. 40 | \end{itemize} 41 | 若子集 $A\subset\mathbb{N}$ 的特征函数 $1_A$ 是基础递归函数, 则称之为\emph{基础递归谓词}. 42 | \end{definition} 43 | 44 | \begin{example} 45 | {(常见的基础递归函数)} 46 | 很多函数都是基础递归函数. 47 | \begin{multicols} 48 | {2} 49 | \begin{itemize} 50 | \item 加法 $+\colon \mathbb{N}^2\to \mathbb{N}$ 是基础递归函数, 因为它有如下定义: 51 | $$ 52 | \begin{aligned} 53 | 0+x &= x, 54 | \\ 55 | (n+1)+x &= \mathsf {succ}(n+x). 56 | \end{aligned} 57 | $$ 58 | \item 乘法, 幂, 阶乘都是基础递归函数. 59 | \item 前驱 $\mathsf {pred}\colon n\mapsto 60 | \begin{cases} 61 | 0 & n=0\\ 62 | n-1 & n>0 63 | \end{cases} 64 | $ 是基础递归函数. 65 | \item 谓词 ``等于零'' $\mathsf {iszero}$ 是基础递归谓词: 66 | $$ 67 | \begin{aligned} 68 | \mathsf{iszero}(0) &= 1, 69 | \\ 70 | \mathsf{iszero}(n+1) &= 0. 71 | \end{aligned} 72 | $$ 73 | \item 谓词 ``大于等于'', ``等于'', ``是素数'' 都是基础递归谓词. 74 | \end{itemize} 75 | \end{multicols} 76 | \end{example} 77 | 78 | \begin{definition} 79 | {(部分递归函数)} 80 | \emph{部分递归函数}的定义是在基础递归函数的基础上增加如下操作: 给定\todo{} 81 | \end{definition} 82 | 83 | %其中\emph{部分函数}是可能对某些输入没有输出的函数, 而\emph{递归} (recursion) 是可计算函数的一种形式化的定义方法. 84 | 85 | \begin{definition} 86 | {(实现)} 87 | 归纳地定义 ``自然数 $n$ 实现公式 $\varphi$'' 如下 (关于公式, 见定义 \ref{formula}, \ref{kinds-of-formulae}): 88 | \begin{itemize} 89 | \item $\varphi$ 为原子公式 (关系或等式). 当 $\varphi$ 取值为真时, 我们称 $0$ 实现 $\varphi$; 90 | \item $\varphi = (\psi \land \chi)$. 当 $m$ 实现 $\psi$ 且 $n$ 实现 $\chi$ 时, 我们称 $\langle m,n\rangle$ 实现 $\varphi$; 91 | \item $\varphi = (\psi \lor \chi)$. 当 $m$ 实现 $\psi$ 时, 我们称 $\langle 0,m\rangle$ 实现 $\varphi$, 当 $n$ 实现 $\chi$ 时, 我们称 $\langle 1,n\rangle$ 实现 $\varphi$; (注意: $\psi\lor\chi$ 要被实现, 必须要明确指出 $\psi,\chi$ 中的某一个被实现) 92 | \item $\varphi = (\psi \Rightarrow \chi)$. 若每当 $m$ 实现 $\psi$ 时, 都有 $n(m)$ 良定义且 $n(m)$ 实现 $\chi$, 我们称 $n$ 实现 $\varphi$; 93 | \item \todo{} 94 | \end{itemize} 95 | \end{definition} -------------------------------------------------------------------------------- /figure-sieve.tex: -------------------------------------------------------------------------------- 1 | 2 | 3 | \tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt 4 | 5 | \begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1] 6 | %uncomment if require: \path (0,300); %set diagram left start at 0, and has height of 300 7 | 8 | %Shape: Ellipse [id:dp10401378338786049] 9 | \draw (74,113.39) .. controls (74,106.67) and (84.1,101.22) .. (96.56,101.22) .. controls (109.01,101.22) and (119.11,106.67) .. (119.11,113.39) .. controls (119.11,120.11) and (109.01,125.56) .. (96.56,125.56) .. controls (84.1,125.56) and (74,120.11) .. (74,113.39) -- cycle ; 10 | %Shape: Grid [id:dp09583060841478219] 11 | \draw [draw opacity=0] (81.95,70.67) -- (126.56,70.67) -- (113.72,94.22) -- (69.11,94.22) -- cycle ; \draw (84.95,70.67) -- (72.11,94.22)(92.95,70.67) -- (80.11,94.22)(100.95,70.67) -- (88.11,94.22)(108.95,70.67) -- (96.11,94.22)(116.95,70.67) -- (104.11,94.22)(124.95,70.67) -- (112.11,94.22) ; \draw (80.31,73.67) -- (124.92,73.67)(75.95,81.67) -- (120.56,81.67)(71.59,89.67) -- (116.2,89.67) ; \draw ; 12 | %Shape: Circle [id:dp31189924819685966] 13 | \draw (80.33,36.39) .. controls (80.33,34.7) and (81.7,33.33) .. (83.39,33.33) .. controls (85.08,33.33) and (86.44,34.7) .. (86.44,36.39) .. controls (86.44,38.08) and (85.08,39.44) .. (83.39,39.44) .. controls (81.7,39.44) and (80.33,38.08) .. (80.33,36.39) -- cycle ; 14 | %Straight Lines [id:da12935033787340267] 15 | \draw (87.72,42.11) -- (94.97,52.58) ; 16 | \draw [shift={(96.11,54.22)}, rotate = 235.29] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; 17 | %Shape: Circle [id:dp7222065551952845] 18 | \draw (96.67,60.56) .. controls (96.67,58.41) and (98.41,56.67) .. (100.56,56.67) .. controls (102.7,56.67) and (104.44,58.41) .. (104.44,60.56) .. controls (104.44,62.7) and (102.7,64.44) .. (100.56,64.44) .. controls (98.41,64.44) and (96.67,62.7) .. (96.67,60.56) -- cycle ; 19 | %Straight Lines [id:da7276461675375434] 20 | \draw (100.56,66.44) -- (100.56,115.89) ; 21 | \draw [shift={(100.56,117.89)}, rotate = 270] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; 22 | %Shape: Ellipse [id:dp12697611260252684] 23 | \draw (156.67,113.39) .. controls (156.67,106.67) and (166.77,101.22) .. (179.22,101.22) .. controls (191.68,101.22) and (201.78,106.67) .. (201.78,113.39) .. controls (201.78,120.11) and (191.68,125.56) .. (179.22,125.56) .. controls (166.77,125.56) and (156.67,120.11) .. (156.67,113.39) -- cycle ; 24 | %Shape: Grid [id:dp9936460967678797] 25 | \draw [draw opacity=0] (164.61,70.67) -- (209.22,70.67) -- (196.39,94.22) -- (151.78,94.22) -- cycle ; \draw (167.61,70.67) -- (154.78,94.22)(175.61,70.67) -- (162.78,94.22)(183.61,70.67) -- (170.78,94.22)(191.61,70.67) -- (178.78,94.22)(199.61,70.67) -- (186.78,94.22)(207.61,70.67) -- (194.78,94.22) ; \draw (162.98,73.67) -- (207.59,73.67)(158.62,81.67) -- (203.23,81.67)(154.26,89.67) -- (198.87,89.67) ; \draw ; 26 | %Shape: Circle [id:dp838751100400853] 27 | \draw (168,49.06) .. controls (168,42.77) and (173.1,37.67) .. (179.39,37.67) .. controls (185.68,37.67) and (190.78,42.77) .. (190.78,49.06) .. controls (190.78,55.35) and (185.68,60.44) .. (179.39,60.44) .. controls (173.1,60.44) and (168,55.35) .. (168,49.06) -- cycle ; 28 | %Straight Lines [id:da5932706956135338] 29 | \draw (179.39,62.44) -- (179.39,111.89) ; 30 | \draw [shift={(179.39,113.89)}, rotate = 270] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; 31 | 32 | % Text Node 33 | \draw (43.33,109) node [anchor=north west][inner sep=0.75pt] [align=left] {$\displaystyle c$}; 34 | % Text Node 35 | \draw (41.33,75) node [anchor=north west][inner sep=0.75pt] [align=left] {$\displaystyle S$}; 36 | % Text Node 37 | \draw (113,123.67) node [anchor=north west][inner sep=0.75pt] [align=left] {$\displaystyle \checkmark$}; 38 | % Text Node 39 | \draw (197,123) node [anchor=north west][inner sep=0.75pt] [align=left] {$\displaystyle \times $}; 40 | % Text Node 41 | \draw (108.33,46) node [anchor=north west][inner sep=0.75pt] [align=left] {$\displaystyle d$}; 42 | % Text Node 43 | \draw (89.67,22) node [anchor=north west][inner sep=0.75pt] [align=left] {$\displaystyle e$}; 44 | 45 | 46 | \end{tikzpicture} -------------------------------------------------------------------------------- /foreword.tex: -------------------------------------------------------------------------------- 1 | \chapter{前言} 2 | 3 | %\todo{重写} 4 | 5 | \philoquote{[A] ``set 6 | theory'' for geometry should apply not only to abstract sets divorced from time, space, 7 | ring of definition, etc., but also to more general sets which do in fact develop along 8 | such parameters. For such sets, usually logic is ``intuitionistic'' (in its formal properties) usually the axiom of choice is false, and usually a set is not determined by its points defined over 1 only.}{William Lawvere, \emph{Quantifiers and Sheaves}} 9 | 10 | 每一个\topos{} (topos) 都是一个数学宇宙. 集合范畴 $\mathsf {Set}$ 是最简单和最重要的\topos{}, 对应着 ``通常数学'' 的宇宙. 一般的\topos{}从外部看有远比集合范畴丰富的结构, 其范畴论性质却与 $\mathsf {Set}$ 几乎相同. 逻辑学上, 每个\topos{}都提供了一种语言以完全在范畴 ``内部'' 进行推理, 仿佛所处理的对象是普通集合一样. 对于熟悉的数学对象, \topos{}也可给我们新的视角; 一些关系在特定\topos{}的语言中很简洁, 而在通常数学语言中则不然. 拓扑空间 $X$ 上的层构成一个\topos{} $\operatorname{Sh}(X)$; 由此, \topos{}可视为空间的概念的推广. 11 | %\topos{}理论的经典文献 \textit{Sketches of an Elephant} \cite{Elephant} 的开头列举了意象更多的解读方式, 表明人们对它的理解正如盲人摸象一样. 12 | 13 | %\section{\topos{}理论简史} 14 | 15 | %本节用尽可能简要的语言概括\topos{}的历史. 16 | 17 | \topos{}的概念主要经过了两批数学家的发展. 时间上较早的是以 Alexander Grothendieck, Jean-Louis Verdier 为首的代数几何的学派; 他们在 1960 年代著名的 ``代数几何学研讨班'' (Séminaire de Géometrie Algébrique, SGA) 中发展了现在称为 ``Grothendieck \topos{}'' 的概念, 作为一种广义的空间, 用来研究代数几何中涌现的各种上同调理论. 而在 1970 年左右, 以 William Lawvere, Myles Tierney 为代表的一众范畴逻辑学家, 在研究范畴论作为数学基础的过程中, 发展了现在称为\topos{}的概念, 揭示了其与逻辑的关系. 1980-2000 年之间出现了一些在不同方面谈论\topos{}理论的著作, 如 Ieke Moerdijk 与 Gonzalo Reyes 的 \emph{Models for Smooth Infinitesimal Analysis} \cite{MSIA}, Saunders Maclane 与 Ieke Moerdijk 的 \emph{Sheaves in Geometry and Logic} \cite{SGL}, Francis Borceux 的三卷 \emph{Handbook of Categorical Algebra} \cite{HCA1},\cite{HCA2},\cite{HCA3}, 进一步展现出这个概念在数学诸多分支的重要地位. 人们意识到, \topos{}的概念正如一头大象, 无论从几何, 逻辑中的哪个侧面去了解它, 只能得到片面的观点; 它的覆盖面如此之广, 以至于迄今为止只有 Peter Johnstone 一千多页的 \textit{Sketches of an Elephant} \cite{Elephant} 曾试图呈现\topos{}理论的全貌. 18 | 19 | % Lawvere 还致力于用\topos{}的语言 (``综合微分几何'') 表述物理的直观, 以绕过分析学上的困难. 20 | 21 | % Lawvere continues to work on his 50-year quest for a rigorous flexible base for physical ideas, free of unnecessary analytic complications. 22 | 23 | %\section{本书的编排原则} 24 | 25 | 本书对数学没有原创性的贡献, 其中几乎所有结果在 20 世纪已经为该领域的专家所熟知. 但是大部分内容找不到成体系的中文资料 (除了李文威老师的\emph{代数学方法} \cite{lww2}), 这是我编写本书的动机之一. 内容的编排原则大致是使初学者最易接受, 并且提供启发性的观点, 从而使人能更快入门去看更多的资料. 本书收录的论证都是十分简单而直观的; 阅读本书虽不能让人成为本领域的专家, 但能让人发现一些事情并非想象的那样困难, 在将玄妙的概念祛魅的过程中产生信心和乐趣. 书中许多内容的含入仅仅是由于个人的品味, 例子的选取受到了本人的微分几何与代数拓扑背景的影响. 一些证明出自本人的思考, 因此错漏是难免的. 限于水平, 书中许多细节无法深究, 包括有关集合论的 ``小'' 性的问题, 以及其它涉及到数学基础的问题. 许多术语没有通行的中文版本, 姑且使用了本人的翻译, 书后附有一个简短的术语表. 26 | 27 | \section*{致谢} 28 | 29 | 在编写本书的过程中, 我得到了 Olivia Caramello 教授, Laurent Lafforgue 教授, 以及杨家同, 陈潇扬等友人的帮助和鼓励. 30 | %在本书参考的文献中, 最重要的是一个名叫 \nlab 的网站, 而其中最主要的贡献者是 Urs Schreiber 教授. 31 | 向他们表达诚挚的感谢. 32 | 33 | %\section*{内容提要} 34 | 35 | %附录 A 包含了本书中多处用到的一些范畴论知识. 36 | 37 | \newpage 38 | 39 | ~\vspace{4em} 40 | 41 | 42 | % TODO 详细介绍参考文献 -------------------------------------------------------------------------------- /glossaries.tex: -------------------------------------------------------------------------------- 1 | \chapter*{术语和符号表} 2 | 3 | 4 | \section*{术语的翻译} 5 | 6 | 如下是一些尚未广为流传的中文术语 (其中一部分是本人的翻译) 与外文的对应. 7 | 8 | \begin{center} 9 | \begin{tabular} 10 | {llr} 11 | 中文&外文&定义\\\hline 12 | \coherent{}逻辑 & coherent logic & \ref{kinds-of-theories}, \ref{inference-rules}\\ 13 | %\cohesive{}\topos{} & cohesive topos \\ 14 | 教条 & doctrine & \ref{definition-doctrine}\\ 15 | \fm{} & frame & \ref{frame-definition} \\ 16 | 联合满射族 & jointly epimorphic/surjective family & \ref{canonical-topology-on-topos} \\ 17 | 位象 & locale & \ref{locale-definition} \\ 18 | \nc{} & nucleus & \ref{nuclei}\\ 19 | \regular{}逻辑 & regular logic & \ref{kinds-of-theories}, \ref{inference-rules} \\ 20 | 景 & site & \ref{site-definition}\\ 21 | 提纲 & sketch & \ref{sketches} \\ 22 | 清晰空间 & sober space & \ref{sober-space} \\ 23 | \topos{} & topos & \ref{topos-definition} \\ 24 | 旋子 & torsor & \ref{G-torsors-over-topos} 25 | \end{tabular} 26 | \end{center} 27 | 28 | \section*{符号} 29 | 30 | 如下是一些可能不通用, 或不广为人知的数学符号. 31 | 32 | \begin{center} 33 | \begin{tabular} 34 | {ll} 35 | 符号&含义\\ \hline 36 | $\internalprop{-}$&借用自然语言表达的内语言中的命题\\ 37 | $\interpretation{-}$&逻辑公式或类型论陈述在范畴语义中的解释\\ 38 | $\square$&Lawvere--Tierney 拓扑 (定义 \ref{Lawvere--Tierney-topology-internal-definition})\\ 39 | $\yo$&米田嵌入 (定义 \ref{definition-yoneda-embedding})\\ 40 | $\yo_{(\mathcal C,J)}$& 景到层范畴的米田嵌入 (定义 \ref{sheafified-yoneda}) 41 | \end{tabular} 42 | \end{center} -------------------------------------------------------------------------------- /infty-bun.tex: -------------------------------------------------------------------------------- 1 | \chapter{$\infty$-\topos{}与 $\infty$-丛} 2 | 3 | % 1-意象中的子对象分类子就是 (-1)-截断丛的分类空间 -------------------------------------------------------------------------------- /infty-cat-appendix.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/SimplicialCat/topos/59a726f343732ec6c90a22ae3c1f4d5fc29fe663/infty-cat-appendix.pdf -------------------------------------------------------------------------------- /infty-cat-appendix.synctex.gz: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/SimplicialCat/topos/59a726f343732ec6c90a22ae3c1f4d5fc29fe663/infty-cat-appendix.synctex.gz -------------------------------------------------------------------------------- /infty-cat.tex: -------------------------------------------------------------------------------- 1 | \chapter{$\infty$-层与 Grothendieck $\infty$-\topos{}} 2 | 3 | %\philoquote{Quite contrary to superficial perception, higher topos theory provides just the mathematical context that physicists are often intuitively but informally assuming anyway.}{Urs Schreiber, \cite{HTTP}} 4 | 5 | 6 | %\[\begin{tikzcd}[ampersand replacement=\&] 7 | % {\text{集合}} \& {\text{\topos{}}} \\ 8 | % {\text{空间}} \& {\infty \text{-\topos{}}} 9 | % \arrow[rightsquigarrow, from=1-1, to=2-1] 10 | % \arrow[rightsquigarrow, from=1-2, to=2-2] 11 | %\end{tikzcd}\] 12 | 13 | \minitoc 14 | 15 | 16 | 17 | 18 | %\section{$\infty$-\topos{}} 19 | 20 | \todo{HTT Ch.6 $\infty$-\topos{}} 21 | 22 | 23 | 24 | \begin{definition} 25 | {(自反局部化)} 26 | 设 $\mathcal C$ 为 $\infty$-范畴, 定义 $\mathcal C$ 的一个\emph{自反局部化}为函子 $a\colon \mathcal C\to \mathcal D$, 其具有全忠实的右伴随. 27 | 进一步, 若 $a$ 为正合函子 (保持有限极限), 则称之为\emph{正合局部化}. 28 | 这与普通范畴中的自反局部化在语法上完全相同. 29 | %(定义 \ref{reflective-subcategory}) 30 | \end{definition} 31 | 32 | 如下是 Grothendieck \topos{}的 $\infty$ 版本. 33 | 34 | \begin{definition} 35 | {($\infty$-\topos{})} 36 | 对于 $\infty$-范畴 $\mathcal X$, 若存在 $\infty$-范畴 $\mathcal C$ 以及一个正合局部化 37 | $$ \widehat {\mathcal C} \to \mathcal X, $$ 则称 $\mathcal X$ 为 (Grothendieck) \emph{$\infty$-\topos{}}. 38 | \end{definition} 39 | 40 | 41 | 42 | 43 | 44 | \section{Grothendieck 拓扑与层} 45 | 46 | \todo{层, HTT 6.2.2} 47 | 48 | \section{Giraud 定理} 49 | 50 | \begin{prop} 51 | {($\infty$-\topos{}的等价定义, $\infty$-Giraud 公理)} 52 | $\infty$-\topos{}等价于局部小, 可表现, 余完备, 拉回保持余极限, 且内群胚有效的 $\infty$-范畴. 53 | % Cisinski CSTT: 局部小, 小余极限, 小可达生成, 拉回保持余极限, 和无交, 内群胚有效. 54 | % nLab: 可表现, 拉回保持余极限, 和无交, 内群胚有效. 55 | \end{prop} 56 | 57 | %\cite{DCCT} -------------------------------------------------------------------------------- /infty-coh.tex: -------------------------------------------------------------------------------- 1 | \chapter{$\infty$-\topos{}与上同调} 2 | 3 | 数学中许多名为某某上同调的概念可以在同一个框架下谈论. 4 | 5 | \begin{definition} 6 | {(上同调)} 7 | 给定 $\infty$-范畴 $\mathcal C$ 及其对象 $X,A$, 定义 $X$ 的\emph{取值于 $A$ 的 $0$ 阶上同调}为 8 | $$ 9 | H^0(X,A):=\pi_0\operatorname{Hom}(X,A). 10 | $$ 11 | 态射 $c\colon X\to A$ 称为\emph{上圈} (cocycle), 12 | 态射的同伦 $c_1\to c_2$ 称为\emph{上边界} (coboundary), 13 | 等价类 $[c]\in\pi_0\operatorname{Hom}_{\mathcal C}(X,A)$ 称为\emph{上同调类} (cohomology class). 通常我们考虑的范畴 $\mathcal C$ 是 $\infty$-\topos{}. 14 | \end{definition} 15 | 16 | \subsection{空间的奇异上同调} 17 | 18 | % $A = K(\mathbb{Z},n)$ 19 | 20 | % $X$ 的 $G$-系数同调是 ``$X$ 那么多个 $G$ 的和'', 21 | % $X$ 的 $G$-系数上同调是 ``$X$ 那么多个 $G$ 的积''. 22 | % tensoring and cotensoring? 23 | 24 | % 本小节目标: 以 ∞-范畴语言简述代数拓扑主要结论 25 | 26 | 27 | \subsection{等变上同调} 28 | 29 | % X 的 G-等变上同调即是 X//G 的上同调. 30 | 31 | \subsection{群上同调} 32 | 33 | % % https://ncatlab.org/nlab/show/group+cohomology 34 | % % delooping 35 | 36 | \subsection{层上同调} 37 | %\begin{example} 38 | % {(层上同调)} 39 | % % https://ncatlab.org/nlab/show/cohomology#Overview 40 | % % 表现 41 | %\end{example} -------------------------------------------------------------------------------- /infty-cohesion.tex: -------------------------------------------------------------------------------- 1 | \chapter{\cohesive{}\topos{}} 2 | 3 | \philoquote{[T]he existence 4 | of a nontrivial shape operation on types is what reflects that types may carry a nontrivial topological (or more generally: cohesive) quality in the first place.}{Urs Schreiber, \cite{DCCT}} 5 | 6 | \minitoc 7 | 8 | \section{\cohesion{}的动机, 基本概念} 9 | 10 | \label{cohesion-basics} 11 | 12 | 拓扑空间范畴 $\mathsf {Top}$ 与集合范畴 $\mathsf {Set}$ 之间存在如下的伴随四元组, 13 | \[\begin{tikzcd}[ampersand replacement=\&] 14 | {\mathsf{Top}} \&\& {\mathsf {Set}} 15 | \arrow[""{name=0, anchor=center, inner sep=0}, "\Gamma"{description, pos=0.7}, shift right=2, from=1-1, to=1-3] 16 | \arrow[""{name=1, anchor=center, inner sep=0}, "{\operatorname{disc}}"{description, pos=0.7}, shift right=2, from=1-3, to=1-1] 17 | \arrow[""{name=2, anchor=center, inner sep=0}, "{\Pi_0}"{description, pos=0.7}, shift left=6, from=1-1, to=1-3] 18 | \arrow[""{name=3, anchor=center, inner sep=0}, "{\operatorname{codisc}}"{description, pos=0.7}, shift left=6, from=1-3, to=1-1] 19 | \arrow["\dashv"{anchor=center, rotate=-90}, draw=none, from=2, to=1] 20 | \arrow["\dashv"{anchor=center, rotate=-90}, draw=none, from=1, to=0] 21 | \arrow["\dashv"{anchor=center, rotate=-90}, draw=none, from=0, to=3] 22 | \end{tikzcd}\] 23 | 其中 24 | \begin{itemize} 25 | \item $\Pi_0$ 给出拓扑空间的\emph{连通分支的集合}; 26 | \item $\operatorname{disc}$ 将集合对应到\emph{离散空间}; 27 | \item $\Gamma$ 将拓扑空间遗忘为其\emph{底层集合}; 28 | \item $\operatorname{codisc}$ 将集合对应到\emph{余离散空间} (即只有空集和全集两个开集的拓扑空间). 29 | \end{itemize} 30 | 31 | 32 | 33 | \begin{definition} 34 | {(\cohesive{}\topos{})} 35 | \emph{\cohesive{}\topos{}} (cohesive topos)\footnotemark{} 是指一个\topos{} $\mathcal E$ 带有如下伴随四元组, 36 | % https://q.uiver.app/#q=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 37 | \[\begin{tikzcd}[ampersand replacement=\&] 38 | {\mathcal E} \&\& {\mathsf {Set}} 39 | \arrow[""{name=0, anchor=center, inner sep=0}, "\Gamma"{description, pos=0.7}, shift right=2, from=1-1, to=1-3] 40 | \arrow[""{name=1, anchor=center, inner sep=0}, "{\operatorname{disc}}"{description, pos=0.7}, shift right=2, from=1-3, to=1-1] 41 | \arrow[""{name=2, anchor=center, inner sep=0}, "{\Pi_0}"{description, pos=0.7}, shift left=6, from=1-1, to=1-3] 42 | \arrow[""{name=3, anchor=center, inner sep=0}, "{\operatorname{codisc}}"{description, pos=0.7}, shift left=6, from=1-3, to=1-1] 43 | \arrow["\dashv"{anchor=center, rotate=-90}, draw=none, from=2, to=1] 44 | \arrow["\dashv"{anchor=center, rotate=-90}, draw=none, from=1, to=0] 45 | \arrow["\dashv"{anchor=center, rotate=-90}, draw=none, from=0, to=3] 46 | \end{tikzcd}\] 47 | 使得 $\Pi_0$ 保持有限乘积. 48 | \end{definition} 49 | 50 | \begin{example} 51 | [label={cohesion-family-of-sets}] 52 | {(集合族)} 53 | % (例 \ref{family-of-sets-fibration}) 54 | % (例 \ref{varying-set-topos}) 55 | % (定义 \ref{Sierpinski-space}) 56 | 考虑 ``集合族范畴'' $\mathsf {Fam}=\mathsf{Fun}(\bullet\to\bullet,\mathsf{Set})$. 将 $\mathsf{Fam}$ 的对象 $W\to X$ 想象为一个大集合 $W$ 分成了 $X$ 那么多组, 每一组是这个映射的一个纤维. $\mathsf {Fam}$ 是一个\cohesive{}\topos{}, 其中 57 | \begin{itemize} 58 | \item $\Pi_0\colon \mathsf {Fam}\to\mathsf {Set}$, $(W\to X)\mapsto X$, ``将每一组捏成一个点''; 59 | \item $\operatorname{disc}\colon\mathsf {Set}\to\mathsf {Fam}, X\mapsto (\operatorname{id}\colon X\to X)$, ``将一个集合每个点当作一组''; 60 | \item $\Gamma \colon \mathsf {Fam}\to\mathsf {Set}$, $(W\to X)\mapsto W$, ``忘记分组''; 61 | \item $\operatorname{codisc}\colon \mathsf {Set}\to \mathsf {Fam}$, $X\mapsto (X\to \{*\})$, ``将一个集合整体当作一组''. 62 | \end{itemize} 63 | \end{example} 64 | 65 | \begin{example} 66 | {(单纯集)} 67 | 单纯集范畴 $\mathsf {sSet}$ 是一个\cohesive{}\topos{}, 其中 68 | \begin{itemize} 69 | \item $\Pi_0\colon \mathsf {sSet}\to\mathsf {Set}$, $X\mapsto \operatorname{coeq}(X_1\rightrightarrows X_0)$, 即 $X$ 的连通分支的集合; 70 | \item $\operatorname{disc}\colon\mathsf {Set}\to\mathsf {sSet}$, 将集合 $X$ 对应到常值单纯集 (也就是离散单纯集) $X$; 71 | \item $\Gamma \colon \mathsf {sSet}\to\mathsf {Set}$, $X\mapsto X_0 = \operatorname{Hom}(\Delta^0,X)$; 72 | \item $\operatorname{codisc}\colon \mathsf {Set}\to \mathsf {sSet}$, $\operatorname{codisc}(X)_n := X^{n+1}$. 73 | \end{itemize} 74 | \end{example} 75 | 76 | \begin{example} 77 | {(光滑空间)} 78 | 光滑空间范畴 $\mathsf {SmoothSp}=\operatorname{Sh}(\mathsf {CartSp})$ (例 \ref{cartsp-site}) 是一个\cohesive{}\topos{}, 其中 79 | \begin{itemize} 80 | \item $\Pi_0\colon \mathsf {SmoothSp}\to\mathsf {Set}$, $X\mapsto \operatorname{coeq}(X(\mathbb{R}^1)\rightrightarrows X(\mathbb{R}^0))$, 其中两个态射分别是层 $X$ 取值于 $0,1\colon \mathbb{R}^0\to \mathbb{R}^1$ (简而言之, $\Pi_0(X)$ 是 $X$ 的道路连通分支的集合); 81 | \item $\operatorname{disc}\colon\mathsf {Set}\to\mathsf {SmoothSp}$, 将集合 $X$ 对应到 ``离散光滑空间'' $X$; 82 | \item $\Gamma \colon \mathsf {SmoothSp}\to\mathsf {Set}$, $X\mapsto X(\mathbb{R}^0)\simeq\operatorname{Hom}_{\mathsf {SmoothSp}}(\mathbb{R}^0,X)$, 将光滑空间对应到其底层集合; 83 | \item $\operatorname{codisc}\colon \mathsf {Set}\to \mathsf {SmoothSp}$, $\operatorname{codisc}(X)(\mathbb{R}^n) := \operatorname{Hom}_{\mathsf {Set}}(\mathbb{R}^n,X)$. 84 | \end{itemize} 85 | \end{example} 86 | 87 | -------------------------------------------------------------------------------- /infty-foreword.tex: -------------------------------------------------------------------------------- 1 | \chapter{前言} 2 | 3 | 本书是\topos{}理论讲义 ``盲人摸象'' 的续篇, 讲述 $\infty$-\topos{}理论, 即\topos{}理论的 $\infty$-范畴版本. 4 | 5 | 6 | 7 | 8 | 范畴论的大部分内容 (包括\topos{}理论) 都有在 $\infty$-范畴中的类比, 但后者包含许多新的现象, 这些新内容是本书的重点. 9 | 10 | 11 | -------------------------------------------------------------------------------- /infty-topos.aux: -------------------------------------------------------------------------------- 1 | \relax 2 | \providecommand\hyper@newdestlabel[2]{} 3 | \providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} 4 | \HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined 5 | \global\let\oldnewlabel\newlabel 6 | \gdef\newlabel#1#2{\newlabelxx{#1}#2} 7 | \gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}} 8 | 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giveni={U\bibinitperiod}}}% 119 | } 120 | \list{publisher}{1}{% 121 | {(尚未出版)}% 122 | } 123 | \strng{namehash}{22bb4311878b35b2db2b84b4f2ade791} 124 | \strng{fullhash}{22bb4311878b35b2db2b84b4f2ade791} 125 | \strng{bibnamehash}{22bb4311878b35b2db2b84b4f2ade791} 126 | \strng{authorbibnamehash}{22bb4311878b35b2db2b84b4f2ade791} 127 | \strng{authornamehash}{22bb4311878b35b2db2b84b4f2ade791} 128 | \strng{authorfullhash}{22bb4311878b35b2db2b84b4f2ade791} 129 | \field{sortinit}{S} 130 | \field{sortinithash}{b164b07b29984b41daf1e85279fbc5ab} 131 | \field{labelnamesource}{author} 132 | \field{labeltitlesource}{title} 133 | \field{title}{Diifferential Cohomology in a Cohesive Topos} 134 | \verb{urlraw} 135 | \verb https://ncatlab.org/schreiber/files/dcct170811.pdf 136 | \endverb 137 | \verb{url} 138 | \verb https://ncatlab.org/schreiber/files/dcct170811.pdf 139 | \endverb 140 | \endentry 141 | \entry{hottbook}{book}{} 142 | \name{author}{1}{}{% 143 | {{hash=eb3cec164c1f4cb5cd7394d2677e2a1c}{% 144 | family={{Univalent Foundations Program}}, 145 | familyi={U\bibinitperiod}, 146 | given={The}, 147 | giveni={T\bibinitperiod}}}% 148 | } 149 | \list{location}{1}{% 150 | {Institute for Advanced Study}% 151 | } 152 | \list{publisher}{1}{% 153 | {\url{https://homotopytypetheory.org/book}}% 154 | } 155 | \strng{namehash}{eb3cec164c1f4cb5cd7394d2677e2a1c} 156 | \strng{fullhash}{eb3cec164c1f4cb5cd7394d2677e2a1c} 157 | \strng{bibnamehash}{eb3cec164c1f4cb5cd7394d2677e2a1c} 158 | \strng{authorbibnamehash}{eb3cec164c1f4cb5cd7394d2677e2a1c} 159 | \strng{authornamehash}{eb3cec164c1f4cb5cd7394d2677e2a1c} 160 | \strng{authorfullhash}{eb3cec164c1f4cb5cd7394d2677e2a1c} 161 | \field{sortinit}{U} 162 | \field{sortinithash}{6901a00e45705986ee5e7ca9fd39adca} 163 | \field{labelnamesource}{author} 164 | \field{labeltitlesource}{title} 165 | \field{title}{Homotopy Type Theory: Univalent Foundations of Mathematics} 166 | \field{year}{2013} 167 | \endentry 168 | \enddatalist 169 | \endrefsection 170 | \endinput 171 | 172 | -------------------------------------------------------------------------------- /infty-topos.bib: -------------------------------------------------------------------------------- 1 | @misc{HTTP, 2 | author={Urs Schreiber}, 3 | title={Higher Topos Theory in Physics}, 4 | book={Encyclopedia of Mathematical Physics 2nd ed (尚未出版)}, 5 | year={2023}, 6 | url={https://ncatlab.org/schreiber/show/Higher+Topos+Theory+in+Physics} 7 | } 8 | 9 | @book{HTT, 10 | author = {Jacob Lurie}, 11 | title = {Higher Topos Theory}, 12 | publisher= {Princeton University Press}, 13 | year = {2009} 14 | } 15 | 16 | @book{DCCT, 17 | author = {Urs Schreiber}, 18 | title = {Diifferential Cohomology in a Cohesive Topos}, 19 | publisher = {(尚未出版)}, 20 | url = {https://ncatlab.org/schreiber/files/dcct170811.pdf} 21 | } 22 | 23 | 24 | @misc{FHC, 25 | author = {Denis-Charles Cisinski}, 26 | title = {Formalization of Higher Category Theory}, 27 | note = {记录人: Bastiaan Cnossen}, 28 | url = {https://elearning.uni-regensburg.de/course/view.php?id=64170}, 29 | type = {课程笔记}, 30 | year = {2023} 31 | } 32 | 33 | @misc{riehl2023typetheorysyntheticinftycategories, 34 | title={A type theory for synthetic $\infty$-categories}, 35 | author={Emily Riehl and Michael Shulman}, 36 | year={2023}, 37 | eprint={1705.07442}, 38 | archivePrefix={arXiv}, 39 | primaryClass={math.CT}, 40 | url={https://arxiv.org/abs/1705.07442}, 41 | } 42 | 43 | @Book{hottbook, 44 | author = {The {Univalent Foundations Program}}, 45 | title = {Homotopy Type Theory: Univalent Foundations of Mathematics}, 46 | publisher = {\url{https://homotopytypetheory.org/book}}, 47 | address = {Institute for Advanced Study}, 48 | year = 2013} -------------------------------------------------------------------------------- /infty-topos.blg: -------------------------------------------------------------------------------- 1 | [0] Config.pm:307> INFO - This is Biber 2.19 2 | [0] Config.pm:310> INFO - Logfile is 'infty-topos.blg' 3 | [97] biber-MSWIN64:340> INFO - === 4 | [133] Biber.pm:419> INFO - Reading 'infty-topos.bcf' 5 | [214] Biber.pm:979> INFO - Found 5 citekeys in bib section 0 6 | [224] Biber.pm:4419> INFO - Processing section 0 7 | [234] Biber.pm:4610> INFO - Looking for bibtex file 'infty-topos.bib' for section 0 8 | [286] bibtex.pm:1713> INFO - LaTeX decoding ... 9 | [295] bibtex.pm:1519> INFO - Found BibTeX data source 'infty-topos.bib' 10 | [329] UCollate.pm:68> INFO - Overriding locale 'en-US' defaults 'variable = shifted' with 'variable = non-ignorable' 11 | [329] UCollate.pm:68> INFO - Overriding locale 'en-US' defaults 'normalization = NFD' with 'normalization = prenormalized' 12 | [329] Biber.pm:4239> INFO - Sorting list 'nty/global//global/global' of type 'entry' with template 'nty' and locale 'en-US' 13 | [329] Biber.pm:4245> INFO - No sort tailoring available for locale 'en-US' 14 | [337] bbl.pm:660> INFO - Writing 'infty-topos.bbl' with encoding 'UTF-8' 15 | [339] 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16 17 | -------------------------------------------------------------------------------- /infty-topos.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/SimplicialCat/topos/59a726f343732ec6c90a22ae3c1f4d5fc29fe663/infty-topos.pdf -------------------------------------------------------------------------------- /infty-topos.run.xml: -------------------------------------------------------------------------------- 1 | 2 | 3 | 4 | 5 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 23 | 28 | 33 | 36 | 39 | 42 | ]> 43 | 44 | 45 | latex 46 | 47 | infty-topos.bcf 48 | 49 | 50 | infty-topos.bbl 51 | 52 | 53 | blx-dm.def 54 | blx-unicode.def 55 | blx-compat.def 56 | biblatex.def 57 | standard.bbx 58 | numeric.bbx 59 | numeric.cbx 60 | biblatex.cfg 61 | english.lbx 62 | 63 | 64 | 65 | biber 66 | 67 | biber 68 | infty-topos 69 | 70 | 71 | infty-topos.bcf 72 | 73 | 74 | infty-topos.bbl 75 | 76 | 77 | infty-topos.bbl 78 | 79 | 80 | infty-topos.bcf 81 | 82 | 83 | infty-topos.bib 84 | 85 | 86 | 87 | -------------------------------------------------------------------------------- /infty-topos.synctex.gz: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/SimplicialCat/topos/59a726f343732ec6c90a22ae3c1f4d5fc29fe663/infty-topos.synctex.gz -------------------------------------------------------------------------------- /infty-topos.tex: -------------------------------------------------------------------------------- 1 | % 2 | 3 | \documentclass{book} 4 | \usepackage[UTF8]{ctex} 5 | \usepackage{fontspec} 6 | %\usepackage{xeCJK} 7 | \newCJKfontfamily\mincho{IPAexMincho} 8 | \setCJKmainfont{SimSun}[ItalicFont=KaiTi, BoldFont=SimHei] 9 | 10 | % SimSun 11 | % STSong 12 | % SimHei 13 | % KaiTi 14 | % DengXian 15 | 16 | \def\topos{意象} 17 | \def\nlab{$n$Lab} 18 | \def\regular{常理} 19 | \def\coherent{贯理} 20 | \def\cohesive{凝集} 21 | \def\cohesion{凝集} 22 | 23 | \newcommand{\tomeun}{{卷 1 }} 24 | 25 | \title{\textbf{\huge 盲人摸象 ($\infty$)}\\\emph{$\infty$-\topos{}理论讲义}} 26 | \author{\emph{王进一}\\\href{我的邮箱}{jin12003@163.com}\\QQ 2917905525} 27 | \date{2024 年夏至今\\~\\~此版本编译时间: \today{}~\\~\\这是一本正在施工的讲义. 目前我迫切需要读者的意见! 28 | \\~\\ 29 | %在非正式版本中, 我故意将 topos 译为 ``\topos{}''. 30 | } 31 | 32 | \usepackage{amsthm} 33 | \usepackage{amsmath} 34 | \usepackage{amssymb} 35 | \usepackage{mathrsfs} 36 | \usepackage{hyperref} 37 | \usepackage{stmaryrd} 38 | 39 | %\usepackage{wrapfig} 40 | 41 | \usepackage{epigraph} 42 | 43 | \usepackage{tikz} 44 | \usetikzlibrary{cd} 45 | \usetikzlibrary{decorations.pathmorphing} 46 | 47 | \usepackage{enumerate} 48 | 49 | \setcounter{secnumdepth}{1} 50 | \setcounter{tocdepth}{1} 51 | 52 | % 哲思 53 | 54 | \newcommand{\philoquote}[2]{ 55 | ~\\ 56 | \begin{center} 57 | \begin{minipage}{0.7\linewidth} 58 | {\quad\sffamily #1}\\ 59 | \begin{flushright} 60 | \textsf{#2} 61 | \end{flushright} 62 | \end{minipage} 63 | \end{center} 64 | ~\\ 65 | } 66 | 67 | % 彩色方框 68 | \input{colorboxes-printing} 69 | 70 | % 分栏 71 | \usepackage{multicol} 72 | 73 | % 页边距 74 | \usepackage{geometry} 75 | \geometry{ 76 | b5paper, 77 | left=15mm, 78 | right=15mm, 79 | top=20mm, 80 | bottom=20mm 81 | } 82 | %\renewcommand{\baselinestretch}{0.9} 83 | 84 | % 页眉与页脚样式 85 | \usepackage{fancyhdr} 86 | 87 | \newcommand{\chaptert}{} 88 | \newcommand{\sectiont}{} 89 | 90 | \renewcommand{\chaptermark}[1]{\renewcommand{\chaptert}{第\ \thechapter\ 章\ #1}} 91 | \renewcommand{\sectionmark}[1]{\renewcommand{\sectiont}{\thesection\ #1}} 92 | 93 | \fancypagestyle{plain}{ 94 | \fancyhf{} 95 | \renewcommand{\headrulewidth}{0pt} 96 | \renewcommand{\footrulewidth}{0pt} 97 | \fancyfoot[LE,RO]{\thepage} 98 | } 99 | 100 | \fancypagestyle{Jfancy}{ 101 | \fancyhead[LE]{\chaptert{}} 102 | \fancyhead[RO]{\sectiont{}} 103 | \fancyhead[RE,LO]{} 104 | \fancyfoot[C]{} 105 | \fancyfoot[LE,RO]{\thepage} 106 | \renewcommand{\headrulewidth}{0.4pt}% Line at the header visible 107 | \renewcommand{\footrulewidth}{0pt}% Line at the footer visible 108 | } 109 | 110 | \pagestyle{Jfancy} 111 | 112 | % 参考文献 113 | \usepackage{biblatex} 114 | \addbibresource{infty-topos.bib} 115 | 116 | % 跨文档引用 (引用第一卷) 117 | \usepackage{xr} 118 | \externaldocument[t1-]{topos} 119 | 120 | % 章节样式 121 | \usepackage{titlesec} 122 | \titleformat{\chapter}{\huge\bfseries}{第\, \thechapter\, 章}{1em}{} 123 | 124 | \usepackage{minitoc} 125 | \renewcommand{\mtctitle}{} 126 | 127 | \usepackage{multirow} 128 | 129 | % 常用记号 130 | 131 | \newcommand{\op}{\text{op}} % 对偶范畴 132 | \newcommand{\yo}{\!\text{{\mincho よ}}} % 米田嵌入 133 | \newcommand{\Top}{\mathcal T\hspace{-3pt}opos} 134 | \newcommand{\interpretation}[1]{{[\![#1]\!]}} 135 | \newcommand{\upward}[1]{\uparrow{\!}{#1}} 136 | \newcommand{\Psh}[1]{\mathsf{Psh}({#1})} 137 | 138 | \newcommand{\Ho}{\mathrm{Ho}} 139 | \newcommand{\infCatinfcat}{{\infty\mathcal {C}\hspace{-1pt}at}} 140 | \newcommand{\infGpdinfcat}{{\infty\mathcal {G}\hspace{-1pt}pd}} 141 | \newcommand{\Grpdinf}{\mathsf{Grpd}_{\infty}} 142 | 143 | % sequent calculus 144 | \newcommand{\sqc}[2]{\dfrac{\quad #1 \quad}{\quad #2 \quad}} 145 | \newcommand{\sqqc}[2]{ 146 | \begin{array} 147 | {c} 148 | #1 \\ \hline \hline #2 149 | \end{array} 150 | } 151 | 152 | \newcommand{\todo}[1]{{\color{red} [\textbf{未完成: #1}]}} 153 | \newcommand{\internalprop}[1]{{\ulcorner {#1} \urcorner}} 154 | 155 | \begin{document} 156 | \dominitoc 157 | 158 | \maketitle 159 | 160 | \tableofcontents 161 | 162 | \setcounter{chapter}{-1} % 这样下面一章就是第 0 章 163 | 164 | % 第零章 前言 165 | \input{infty-foreword} 166 | 167 | \input{infty-lan} 168 | 169 | \input{infty-cat} 170 | 171 | \input{infty-coh} 172 | 173 | \input{infty-bun} 174 | 175 | \input{infty-cohesion} 176 | 177 | \appendix 178 | 179 | % 附录 范畴论 180 | \input{infty-cat-appendix} 181 | 182 | %\input{infty-glossaries} 183 | 184 | \printbibliography[title=参考文献] 185 | 186 | \end{document} 187 | -------------------------------------------------------------------------------- /infty-topos.toc: -------------------------------------------------------------------------------- 1 | \contentsline {chapter}{\numberline {0}前言}{5}{chapter.0}% 2 | \contentsline {chapter}{\numberline {1}$\infty $-范畴的语言}{7}{chapter.1}% 3 | \contentsline {section}{\numberline {1.0}同伦类型论基础}{8}{section.1.0}% 4 | \contentsline {subsection}{集合, 逻辑与截断性}{11}{tcb@cnt@remark.1.0.12}% 5 | \contentsline {section}{\numberline {1.1}基本概念}{12}{section.1.1}% 6 | \contentsline {section}{\numberline {1.2}$\infty $-范畴中的结构与性质}{13}{section.1.2}% 7 | \contentsline {subsection}{连通性与截断性}{13}{section.1.2}% 8 | \contentsline {subsection}{$n$-范畴}{14}{tcb@cnt@remark.1.2.3}% 9 | \contentsline {subsection}{东西, 结构, 性质}{15}{tcb@cnt@remark.1.2.6}% 10 | \contentsline {subsection}{伴随}{15}{tcb@cnt@remark.1.2.6}% 11 | \contentsline {subsection}{极限与余极限}{16}{tcb@cnt@remark.1.2.8}% 12 | \contentsline {subsection}{$\operatorname {Hom}$ 函子, 预层与 $\infty $-米田引理}{19}{tcb@cnt@remark.1.2.15}% 13 | \contentsline {chapter}{\numberline {2}$\infty $-层与 Grothendieck $\infty $-意象{}}{21}{chapter.2}% 14 | \contentsline {section}{\numberline {2.1}Grothendieck 拓扑与层}{21}{section.2.1}% 15 | \contentsline {section}{\numberline {2.2}Giraud 定理}{22}{section.2.2}% 16 | \contentsline {chapter}{\numberline {3}$\infty $-意象{}与上同调}{23}{chapter.3}% 17 | \contentsline {subsection}{空间的奇异上同调}{23}{tcb@cnt@remark.3.0.1}% 18 | \contentsline {subsection}{等变上同调}{23}{tcb@cnt@remark.3.0.1}% 19 | \contentsline {subsection}{群上同调}{23}{tcb@cnt@remark.3.0.1}% 20 | \contentsline {subsection}{层上同调}{23}{tcb@cnt@remark.3.0.1}% 21 | \contentsline {chapter}{\numberline {4}$\infty $-意象{}与 $\infty $-丛}{25}{chapter.4}% 22 | \contentsline {chapter}{\numberline {5}凝集{}意象{}}{27}{chapter.5}% 23 | \contentsline {section}{\numberline {5.1}凝集{}的动机, 基本概念}{27}{section.5.1}% 24 | \contentsline {chapter}{\numberline {A}$\infty $-范畴论的补充知识}{31}{appendix.A}% 25 | \contentsline 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https://raw.githubusercontent.com/SimplicialCat/topos/59a726f343732ec6c90a22ae3c1f4d5fc29fe663/main.pdf -------------------------------------------------------------------------------- /main.run.xml: -------------------------------------------------------------------------------- 1 | 2 | 3 | 4 | 5 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 23 | 28 | 33 | 36 | 39 | 42 | ]> 43 | 44 | 45 | latex 46 | 47 | main.bcf 48 | 49 | 50 | main.bbl 51 | 52 | 53 | blx-dm.def 54 | blx-unicode.def 55 | blx-compat.def 56 | biblatex.def 57 | standard.bbx 58 | numeric.bbx 59 | numeric.cbx 60 | biblatex.cfg 61 | english.lbx 62 | 63 | 64 | 65 | biber 66 | 67 | biber 68 | main 69 | 70 | 71 | main.bcf 72 | 73 | 74 | main.bbl 75 | 76 | 77 | main.bbl 78 | 79 | 80 | main.bcf 81 | 82 | 83 | Topos.bib 84 | 85 | 86 | 87 | -------------------------------------------------------------------------------- /main.synctex.gz: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/SimplicialCat/topos/59a726f343732ec6c90a22ae3c1f4d5fc29fe663/main.synctex.gz -------------------------------------------------------------------------------- /main.tex: -------------------------------------------------------------------------------- 1 | % 2 | 3 | \documentclass{book} 4 | \usepackage[UTF8]{ctex} 5 | \usepackage{fontspec} 6 | %\usepackage{xeCJK} 7 | \newCJKfontfamily\mincho{IPAexMincho} 8 | \setCJKmainfont{SimSun}[ItalicFont=KaiTi, BoldFont=SimHei] 9 | 10 | % SimSun 11 | % STSong 12 | % SimHei 13 | % KaiTi 14 | % DengXian 15 | 16 | \def\topos{意象} 17 | \def\nlab{$n$Lab} 18 | 19 | \title{\textbf{\huge 盲人摸象}\\\emph{\topos{}理论讲义}} 20 | \author{\emph{王进一}\\\href{我的邮箱}{jin12003@163.com}\\QQ 2917905525} 21 | \date{2023 年夏至今\\~\\~此版本编译时间: \today{}~\\~\\这是一本正在施工的讲义. 目前我迫切需要读者的意见! 22 | \\~\\ 23 | %在非正式版本中, 我故意将 topos 译为 ``\topos{}''. 24 | } 25 | 26 | \usepackage{amsthm} 27 | \usepackage{amsmath} 28 | \usepackage{amssymb} 29 | \usepackage{mathrsfs} 30 | \usepackage{hyperref} 31 | \usepackage{stmaryrd} 32 | 33 | %\usepackage{wrapfig} 34 | 35 | \usepackage{epigraph} 36 | 37 | \usepackage{tikz} 38 | \usetikzlibrary{cd} 39 | \usetikzlibrary{decorations.pathmorphing} 40 | 41 | \usepackage{enumerate} 42 | 43 | \setcounter{secnumdepth}{1} 44 | \setcounter{tocdepth}{1} 45 | 46 | % 哲思 47 | 48 | \newcommand{\philoquote}[2]{ 49 | ~\\ 50 | \begin{center} 51 | \begin{minipage}{0.7\linewidth} 52 | {\quad\sffamily #1}\\ 53 | \begin{flushright} 54 | \textsf{#2} 55 | \end{flushright} 56 | \end{minipage} 57 | \end{center} 58 | ~\\ 59 | } 60 | 61 | % 彩色方框 62 | \input{colorboxes-printing} 63 | 64 | % 分栏 65 | \usepackage{multicol} 66 | 67 | % 页边距 68 | \usepackage{geometry} 69 | \geometry{ 70 | b5paper, 71 | left=15mm, 72 | right=15mm, 73 | top=20mm, 74 | bottom=20mm 75 | } 76 | %\renewcommand{\baselinestretch}{0.9} 77 | 78 | % 页眉与页脚样式 79 | \usepackage{fancyhdr} 80 | 81 | \newcommand{\chaptert}{} 82 | \newcommand{\sectiont}{} 83 | 84 | \renewcommand{\chaptermark}[1]{\renewcommand{\chaptert}{第\ \thechapter\ 章\ #1}} 85 | \renewcommand{\sectionmark}[1]{\renewcommand{\sectiont}{\thesection\ #1}} 86 | 87 | \fancypagestyle{plain}{ 88 | \fancyhf{} 89 | \renewcommand{\headrulewidth}{0pt} 90 | \renewcommand{\footrulewidth}{0pt} 91 | \fancyfoot[LE,RO]{\thepage} 92 | } 93 | 94 | \fancypagestyle{Jfancy}{ 95 | \fancyhead[LE]{\chaptert{}} 96 | \fancyhead[RO]{\sectiont{}} 97 | \fancyhead[RE,LO]{} 98 | \fancyfoot[C]{} 99 | \fancyfoot[LE,RO]{\thepage} 100 | \renewcommand{\headrulewidth}{0.4pt}% Line at the header visible 101 | \renewcommand{\footrulewidth}{0pt}% Line at the footer visible 102 | } 103 | 104 | \pagestyle{Jfancy} 105 | 106 | % 参考文献 107 | \usepackage{biblatex} 108 | \addbibresource{Topos.bib} 109 | 110 | % 章节样式 111 | \usepackage{titlesec} 112 | \titleformat{\chapter}{\huge\bfseries}{第\, \thechapter\, 章}{1em}{} 113 | 114 | \usepackage{minitoc} 115 | \renewcommand{\mtctitle}{} 116 | 117 | % 常用记号 118 | 119 | \newcommand{\op}{\text{op}} % 对偶范畴 120 | \newcommand{\yo}{\!\text{{\mincho よ}}} % 米田嵌入 121 | \newcommand{\Top}{\mathcal T\hspace{-3pt}opos} 122 | \newcommand{\interpretation}[1]{{[\![#1]\!]}} 123 | 124 | % sequent calculus 125 | \newcommand{\sqc}[2]{\dfrac{\quad #1 \quad}{\quad #2 \quad}} 126 | \newcommand{\sqqc}[2]{ 127 | \begin{array} 128 | {c} 129 | #1 \\ \hline \hline #2 130 | \end{array} 131 | } 132 | 133 | \newcommand{\todo}[1]{{\color{red} [\textbf{未完成: #1}]}} 134 | \newcommand{\internalprop}[1]{{\ulcorner {#1} \urcorner}} 135 | 136 | \begin{document} 137 | \dominitoc 138 | 139 | \maketitle 140 | 141 | \tableofcontents 142 | 143 | \setcounter{chapter}{-1} % 这样下面一章就是第 0 章 144 | 145 | % 第零章 前言 146 | \input{foreword} 147 | 148 | % 第一章 范畴论 149 | \input{cat} 150 | 151 | % 第二章 位象理论 152 | \input{loc} 153 | 154 | % 第三章 Grothendieck \topos{} 155 | \input{Gro} 156 | 157 | % 第四章 内语言 (意象与逻辑?) 158 | \input{lan} 159 | 160 | % 第五章 分类\topos{} 161 | \input{cla} 162 | 163 | % 第六章 相对\topos{} 164 | \input{rel} 165 | 166 | % 第七章 高阶范畴与高阶\topos{} 167 | \input{higher} 168 | 169 | % 第八章 凝聚\topos{} 170 | \input{cohesion} 171 | 172 | \appendix 173 | 174 | % 附录 范畴论 175 | \input{cat-appendix} 176 | 177 | % 附录 形式逻辑 178 | \input{logic-appendix} 179 | 180 | %\input{colimit-appendix} 181 | 182 | \printbibliography[title=参考文献] 183 | 184 | \end{document} 185 | -------------------------------------------------------------------------------- /main.toc: -------------------------------------------------------------------------------- 1 | \contentsline {chapter}{\numberline {0}前言}{7}{chapter.0}% 2 | \contentsline {chapter}{\numberline {1}意象{}的范畴论性质}{9}{chapter.1}% 3 | \contentsline {section}{\numberline {1.1}范畴论基本概念}{10}{section.1.1}% 4 | \contentsline {subsection}{极限与余极限}{10}{section.1.1}% 5 | \contentsline {subsection}{指数对象与积闭范畴}{11}{tcb@cnt@remark.1.1.1}% 6 | \contentsline {subsection}{子对象分类子}{14}{tcb@cnt@remark.1.1.10}% 7 | \contentsline {subsection}{幂对象}{18}{tcb@cnt@remark.1.1.24}% 8 | \contentsline {subsection}{俯范畴与局部积闭性}{20}{tcb@cnt@remark.1.1.26}% 9 | \contentsline {section}{\numberline {1.2}意象{}}{25}{section.1.2}% 10 | \contentsline {section}{\numberline {1.3}更多范畴论结构}{27}{section.1.3}% 11 | \contentsline {subsection}{0 和 1}{27}{section.1.3}% 12 | \contentsline {subsection}{单射与满射}{28}{tcb@cnt@remark.1.3.3}% 13 | \contentsline {subsection}{正则单射与满射, 等价关系}{28}{tcb@cnt@remark.1.3.7}% 14 | \contentsline {subsection}{像}{31}{tcb@cnt@remark.1.3.17}% 15 | \contentsline {subsection}{满--单分解}{33}{tcb@cnt@remark.1.3.23}% 16 | \contentsline {subsection}{子终对象}{34}{tcb@cnt@remark.1.3.25}% 17 | \contentsline {subsection}{子对象的格与 Heyting 代数}{36}{tcb@cnt@remark.1.3.31}% 18 | \contentsline {subsection}{自然数对象}{42}{tcb@cnt@remark.1.3.55}% 19 | \contentsline {subsection}{无交和}{43}{tcb@cnt@remark.1.3.59}% 20 | \contentsline {subsection}{Boole 意象{}与选择公理}{45}{tcb@cnt@remark.1.3.62}% 21 | \contentsline {chapter}{\numberline {2}位象: 无点拓扑学}{51}{chapter.2}% 22 | \contentsline {section}{\numberline {2.1}基本概念}{52}{section.2.1}% 23 | \contentsline {section}{\numberline {2.2}位象的几何性质}{57}{section.2.2}% 24 | \contentsline {subsection}{子位象}{57}{section.2.2}% 25 | \contentsline {subsubsection}{开子位象与闭子位象}{60}{tcb@cnt@remark.2.2.5}% 26 | \contentsline {subsubsection}{子位象与内核{}}{61}{tcb@cnt@remark.2.2.8}% 27 | \contentsline {subsection}{Boole 位象}{62}{tcb@cnt@remark.2.2.13}% 28 | \contentsline {subsection}{位象的满射}{65}{tcb@cnt@remark.2.2.19}% 29 | \contentsline {subsection}{开映射}{66}{tcb@cnt@remark.2.2.22}% 30 | \contentsline {subsection}{局部位象}{68}{tcb@cnt@remark.2.2.27}% 31 | \contentsline {subsection}{局部连通位象}{68}{tcb@cnt@remark.2.2.31}% 32 | \contentsline {section}{\numberline {2.3}位象与逻辑}{69}{section.2.3}% 33 | \contentsline {subsection}{经典命题逻辑与 Boole 代数}{69}{section.2.3}% 34 | \contentsline {subsection}{几何逻辑与位格{}}{73}{tcb@cnt@remark.2.3.14}% 35 | \contentsline {chapter}{\numberline {3}意象{}与空间的概念}{77}{chapter.3}% 36 | \contentsline {section}{\numberline {3.1}拓扑空间上的层与平展空间}{79}{section.3.1}% 37 | \contentsline {subsection}{拓扑空间上层的直像与逆像}{83}{tcb@cnt@remark.3.1.14}% 38 | \contentsline {section}{\numberline {3.2}位象上的层与平展空间}{87}{section.3.2}% 39 | \contentsline {subsection}{位格{}取值的集合}{88}{tcb@cnt@remark.3.2.2}% 40 | \contentsline {section}{\numberline {3.3}范畴上的预层}{97}{section.3.3}% 41 | \contentsline {subsection}{筛与预层范畴中的子对象}{99}{tcb@cnt@remark.3.3.8}% 42 | \contentsline {subsubsection}{预层范畴的子对象分类子}{102}{tcb@cnt@remark.3.3.17}% 43 | \contentsline {section}{\numberline {3.4}景}{103}{section.3.4}% 44 | \contentsline {subsection}{从覆盖到 Grothendieck 拓扑}{104}{section.3.4}% 45 | \contentsline {subsection}{常见的景}{109}{tcb@cnt@remark.3.4.16}% 46 | \contentsline {subsection}{典范与次典范拓扑}{112}{tcb@cnt@remark.3.4.24}% 47 | \contentsline {section}{\numberline {3.5}层化与 Grothendieck $+$构造}{113}{section.3.5}% 48 | \contentsline {section}{\numberline {3.6}Grothendieck 意象{}}{116}{section.3.6}% 49 | \contentsline {subsection}{层范畴的性质}{116}{section.3.6}% 50 | \contentsline {subsubsection}{层范畴的子对象分类子}{117}{tcb@cnt@remark.3.6.2}% 51 | \contentsline {subsubsection}{层范畴中的指数对象}{119}{tcb@cnt@remark.3.6.6}% 52 | \contentsline {subsection}{Grothendieck 意象{}}{119}{tcb@cnt@remark.3.6.8}% 53 | \contentsline {subsection}{位象型意象{}}{120}{tcb@cnt@remark.3.6.12}% 54 | \contentsline {section}{\numberline {3.7}Lawvere--Tierney 拓扑, 内蕴层化与局部化}{121}{section.3.7}% 55 | \contentsline {subsection}{Lawvere--Tierney 拓扑}{121}{section.3.7}% 56 | \contentsline {subsection}{层范畴的性质}{125}{tcb@cnt@remark.3.7.10}% 57 | \contentsline {subsubsection}{层范畴中的有限极限}{125}{tcb@cnt@remark.3.7.10}% 58 | \contentsline {subsubsection}{层范畴中的子对象}{125}{tcb@cnt@remark.3.7.12}% 59 | \contentsline {subsubsection}{层范畴中的指数对象}{127}{tcb@cnt@remark.3.7.16}% 60 | \contentsline {subsection}{层化与局部化}{127}{tcb@cnt@remark.3.7.17}% 61 | \contentsline {section}{\numberline {3.8}意象{}之间的态射}{127}{section.3.8}% 62 | \contentsline {subsection}{几何态射}{127}{section.3.8}% 63 | \contentsline {subsection}{逻辑态射}{130}{tcb@cnt@remark.3.8.9}% 64 | \contentsline {subsection}{嵌入与满射}{130}{tcb@cnt@remark.3.8.11}% 65 | \contentsline {subsection}{满--单分解}{131}{tcb@cnt@remark.3.8.15}% 66 | \contentsline {subsection}{群作用与张量--同态伴随}{133}{tcb@cnt@remark.3.8.17}% 67 | \contentsline {section}{\numberline {3.9}景之间的态射}{137}{section.3.9}% 68 | \contentsline {subsection}{$\mathsf {Set}$-值平坦函子}{139}{tcb@cnt@remark.3.9.4}% 69 | \contentsline {subsection}{预层意象{}的点}{140}{tcb@cnt@remark.3.9.6}% 70 | \contentsline {subsection}{景取值的平坦函子}{141}{tcb@cnt@remark.3.9.9}% 71 | \contentsline {subsection}{层意象{}的点}{142}{tcb@cnt@remark.3.9.11}% 72 | \contentsline {subsection}{景之间的态射}{142}{tcb@cnt@remark.3.9.14}% 73 | \contentsline {subsection}{比较原理}{143}{tcb@cnt@remark.3.9.16}% 74 | \contentsline {section}{\numberline {3.10}意象{}的几何性质}{144}{section.3.10}% 75 | \contentsline {subsection}{平展性}{144}{section.3.10}% 76 | \contentsline {subsection}{连通性}{144}{tcb@cnt@remark.3.10.1}% 77 | \contentsline {subsection}{开几何态射}{146}{Item.55}% 78 | \contentsline {subsection}{本质几何态射}{146}{tcb@cnt@remark.3.10.6}% 79 | \contentsline {subsection}{紧合几何态射}{147}{tcb@cnt@remark.3.10.11}% 80 | \contentsline {section}{\numberline {3.11}Giraud 定理}{147}{section.3.11}% 81 | \contentsline {paragraph}{第一步, 景的构造}{148}{tcb@cnt@remark.3.11.2}% 82 | \contentsline {paragraph}{第二步, 层条件的验证}{149}{tcb@cnt@remark.3.11.2}% 83 | \contentsline {paragraph}{第三步, 范畴等价的证明}{149}{tcb@cnt@remark.3.11.2}% 84 | \contentsline {section}{\numberline {3.12}等变层与拓扑群胚}{150}{section.3.12}% 85 | \contentsline {chapter}{\numberline {4}意象{}的内语言}{153}{chapter.4}% 86 | \contentsline {section}{\numberline {4.1}Mitchell--B\'enabou 语言}{153}{section.4.1}% 87 | \contentsline {section}{\numberline {4.2}Kripke--Joyal 语义}{160}{section.4.2}% 88 | \contentsline {subsection}{层语义}{162}{Item.64}% 89 | \contentsline {section}{\numberline {4.3}模态与层化}{164}{section.4.3}% 90 | \contentsline {section}{\numberline {4.4}内位象}{167}{section.4.4}% 91 | \contentsline {chapter}{\numberline {5}语法景与分类意象{}}{169}{chapter.5}% 92 | \contentsline {section}{\numberline {5.1}语法范畴: 语法--语义对偶}{169}{section.5.1}% 93 | \contentsline {subsection}{类型论的语境范畴}{171}{tcb@cnt@remark.5.1.5}% 94 | \contentsline {subsection}{语法景}{172}{tcb@cnt@remark.5.1.6}% 95 | \contentsline {section}{\numberline {5.2}分类意象{}}{172}{section.5.2}% 96 | \contentsline {subsection}{$G$-旋子的分类意象{}}{173}{section.5.2}% 97 | \contentsline {subsection}{对象的分类意象}{175}{tcb@cnt@remark.5.2.6}% 98 | \contentsline {subsection}{子终对象的分类意象}{176}{tcb@cnt@remark.5.2.7}% 99 | \contentsline {subsection}{群的分类意象{}}{176}{tcb@cnt@remark.5.2.8}% 100 | \contentsline {subsection}{环的分类意象{}}{176}{tcb@cnt@remark.5.2.8}% 101 | \contentsline {subsection}{几何理论的分类意象{}}{176}{tcb@cnt@remark.5.2.8}% 102 | \contentsline {chapter}{\numberline {6}高阶意象{}}{177}{chapter.6}% 103 | \contentsline {section}{\numberline {6.1}$\infty $-范畴: 单纯集模型}{178}{section.6.1}% 104 | \contentsline {subsection}{同伦}{181}{tcb@cnt@remark.6.1.12}% 105 | \contentsline {subsection}{单纯范畴}{182}{tcb@cnt@remark.6.1.15}% 106 | \contentsline {subsection}{伴随}{185}{tcb@cnt@remark.6.1.25}% 107 | \contentsline {subsection}{极限与余极限}{186}{tcb@cnt@remark.6.1.27}% 108 | \contentsline {subsection}{$\operatorname {Hom}$ 函子, 预层与 $\infty $-米田引理}{189}{tcb@cnt@remark.6.1.34}% 109 | \contentsline {section}{\numberline {6.2}$\infty $-意象{}}{190}{section.6.2}% 110 | \contentsline {section}{\numberline {6.3}$\infty $-层 $\infty $-意象{}及其表现}{191}{section.6.3}% 111 | \contentsline {subsection}{Grothendieck 拓扑与层}{191}{section.6.3}% 112 | \contentsline {section}{\numberline {6.4}上同调}{191}{section.6.4}% 113 | \contentsline {subsection}{例}{192}{tcb@cnt@remark.6.4.1}% 114 | \contentsline {section}{\numberline {6.5}$n$-范畴}{192}{section.6.5}% 115 | \contentsline {subsection}{东西, 结构, 性质}{193}{tcb@cnt@remark.6.5.4}% 116 | \contentsline {chapter}{\numberline {7}凝集{}意象{}}{195}{chapter.7}% 117 | \contentsline {section}{\numberline {7.1}凝集{}的动机, 基本概念}{195}{section.7.1}% 118 | \contentsline {chapter}{\numberline {8}意象{}理论的应用}{199}{chapter.8}% 119 | \contentsline {section}{\numberline {8.1}非标准分析}{199}{section.8.1}% 120 | \contentsline {subsection}{基本概念}{199}{section.8.1}% 121 | \contentsline {subsection}{滤商}{200}{tcb@cnt@remark.8.1.2}% 122 | \contentsline {subsection}{超滤范畴}{200}{tcb@cnt@remark.8.1.2}% 123 | \contentsline {section}{\numberline {8.2}可计算性理论与有效意象}{200}{section.8.2}% 124 | \contentsline {subsection}{基础知识}{200}{section.8.2}% 125 | \contentsline {section}{\numberline {8.3}综合微分几何与光滑无穷小分析}{202}{section.8.3}% 126 | \contentsline {subsection}{综合微分几何的理论}{202}{section.8.3}% 127 | \contentsline {paragraph}{Kock--Lawvere 公理与导数}{203}{tcb@cnt@remark.8.3.5}% 128 | \contentsline {paragraph}{Weil 代数与无穷小几何对象}{203}{tcb@cnt@remark.8.3.7}% 129 | \contentsline {subsection}{综合微分几何的模型}{206}{tcb@cnt@remark.8.3.16}% 130 | \contentsline {subsubsection}{``代数'' 模型}{206}{tcb@cnt@remark.8.3.16}% 131 | \contentsline {subsubsection}{光滑代数}{208}{tcb@cnt@remark.8.3.23}% 132 | \contentsline {section}{\numberline {8.4}量子理论与 Bohr 意象}{210}{section.8.4}% 133 | \contentsline {subsection}{$C^*$-代数, 经典语境与 Bohr 景}{211}{section.8.4}% 134 | \contentsline {subsubsection}{偏序集上的层}{213}{tcb@cnt@remark.8.4.9}% 135 | \contentsline {subsection}{Bohr 意象}{214}{tcb@cnt@remark.8.4.13}% 136 | \contentsline {subsubsection}{Gelfand 对偶}{214}{tcb@cnt@remark.8.4.14}% 137 | \contentsline {subsection}{Bohr 意象中的命题}{215}{tcb@cnt@remark.8.4.19}% 138 | \contentsline {section}{\numberline {8.5}Cohen 力迫法}{215}{section.8.5}% 139 | \contentsline {subsection}{双重否定与稠密拓扑}{216}{section.8.5}% 140 | \contentsline {subsection}{意象{}中基数的比较}{217}{tcb@cnt@remark.8.5.4}% 141 | \contentsline {subsection}{连续统假设反例的构造}{218}{tcb@cnt@remark.8.5.6}% 142 | \contentsline {section}{\numberline {8.6}凝聚态数学}{218}{section.8.6}% 143 | \contentsline {chapter}{\numberline {A}范畴论基础}{221}{appendix.A}% 144 | \contentsline {section}{\numberline {A.1}$2$-范畴}{222}{section.A.1}% 145 | \contentsline {subsection}{$2$-范畴中的万有性质}{227}{tcb@cnt@remark.A.1.11}% 146 | \contentsline {subsection}{俯 $2$-范畴}{228}{tcb@cnt@remark.A.1.12}% 147 | \contentsline {section}{\numberline {A.2}伴随}{228}{section.A.2}% 148 | \contentsline {subsection}{伴随保持极限}{230}{tcb@cnt@remark.A.2.4}% 149 | \contentsline {subsection}{伴随的自然变换}{233}{tcb@cnt@remark.A.2.10}% 150 | \contentsline {subsection}{伴随三元组}{234}{tcb@cnt@remark.A.2.12}% 151 | \contentsline {subsection}{伴随函子的 Frobenius 互反律}{235}{tcb@cnt@remark.A.2.15}% 152 | \contentsline {section}{\numberline {A.3}自反子范畴与局部化}{236}{section.A.3}% 153 | \contentsline {subsection}{局部对象}{240}{tcb@cnt@remark.A.3.8}% 154 | \contentsline {subsection}{分式计算}{244}{tcb@cnt@remark.A.3.19}% 155 | \contentsline {section}{\numberline {A.4}预层范畴与米田嵌入}{247}{section.A.4}% 156 | \contentsline {subsection}{米田引理}{247}{section.A.4}% 157 | \contentsline {subsection}{可表函子的余极限}{248}{tcb@cnt@remark.A.4.4}% 158 | \contentsline {subsection}{自由余完备化, 脉与几何实现}{249}{tcb@cnt@remark.A.4.8}% 159 | \contentsline {subsection}{预层范畴的俯范畴}{251}{tcb@cnt@remark.A.4.15}% 160 | \contentsline {section}{\numberline {A.5}(余) 滤范畴和 (余) 滤 (余) 极限}{252}{section.A.5}% 161 | \contentsline {section}{\numberline {A.6}可表现范畴}{256}{section.A.6}% 162 | \contentsline {subsection}{可表现对象}{256}{section.A.6}% 163 | \contentsline {subsection}{稠密子范畴}{261}{tcb@cnt@remark.A.6.17}% 164 | \contentsline {subsection}{可表现范畴的性质与判定}{262}{tcb@cnt@remark.A.6.23}% 165 | \contentsline {subsection}{可表现范畴的伴随函子定理}{263}{Item.77}% 166 | \contentsline {section}{\numberline {A.7}Kan 扩张}{263}{section.A.7}% 167 | \contentsline {section}{\numberline {A.8}单子论}{265}{section.A.8}% 168 | \contentsline {section}{\numberline {A.9}万有代数}{272}{section.A.9}% 169 | \contentsline {subsection}{Lawvere 理论}{273}{section.A.9}% 170 | \contentsline {subsection}{模型的表现}{277}{tcb@cnt@remark.A.9.14}% 171 | \contentsline {subsection}{代数理论之间的态射}{278}{tcb@cnt@remark.A.9.16}% 172 | \contentsline {subsection}{单子与代数理论}{281}{Item.91}% 173 | \contentsline {section}{\numberline {A.10}纤维范畴与索引范畴}{281}{section.A.10}% 174 | \contentsline {subsection}{等变对象}{289}{tcb@cnt@remark.A.10.20}% 175 | \contentsline {section}{\numberline {A.11}下降}{289}{section.A.11}% 176 | \contentsline {section}{\numberline {A.12}内范畴}{292}{section.A.12}% 177 | \contentsline {chapter}{\numberline {B}形式逻辑基础}{295}{appendix.B}% 178 | \contentsline {section}{\numberline {B.1}一阶逻辑}{295}{section.B.1}% 179 | \contentsline {subsection}{一阶语言的基本要件}{297}{tcb@cnt@remark.B.1.4}% 180 | \contentsline {subsubsection}{符号表}{297}{tcb@cnt@remark.B.1.4}% 181 | \contentsline {subsubsection}{项, 公式}{299}{tcb@cnt@remark.B.1.10}% 182 | \contentsline {subsection}{一阶理论}{301}{tcb@cnt@remark.B.1.16}% 183 | \contentsline {section}{\numberline {B.2}一阶逻辑的范畴语义}{308}{section.B.2}% 184 | \contentsline {subsection}{一阶语言在范畴中的解释}{308}{section.B.2}% 185 | \contentsline {subsection}{一阶理论在范畴中的模型}{312}{tcb@cnt@remark.B.2.11}% 186 | \contentsline {subsection}{提纲}{312}{tcb@cnt@remark.B.2.15}% 187 | \contentsline {subsection}{教条}{313}{tcb@cnt@remark.B.2.19}% 188 | \contentsline {section}{\numberline {B.3}高阶逻辑}{315}{section.B.3}% 189 | \contentsline {subsection}{高阶语言的基本要件}{315}{section.B.3}% 190 | \contentsline {section}{\numberline {B.4}类型论}{318}{section.B.4}% 191 | \contentsline {subsection}{命题是类型: Curry--Howard 同构}{320}{tcb@cnt@remark.B.4.4}% 192 | \contentsline {subsection}{类型论的范畴语义}{321}{tcb@cnt@remark.B.4.4}% 193 | \contentsline {section}{\numberline {B.5}模态逻辑}{322}{section.B.5}% 194 | -------------------------------------------------------------------------------- /rel.log: -------------------------------------------------------------------------------- 1 | This is XeTeX, Version 3.141592653-2.6-0.999995 (TeX Live 2023) (preloaded format=xelatex 2023.4.3) 29 JAN 2024 09:00 2 | entering extended mode 3 | restricted \write18 enabled. 4 | %&-line parsing enabled. 5 | **rel.tex 6 | (./rel.tex 7 | LaTeX2e <2022-11-01> patch level 1 8 | L3 programming layer <2023-03-30> 9 | ! 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Otherwise just continue, 252 | and I'll forget about whatever was undefined. 253 | 254 | Missing character: There is no , ("2C) in font nullfont! 255 | Missing character: There is no 为 ("4E3A) in font nullfont! 256 | Missing character: There is no 的 ("7684) in font nullfont! 257 | Missing character: There is no 内 ("5185) in font nullfont! 258 | Missing character: There is no 蕴 ("8574) in font nullfont! 259 | Missing character: There is no 范 ("8303) in font nullfont! 260 | Missing character: There is no 畴 ("7574) in font nullfont! 261 | Missing character: There is no , ("2C) in font nullfont! 262 | Missing character: There is no 为 ("4E3A) in font nullfont! 263 | Missing character: There is no 相 ("76F8) in font nullfont! 264 | Missing character: There is no 对 ("5BF9) in font nullfont! 265 | ! Undefined control sequence. 266 | l.10 ... $p\colon \mathsf E\to\mathsf S$ 为相对\topos 267 | {}. 那么有范畴等价 268 | The control sequence at the end of the top line 269 | of your error message was never \def'ed. If you have 270 | misspelled it (e.g., `\hobx'), type `I' and the correct 271 | spelling (e.g., `I\hbox'). Otherwise just continue, 272 | and I'll forget about whatever was undefined. 273 | 274 | Missing character: There is no . ("2E) in font nullfont! 275 | Missing character: There is no 那 ("90A3) in font nullfont! 276 | Missing character: There is no 么 ("4E48) in font nullfont! 277 | Missing character: There is no 有 ("6709) in font nullfont! 278 | Missing character: There is no 范 ("8303) in font nullfont! 279 | Missing character: There is no 畴 ("7574) in font nullfont! 280 | Missing character: There is no 等 ("7B49) in font nullfont! 281 | Missing character: There is no 价 ("4EF7) in font nullfont! 282 | 283 | Overfull \hbox (20.0pt too wide) in paragraph at lines 9--11 284 | [] 285 | [] 286 | 287 | 288 | Overfull \hbox (5.55557pt too wide) in paragraph at lines 9--11 289 | \OT1/cmss/m/n/10 S$ 290 | [] 291 | 292 | 293 | Overfull \hbox (5.84865pt too wide) in paragraph at lines 9--11 294 | \OMS/cmsy/m/n/10 C$ 295 | [] 296 | 297 | 298 | Overfull \hbox (5.55557pt too wide) in paragraph at lines 9--11 299 | \OT1/cmss/m/n/10 S$ 300 | [] 301 | 302 | 303 | Overfull \hbox (28.22562pt too wide) in paragraph at lines 9--11 304 | \OML/cmm/m/it/10 p\OT1/cmr/m/n/10 : \OT1/cmss/m/n/10 E \OMS/cmsy/m/n/10 ! 305 | [] 306 | 307 | 308 | Overfull \hbox (5.55557pt too wide) in paragraph at lines 9--11 309 | \OT1/cmss/m/n/10 S$ 310 | [] 311 | 312 | ! Undefined control sequence. 313 | l.12 ^^I\operatorname 314 | {Hom}_{\Top/\mathsf S} 315 | The control sequence at the end of the top line 316 | of your error message was never \def'ed. If you have 317 | misspelled it (e.g., `\hobx'), type `I' and the correct 318 | spelling (e.g., `I\hbox'). Otherwise just continue, 319 | and I'll forget about whatever was undefined. 320 | 321 | ! Undefined control sequence. 322 | l.12 ^^I\operatorname{Hom}_{\Top 323 | /\mathsf S} 324 | The control sequence at the end of the top line 325 | of your error message was never \def'ed. If you have 326 | misspelled it (e.g., `\hobx'), type `I' and the correct 327 | spelling (e.g., `I\hbox'). Otherwise just continue, 328 | and I'll forget about whatever was undefined. 329 | 330 | 331 | Overfull \hbox (147.15105pt too wide) detected at line 15 332 | [][]\OT1/cmr/m/n/10 (\OT1/cmss/m/n/10 E\OML/cmm/m/it/10 ; []\OT1/cmr/m/n/10 (\O 333 | MS/cmsy/m/n/10 C\OML/cmm/m/it/10 ; \OT1/cmss/m/n/10 S\OT1/cmr/m/n/10 )) \OMS/cm 334 | sy/m/n/10 ' []\OT1/cmr/m/n/10 (\OMS/cmsy/m/n/10 C\OML/cmm/m/it/10 ; \OT1/cmss/m 335 | /n/10 E\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 : 336 | [] 337 | 338 | 339 | ! LaTeX Error: \begin{document} ended by \end{prop}. 340 | 341 | See the LaTeX manual or LaTeX Companion for explanation. 342 | Type H for immediate help. 343 | ... 344 | 345 | l.16 \end{prop} 346 | 347 | Your command was ignored. 348 | Type I to replace it with another command, 349 | or to continue without it. 350 | 351 | ) 352 | ! Emergency stop. 353 | <*> rel.tex 354 | 355 | *** (job aborted, no legal \end found) 356 | 357 | 358 | Here is how much of TeX's memory you used: 359 | 24 strings out of 476675 360 | 383 string characters out of 5807327 361 | 1843290 words of memory out of 5000000 362 | 20378 multiletter control sequences out of 15000+600000 363 | 513192 words of font info for 35 fonts, out of 8000000 for 9000 364 | 1348 hyphenation exceptions out of 8191 365 | 18i,3n,15p,225b,63s stack positions out of 10000i,1000n,20000p,200000b,200000s 366 | No pages of output. 367 | -------------------------------------------------------------------------------- /rel.tex: -------------------------------------------------------------------------------- 1 | \chapter{相对\topos{}} 2 | 3 | \minitoc 4 | 5 | 在相对的观点下, \topos{}之间的几何态射 $\mathcal E\to \mathcal S$, 即 ``相对于'' $\mathcal S$ 的 \topos{}, 可视为 $\mathcal E$ 是某个景上的 $\mathcal S$-值层范畴. 当然, 这里的 ``景'' 与 ``层'' 的概念是在 $\mathcal S$ 的内语言中谈论的. 6 | 7 | % Joyal--Tierney EGG 8 | 9 | % internal locale 写在内语言那一章 10 | 11 | %\section{位象} 12 | % 13 | %\begin{definition} 14 | % {(位象)} 15 | % 16 | %\end{definition} 17 | 18 | \section{} 19 | 20 | \begin{definition} 21 | {(旋子)} 22 | 设 $\mathcal S$ 为\topos{}, $\mathcal C$ 为 $\mathcal S$ 的内蕴范畴, $p\colon \mathcal E\to\mathcal S$ 为相对\topos{}. 定义 $\mathcal E$ 中的 $\mathcal C$-旋子为 $\mathcal S$-索引函子 $F\colon \mathcal C^{\op}\to\mathcal E$, 使得 23 | \end{definition} 24 | 25 | \begin{prop} 26 | {(Diaconescu 定理)} 27 | 设 $\mathcal S$ 为\topos{}, $\mathcal C$ 为 $\mathcal S$ 的内蕴范畴, $p\colon \mathcal E\to\mathcal S$ 为相对\topos{}. 那么有范畴等价 28 | \[ 29 | \operatorname{Hom}_{\Top/\mathcal S} 30 | (\mathcal E,\mathsf {Fun}(\mathcal C,\mathcal S)) 31 | \simeq \mathsf {Tors}(\mathcal C,\mathcal E). 32 | \] 33 | \end{prop} 34 | 35 | -------------------------------------------------------------------------------- /topos.blg: -------------------------------------------------------------------------------- 1 | [0] Config.pm:307> INFO - This is Biber 2.19 2 | [0] Config.pm:310> INFO - Logfile is 'topos.blg' 3 | [121] biber-MSWIN64:340> INFO - === 4 | [155] Biber.pm:419> INFO - Reading 'topos.bcf' 5 | [226] Biber.pm:979> INFO - Found 30 citekeys in bib section 0 6 | [236] Biber.pm:4419> INFO - Processing section 0 7 | [244] Biber.pm:4610> INFO - Looking for bibtex file 'Topos.bib' for section 0 8 | [290] bibtex.pm:1713> INFO - LaTeX decoding ... 9 | [311] bibtex.pm:1519> INFO - Found BibTeX data source 'Topos.bib' 10 | [392] UCollate.pm:68> INFO - Overriding locale 'en-US' defaults 'normalization = NFD' with 'normalization = prenormalized' 11 | [392] UCollate.pm:68> INFO - Overriding locale 'en-US' defaults 'variable = shifted' with 'variable = non-ignorable' 12 | [392] Biber.pm:4239> INFO - Sorting list 'nty/global//global/global' of type 'entry' with template 'nty' and locale 'en-US' 13 | [392] Biber.pm:4245> INFO - No sort tailoring available for locale 'en-US' 14 | [409] bbl.pm:660> INFO - Writing 'topos.bbl' with encoding 'UTF-8' 15 | [416] bbl.pm:763> INFO - Output to topos.bbl 16 | -------------------------------------------------------------------------------- /topos.maf: -------------------------------------------------------------------------------- 1 | topos.mtc 2 | topos.mtc0 3 | topos.mtc9 4 | topos.mtc8 5 | topos.mtc7 6 | topos.mtc6 7 | topos.mtc5 8 | topos.mtc4 9 | topos.mtc3 10 | topos.mtc2 11 | topos.mtc1 12 | -------------------------------------------------------------------------------- /topos.mtc: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/SimplicialCat/topos/59a726f343732ec6c90a22ae3c1f4d5fc29fe663/topos.mtc -------------------------------------------------------------------------------- /topos.mtc0: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/SimplicialCat/topos/59a726f343732ec6c90a22ae3c1f4d5fc29fe663/topos.mtc0 -------------------------------------------------------------------------------- /topos.mtc1: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/SimplicialCat/topos/59a726f343732ec6c90a22ae3c1f4d5fc29fe663/topos.mtc1 -------------------------------------------------------------------------------- /topos.mtc10: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/SimplicialCat/topos/59a726f343732ec6c90a22ae3c1f4d5fc29fe663/topos.mtc10 -------------------------------------------------------------------------------- /topos.mtc11: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/SimplicialCat/topos/59a726f343732ec6c90a22ae3c1f4d5fc29fe663/topos.mtc11 -------------------------------------------------------------------------------- /topos.mtc2: 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景的构造}{\reset@font\mtcPfont 155}{tcb@cnt@remark.3.11.2}} 48 | {\reset@font\mtcPfont\mtc@string\contentsline{paragraph}{\noexpand \leavevmode 第二步, 层条件的验证}{\reset@font\mtcPfont 156}{tcb@cnt@remark.3.11.2}} 49 | {\reset@font\mtcPfont\mtc@string\contentsline{paragraph}{\noexpand \leavevmode 第三步, 范畴等价的证明}{\reset@font\mtcPfont 156}{tcb@cnt@remark.3.11.2}} 50 | {\reset@font\mtcSfont\mtc@string\contentsline{section}{\noexpand \leavevmode \numberline {3.12}等变层与拓扑群胚}{\reset@font\mtcSfont 157}{section.3.12}} 51 | -------------------------------------------------------------------------------- /topos.mtc5: -------------------------------------------------------------------------------- 1 | {\reset@font\mtcSfont\mtc@string\contentsline{section}{\noexpand \leavevmode \numberline {4.1}Mitchell--B\'enabou 语言}{\reset@font\mtcSfont 162}{section.4.1}} 2 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 使用 Mitchell--B\'enabou 语言表达意象{}中的对象和态射}{\reset@font\mtcSSfont 165}{tcb@cnt@remark.4.1.11}} 3 | {\reset@font\mtcSfont\mtc@string\contentsline{section}{\noexpand \leavevmode \numberline {4.2}Kripke--Joyal 语义}{\reset@font\mtcSfont 168}{section.4.2}} 4 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 层语义}{\reset@font\mtcSSfont 170}{Item.61}} 5 | {\reset@font\mtcSfont\mtc@string\contentsline{section}{\noexpand \leavevmode \numberline {4.3}模态与层化}{\reset@font\mtcSfont 172}{section.4.3}} 6 | {\reset@font\mtcSfont\mtc@string\contentsline{section}{\noexpand \leavevmode \numberline {4.4}内位象}{\reset@font\mtcSfont 176}{section.4.4}} 7 | -------------------------------------------------------------------------------- /topos.mtc6: -------------------------------------------------------------------------------- 1 | {\reset@font\mtcSfont\mtc@string\contentsline{section}{\noexpand \leavevmode \numberline {5.1}语法范畴: 语法--语义对偶}{\reset@font\mtcSfont 177}{section.5.1}} 2 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 命题理论的语法范畴}{\reset@font\mtcSSfont 182}{tcb@cnt@remark.5.1.11}} 3 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 类型论的语境范畴}{\reset@font\mtcSSfont 182}{tcb@cnt@remark.5.1.11}} 4 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 语法景}{\reset@font\mtcSSfont 182}{tcb@cnt@remark.5.1.12}} 5 | {\reset@font\mtcSfont\mtc@string\contentsline{section}{\noexpand \leavevmode \numberline {5.2}分类意象{}}{\reset@font\mtcSfont 183}{section.5.2}} 6 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode $G$-旋子的分类意象{}}{\reset@font\mtcSSfont 184}{section.5.2}} 7 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 对象的分类意象}{\reset@font\mtcSSfont 186}{tcb@cnt@remark.5.2.6}} 8 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 子终对象的分类意象}{\reset@font\mtcSSfont 187}{tcb@cnt@remark.5.2.9}} 9 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 命题理论的分类意象{}}{\reset@font\mtcSSfont 187}{tcb@cnt@remark.5.2.10}} 10 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 群的分类意象{}}{\reset@font\mtcSSfont 188}{tcb@cnt@remark.5.2.14}} 11 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 环的分类意象{}}{\reset@font\mtcSSfont 189}{tcb@cnt@remark.5.2.16}} 12 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 向量空间的分类意象{}}{\reset@font\mtcSSfont 189}{tcb@cnt@remark.5.2.16}} 13 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 几何理论的分类意象{}}{\reset@font\mtcSSfont 189}{tcb@cnt@remark.5.2.16}} 14 | -------------------------------------------------------------------------------- /topos.mtc7: -------------------------------------------------------------------------------- 1 | {\reset@font\mtcSfont\mtc@string\contentsline{section}{\noexpand \leavevmode \numberline {6.1}非标准分析}{\reset@font\mtcSfont 191}{section.6.1}} 2 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 基本概念}{\reset@font\mtcSSfont 191}{section.6.1}} 3 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 滤商}{\reset@font\mtcSSfont 191}{tcb@cnt@remark.6.1.2}} 4 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 超滤范畴}{\reset@font\mtcSSfont 191}{tcb@cnt@remark.6.1.2}} 5 | {\reset@font\mtcSfont\mtc@string\contentsline{section}{\noexpand \leavevmode \numberline {6.2}可计算性理论与有效意象}{\reset@font\mtcSfont 192}{section.6.2}} 6 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 基础知识}{\reset@font\mtcSSfont 192}{section.6.2}} 7 | {\reset@font\mtcSfont\mtc@string\contentsline{section}{\noexpand \leavevmode \numberline {6.3}代数几何的函子观点}{\reset@font\mtcSfont 194}{section.6.3}} 8 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode ``小'' 意象{}与 ``大'' 意象{}}{\reset@font\mtcSSfont 196}{tcb@cnt@remark.6.3.7}} 9 | {\reset@font\mtcSfont\mtc@string\contentsline{section}{\noexpand \leavevmode \numberline {6.4}综合微分几何与光滑无穷小分析}{\reset@font\mtcSfont 196}{section.6.4}} 10 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 综合微分几何的理论}{\reset@font\mtcSSfont 196}{section.6.4}} 11 | {\reset@font\mtcPfont\mtc@string\contentsline{paragraph}{\noexpand \leavevmode Kock--Lawvere 公理与导数}{\reset@font\mtcPfont 197}{tcb@cnt@remark.6.4.5}} 12 | {\reset@font\mtcPfont\mtc@string\contentsline{paragraph}{\noexpand \leavevmode Weil 代数与无穷小几何对象}{\reset@font\mtcPfont 198}{tcb@cnt@remark.6.4.7}} 13 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 综合微分几何的模型}{\reset@font\mtcSSfont 200}{tcb@cnt@remark.6.4.16}} 14 | {\reset@font\mtcSSSfont\mtc@string\contentsline{subsubsection}{\noexpand \leavevmode ``代数'' 模型}{\reset@font\mtcSSSfont 200}{tcb@cnt@remark.6.4.16}} 15 | {\reset@font\mtcSSSfont\mtc@string\contentsline{subsubsection}{\noexpand \leavevmode 光滑代数}{\reset@font\mtcSSSfont 202}{tcb@cnt@remark.6.4.22}} 16 | {\reset@font\mtcSfont\mtc@string\contentsline{section}{\noexpand \leavevmode \numberline {6.5}量子理论与 Bohr 意象}{\reset@font\mtcSfont 204}{section.6.5}} 17 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode $C^*$-代数, 经典语境与 Bohr 景}{\reset@font\mtcSSfont 205}{section.6.5}} 18 | {\reset@font\mtcSSSfont\mtc@string\contentsline{subsubsection}{\noexpand \leavevmode 偏序集上的层}{\reset@font\mtcSSSfont 207}{tcb@cnt@remark.6.5.9}} 19 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode Bohr 意象}{\reset@font\mtcSSfont 208}{tcb@cnt@remark.6.5.13}} 20 | {\reset@font\mtcSSSfont\mtc@string\contentsline{subsubsection}{\noexpand \leavevmode Gelfand 对偶}{\reset@font\mtcSSSfont 208}{tcb@cnt@remark.6.5.14}} 21 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode Bohr 意象中的命题}{\reset@font\mtcSSfont 209}{tcb@cnt@remark.6.5.19}} 22 | {\reset@font\mtcSfont\mtc@string\contentsline{section}{\noexpand \leavevmode \numberline {6.6}连续统假设的独立性}{\reset@font\mtcSfont 209}{section.6.6}} 23 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 双重否定与稠密拓扑}{\reset@font\mtcSSfont 210}{section.6.6}} 24 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 意象{}中基数的比较}{\reset@font\mtcSSfont 211}{tcb@cnt@remark.6.6.5}} 25 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 连续统假设反例的构造}{\reset@font\mtcSSfont 212}{tcb@cnt@remark.6.6.8}} 26 | {\reset@font\mtcSfont\mtc@string\contentsline{section}{\noexpand \leavevmode \numberline {6.7}凝聚态数学}{\reset@font\mtcSfont 215}{section.6.7}} 27 | -------------------------------------------------------------------------------- /topos.mtc8: -------------------------------------------------------------------------------- 1 | {\reset@font\mtcSfont\mtc@string\contentsline{section}{\noexpand \leavevmode \numberline {A.1}$2$-范畴}{\reset@font\mtcSfont 218}{section.A.1}} 2 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode $2$-范畴中的万有性质}{\reset@font\mtcSSfont 223}{tcb@cnt@remark.A.1.11}} 3 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 俯 $2$-范畴}{\reset@font\mtcSSfont 224}{tcb@cnt@remark.A.1.12}} 4 | {\reset@font\mtcSfont\mtc@string\contentsline{section}{\noexpand \leavevmode \numberline {A.2}伴随}{\reset@font\mtcSfont 224}{section.A.2}} 5 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 伴随保持极限}{\reset@font\mtcSSfont 226}{tcb@cnt@remark.A.2.4}} 6 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 伴随的自然变换}{\reset@font\mtcSSfont 229}{tcb@cnt@remark.A.2.10}} 7 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 伴随三元组}{\reset@font\mtcSSfont 230}{tcb@cnt@remark.A.2.12}} 8 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 伴随函子的 Frobenius 互反律}{\reset@font\mtcSSfont 231}{tcb@cnt@remark.A.2.16}} 9 | {\reset@font\mtcSfont\mtc@string\contentsline{section}{\noexpand \leavevmode \numberline {A.3}自反子范畴与局部化}{\reset@font\mtcSfont 232}{section.A.3}} 10 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 局部对象}{\reset@font\mtcSSfont 236}{tcb@cnt@remark.A.3.8}} 11 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 分式计算}{\reset@font\mtcSSfont 240}{tcb@cnt@remark.A.3.19}} 12 | {\reset@font\mtcSfont\mtc@string\contentsline{section}{\noexpand \leavevmode \numberline {A.4}预层范畴与米田嵌入}{\reset@font\mtcSfont 243}{section.A.4}} 13 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 米田引理}{\reset@font\mtcSSfont 243}{section.A.4}} 14 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 可表函子的余极限}{\reset@font\mtcSSfont 244}{tcb@cnt@remark.A.4.4}} 15 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 自由余完备化, 脉与几何实现}{\reset@font\mtcSSfont 245}{tcb@cnt@remark.A.4.8}} 16 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 预层范畴的俯范畴}{\reset@font\mtcSSfont 247}{tcb@cnt@remark.A.4.15}} 17 | {\reset@font\mtcSfont\mtc@string\contentsline{section}{\noexpand \leavevmode \numberline {A.5}(余) 滤范畴和 (余) 滤 (余) 极限}{\reset@font\mtcSfont 248}{section.A.5}} 18 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode $\mathsf {Set}$-值平坦函子}{\reset@font\mtcSSfont 249}{tcb@cnt@remark.A.5.4}} 19 | {\reset@font\mtcSfont\mtc@string\contentsline{section}{\noexpand \leavevmode \numberline {A.6}可表现范畴}{\reset@font\mtcSfont 254}{section.A.6}} 20 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 可表现对象}{\reset@font\mtcSSfont 254}{section.A.6}} 21 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 稠密子范畴}{\reset@font\mtcSSfont 259}{tcb@cnt@remark.A.6.17}} 22 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 可表现范畴的性质与判定}{\reset@font\mtcSSfont 260}{tcb@cnt@remark.A.6.23}} 23 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 可表现范畴的伴随函子定理}{\reset@font\mtcSSfont 261}{Item.84}} 24 | {\reset@font\mtcSfont\mtc@string\contentsline{section}{\noexpand \leavevmode \numberline {A.7}Kan 扩张}{\reset@font\mtcSfont 262}{section.A.7}} 25 | {\reset@font\mtcSfont\mtc@string\contentsline{section}{\noexpand \leavevmode \numberline {A.8}单子论}{\reset@font\mtcSfont 265}{section.A.8}} 26 | {\reset@font\mtcSfont\mtc@string\contentsline{section}{\noexpand \leavevmode \numberline {A.9}万有代数}{\reset@font\mtcSfont 272}{section.A.9}} 27 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode Lawvere 理论}{\reset@font\mtcSSfont 273}{section.A.9}} 28 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 模型的表现}{\reset@font\mtcSSfont 278}{tcb@cnt@remark.A.9.15}} 29 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 代数理论之间的态射}{\reset@font\mtcSSfont 279}{tcb@cnt@remark.A.9.17}} 30 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 单子与代数理论}{\reset@font\mtcSSfont 281}{Item.98}} 31 | {\reset@font\mtcSfont\mtc@string\contentsline{section}{\noexpand \leavevmode \numberline {A.10}纤维范畴与索引范畴}{\reset@font\mtcSfont 282}{section.A.10}} 32 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 等变对象}{\reset@font\mtcSSfont 289}{tcb@cnt@remark.A.10.20}} 33 | {\reset@font\mtcSfont\mtc@string\contentsline{section}{\noexpand \leavevmode \numberline {A.11}下降}{\reset@font\mtcSfont 289}{section.A.11}} 34 | -------------------------------------------------------------------------------- /topos.mtc9: -------------------------------------------------------------------------------- 1 | {\reset@font\mtcSfont\mtc@string\contentsline{section}{\noexpand \leavevmode \numberline {B.1}一阶逻辑}{\reset@font\mtcSfont 293}{section.B.1}} 2 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 一阶语言的基本要件}{\reset@font\mtcSSfont 295}{tcb@cnt@remark.B.1.4}} 3 | {\reset@font\mtcSSSfont\mtc@string\contentsline{subsubsection}{\noexpand \leavevmode 符号表}{\reset@font\mtcSSSfont 295}{tcb@cnt@remark.B.1.4}} 4 | {\reset@font\mtcSSSfont\mtc@string\contentsline{subsubsection}{\noexpand \leavevmode 项, 公式}{\reset@font\mtcSSSfont 297}{tcb@cnt@remark.B.1.10}} 5 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 一阶理论}{\reset@font\mtcSSfont 299}{tcb@cnt@remark.B.1.16}} 6 | {\reset@font\mtcSfont\mtc@string\contentsline{section}{\noexpand \leavevmode \numberline {B.2}一阶逻辑的范畴语义}{\reset@font\mtcSfont 306}{section.B.2}} 7 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 一阶语言在范畴中的解释}{\reset@font\mtcSSfont 306}{section.B.2}} 8 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 一阶理论在范畴中的模型}{\reset@font\mtcSSfont 310}{tcb@cnt@remark.B.2.11}} 9 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 提纲}{\reset@font\mtcSSfont 310}{tcb@cnt@remark.B.2.15}} 10 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 教条}{\reset@font\mtcSSfont 311}{tcb@cnt@remark.B.2.19}} 11 | {\reset@font\mtcSfont\mtc@string\contentsline{section}{\noexpand \leavevmode \numberline {B.3}高阶逻辑}{\reset@font\mtcSfont 313}{section.B.3}} 12 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 高阶语言的基本要件}{\reset@font\mtcSSfont 313}{section.B.3}} 13 | {\reset@font\mtcSfont\mtc@string\contentsline{section}{\noexpand \leavevmode \numberline {B.4}类型论}{\reset@font\mtcSfont 316}{section.B.4}} 14 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 命题是类型: Curry--Howard 同构}{\reset@font\mtcSSfont 319}{tcb@cnt@remark.B.4.4}} 15 | {\reset@font\mtcSSfont\mtc@string\contentsline{subsection}{\noexpand \leavevmode 类型论的范畴语义}{\reset@font\mtcSSfont 319}{tcb@cnt@remark.B.4.4}} 16 | {\reset@font\mtcSfont\mtc@string\contentsline{section}{\noexpand \leavevmode \numberline {B.5}模态逻辑}{\reset@font\mtcSfont 320}{section.B.5}} 17 | -------------------------------------------------------------------------------- /topos.out: -------------------------------------------------------------------------------- 1 | \BOOKMARK [0][-]{chapter.0}{\376\377\122\115\212\000}{}% 1 2 | \BOOKMARK [0][-]{chapter.1}{\376\377\141\017\214\141\166\204\203\003\165\164\213\272\140\047\215\050}{}% 2 3 | \BOOKMARK [1][-]{section.1.1}{\376\377\203\003\165\164\213\272\127\372\147\054\151\202\137\365}{chapter.1}% 3 4 | \BOOKMARK 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https://raw.githubusercontent.com/SimplicialCat/topos/59a726f343732ec6c90a22ae3c1f4d5fc29fe663/topos.pdf -------------------------------------------------------------------------------- /topos.run.xml: -------------------------------------------------------------------------------- 1 | 2 | 3 | 4 | 5 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 23 | 28 | 33 | 36 | 39 | 42 | ]> 43 | 44 | 45 | latex 46 | 47 | topos.bcf 48 | 49 | 50 | topos.bbl 51 | 52 | 53 | blx-dm.def 54 | blx-unicode.def 55 | blx-compat.def 56 | biblatex.def 57 | standard.bbx 58 | numeric.bbx 59 | numeric.cbx 60 | biblatex.cfg 61 | english.lbx 62 | 63 | 64 | 65 | biber 66 | 67 | biber 68 | topos 69 | 70 | 71 | topos.bcf 72 | 73 | 74 | topos.bbl 75 | 76 | 77 | topos.bbl 78 | 79 | 80 | topos.bcf 81 | 82 | 83 | Topos.bib 84 | 85 | 86 | 87 | -------------------------------------------------------------------------------- /topos.synctex.gz: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/SimplicialCat/topos/59a726f343732ec6c90a22ae3c1f4d5fc29fe663/topos.synctex.gz -------------------------------------------------------------------------------- /topos.tex: -------------------------------------------------------------------------------- 1 | % 2 | 3 | \documentclass{book} 4 | \usepackage[UTF8]{ctex} 5 | \usepackage{fontspec} 6 | %\usepackage{xeCJK} 7 | \newCJKfontfamily\mincho{IPAexMincho} 8 | \setCJKmainfont{SimSun}[ItalicFont=KaiTi, BoldFont=SimHei] 9 | 10 | % SimSun 11 | % STSong 12 | % SimHei 13 | % KaiTi 14 | % DengXian 15 | 16 | \input{toposCommands} 17 | 18 | 19 | 20 | \title{\textbf{\huge 盲人摸象}\\\emph{\topos{}理论讲义}} 21 | \author{\emph{王进一}\\\href{我的邮箱}{jin12003@163.com}\\QQ 2917905525} 22 | \date{2023 年夏至今\\~\\~此版本编译时间: \today{}~\\~\\这是一本正在施工的讲义. 目前我迫切需要读者的意见! 23 | \\~\\ 24 | %在非正式版本中, 我故意将 topos 译为 ``\topos{}''. 25 | } 26 | 27 | \usepackage{amsthm} 28 | \usepackage{amsmath} 29 | \usepackage{amssymb} 30 | \usepackage{mathrsfs} 31 | \usepackage{hyperref} 32 | \usepackage{stmaryrd} 33 | 34 | %\usepackage{wrapfig} 35 | 36 | \usepackage{epigraph} 37 | 38 | \usepackage{tikz} 39 | \usetikzlibrary{cd} 40 | \usetikzlibrary{decorations.pathmorphing} 41 | 42 | \usepackage{enumerate} 43 | 44 | \setcounter{secnumdepth}{1} 45 | \setcounter{tocdepth}{1} 46 | 47 | % 哲思 48 | 49 | \newcommand{\philoquote}[2]{ 50 | ~\\ 51 | \begin{center} 52 | \begin{minipage}{0.7\linewidth} 53 | {\quad\sffamily #1}\\ 54 | \begin{flushright} 55 | \textsf{#2} 56 | \end{flushright} 57 | \end{minipage} 58 | \end{center} 59 | ~\\ 60 | } 61 | 62 | % 彩色方框 63 | \input{colorboxes} 64 | 65 | % 分栏 66 | \usepackage{multicol} 67 | 68 | % 页边距 69 | \usepackage{geometry} 70 | \geometry{ 71 | b5paper, 72 | left=15mm, 73 | right=15mm, 74 | top=20mm, 75 | bottom=20mm 76 | } 77 | %\renewcommand{\baselinestretch}{0.9} 78 | 79 | % 页眉与页脚样式 80 | \usepackage{fancyhdr} 81 | 82 | \newcommand{\chaptert}{} 83 | \newcommand{\sectiont}{} 84 | 85 | \renewcommand{\chaptermark}[1]{\renewcommand{\chaptert}{第\ \thechapter\ 章\ #1}} 86 | \renewcommand{\sectionmark}[1]{\renewcommand{\sectiont}{\thesection\ #1}} 87 | 88 | \fancypagestyle{plain}{ 89 | \fancyhf{} 90 | \renewcommand{\headrulewidth}{0pt} 91 | \renewcommand{\footrulewidth}{0pt} 92 | \fancyfoot[LE,RO]{\thepage} 93 | } 94 | 95 | \fancypagestyle{Jfancy}{ 96 | \fancyhead[LE]{\chaptert{}} 97 | \fancyhead[RO]{\sectiont{}} 98 | \fancyhead[RE,LO]{} 99 | \fancyfoot[C]{} 100 | \fancyfoot[LE,RO]{\thepage} 101 | \renewcommand{\headrulewidth}{0.4pt}% Line at the header visible 102 | \renewcommand{\footrulewidth}{0pt}% Line at the footer visible 103 | } 104 | 105 | \pagestyle{Jfancy} 106 | 107 | % 参考文献 108 | \usepackage{biblatex} 109 | \addbibresource{Topos.bib} 110 | 111 | % 章节样式 112 | \usepackage{titlesec} 113 | \titleformat{\chapter}{\huge\bfseries}{第\, \thechapter\, 章}{1em}{} 114 | 115 | \usepackage{minitoc} 116 | \renewcommand{\mtctitle}{} 117 | 118 | \usepackage{multirow} 119 | 120 | 121 | 122 | \begin{document} 123 | \dominitoc 124 | 125 | \maketitle 126 | 127 | \newpage 128 | 129 | \vspace{4cm} 130 | 131 | \philoquote{ 132 | Je vous souhaite le meilleur succès. Ce serait magnifique que vous puissiez étudier la théorie des topos de Grothendieck et travailler dans ce domaine. Travaillez beaucoup, soyez patient, ayez bon courage et vos efforts seront récompensés. 133 | }{ 134 | Laurent Lafforgue\footnotemark 135 | } 136 | \footnotetext{这是 Lafforgue 教授在一次讲座之后写给作者的话. ``祝愿你获得最大的成功. 你能够学习 Grothendieck 意象理论并在这个领域工作, 是一件美妙的事情. 努力学习, 保持耐心, 勇往直前, 你的努力将会得到回报.'' Lafforgue 教授是 Caramello 教授多年的合作者.} 137 | 138 | \philoquote{Once you see at least one example and you do it yourself and you experience the kind of enlightenment it brings, you will be convinced forever.}{Olivia Caramello\footnotemark} 139 | \footnotetext{这是 Caramello 教授在与作者的采访中关于 topos 理论的评论. Caramello 教授是意象理论和逻辑学专家.} 140 | 141 | %\vfill 142 | 143 | \tableofcontents 144 | 145 | \setcounter{chapter}{-1} % 这样下面一章就是第 0 章 146 | 147 | % 第零章 前言 148 | \input{foreword} 149 | 150 | % 第一章 范畴论 151 | \input{cat} 152 | 153 | % 第二章 位象理论 154 | \input{loc} 155 | 156 | % 第三章 Grothendieck \topos{} 157 | \input{Gro} 158 | 159 | % 第四章 内语言 (意象与逻辑?) 160 | \input{lan} 161 | 162 | % 第五章 分类\topos{} 163 | \input{cla} 164 | 165 | % 第六章 相对\topos{} 166 | %\input{rel} 167 | 168 | %% 第七章 高阶范畴与高阶\topos{} 169 | %\input{higher} 170 | 171 | %% 第八章 凝聚\topos{} 172 | %\input{cohesion} 173 | 174 | \input{app} 175 | 176 | \appendix 177 | 178 | % 附录 范畴论 179 | \input{cat-appendix} 180 | 181 | % 附录 形式逻辑 182 | \input{logic-appendix} 183 | 184 | %\input{colimit-appendix} 185 | 186 | \input{glossaries} 187 | \addcontentsline{toc}{chapter}{术语和符号表} 188 | 189 | \printbibliography[title=参考文献] 190 | \addcontentsline{toc}{chapter}{参考文献} 191 | 192 | \end{document} 193 | -------------------------------------------------------------------------------- /topos.toc: -------------------------------------------------------------------------------- 1 | \contentsline {chapter}{\numberline {0}前言}{7}{chapter.0}% 2 | \contentsline {chapter}{\numberline {1}意象{}的范畴论性质}{11}{chapter.1}% 3 | \contentsline {section}{\numberline {1.1}范畴论基本概念}{12}{section.1.1}% 4 | \contentsline {subsection}{极限与余极限}{12}{section.1.1}% 5 | \contentsline {subsection}{指数对象与积闭范畴}{13}{tcb@cnt@remark.1.1.1}% 6 | \contentsline {subsection}{子对象分类子}{16}{tcb@cnt@remark.1.1.11}% 7 | \contentsline {subsection}{幂对象}{21}{tcb@cnt@remark.1.1.25}% 8 | \contentsline {subsection}{俯范畴与局部积闭性}{22}{tcb@cnt@remark.1.1.27}% 9 | \contentsline {section}{\numberline {1.2}意象{}}{27}{section.1.2}% 10 | \contentsline {section}{\numberline {1.3}更多范畴论结构}{29}{section.1.3}% 11 | \contentsline {subsection}{0 和 1}{29}{section.1.3}% 12 | \contentsline {subsection}{单射与满射}{30}{tcb@cnt@remark.1.3.3}% 13 | \contentsline {subsection}{正则单射与满射, 等价关系}{31}{tcb@cnt@remark.1.3.7}% 14 | \contentsline {subsection}{像}{33}{tcb@cnt@remark.1.3.17}% 15 | \contentsline {subsection}{满--单分解}{35}{tcb@cnt@remark.1.3.23}% 16 | \contentsline {subsection}{子终对象}{36}{tcb@cnt@remark.1.3.25}% 17 | \contentsline {subsection}{子对象的格与 Heyting 代数}{38}{tcb@cnt@remark.1.3.31}% 18 | \contentsline {subsection}{自然数对象}{45}{tcb@cnt@remark.1.3.55}% 19 | \contentsline {subsection}{无交和}{45}{tcb@cnt@remark.1.3.59}% 20 | \contentsline {subsection}{Boole 意象{}与选择公理}{48}{tcb@cnt@remark.1.3.62}% 21 | \contentsline {chapter}{\numberline {2}位象: 无点拓扑学}{53}{chapter.2}% 22 | \contentsline {section}{\numberline {2.1}基本概念}{54}{section.2.1}% 23 | \contentsline {section}{\numberline {2.2}位象的几何性质}{60}{section.2.2}% 24 | \contentsline {subsection}{子位象}{60}{section.2.2}% 25 | \contentsline {subsubsection}{开子位象与闭子位象}{62}{tcb@cnt@remark.2.2.6}% 26 | \contentsline {subsubsection}{子位象与内核{}}{64}{tcb@cnt@remark.2.2.9}% 27 | \contentsline {subsection}{Boole 位象}{65}{tcb@cnt@remark.2.2.14}% 28 | \contentsline {subsection}{位象的满射}{67}{tcb@cnt@remark.2.2.20}% 29 | \contentsline {subsection}{开映射}{68}{tcb@cnt@remark.2.2.23}% 30 | \contentsline {subsection}{局部位象}{70}{tcb@cnt@remark.2.2.28}% 31 | \contentsline {subsection}{局部连通位象}{71}{tcb@cnt@remark.2.2.32}% 32 | \contentsline {section}{\numberline {2.3}位象与逻辑}{71}{section.2.3}% 33 | \contentsline {subsection}{经典命题逻辑与 Boole 代数}{72}{section.2.3}% 34 | \contentsline {subsection}{几何逻辑与位格{}}{76}{tcb@cnt@remark.2.3.14}% 35 | \contentsline {chapter}{\numberline {3}意象{}与空间的概念}{81}{chapter.3}% 36 | \contentsline {section}{\numberline {3.1}拓扑空间上的层与平展空间}{83}{section.3.1}% 37 | \contentsline {subsection}{拓扑空间上层的直像与逆像}{87}{tcb@cnt@remark.3.1.14}% 38 | \contentsline {section}{\numberline {3.2}位象上的层与平展空间}{91}{section.3.2}% 39 | \contentsline {subsection}{位格{}取值的集合}{92}{tcb@cnt@remark.3.2.2}% 40 | \contentsline {section}{\numberline {3.3}范畴上的预层}{101}{section.3.3}% 41 | \contentsline {subsection}{预层范畴中的极限与余极限}{103}{tcb@cnt@remark.3.3.6}% 42 | \contentsline {subsection}{预层范畴中的指数对象}{103}{tcb@cnt@remark.3.3.8}% 43 | \contentsline {subsection}{预层范畴中的子对象分类子}{104}{tcb@cnt@remark.3.3.9}% 44 | \contentsline {subsubsection}{预层范畴的子对象分类子}{107}{tcb@cnt@remark.3.3.18}% 45 | \contentsline {section}{\numberline {3.4}景}{108}{section.3.4}% 46 | \contentsline {subsection}{从覆盖到 Grothendieck 拓扑}{108}{section.3.4}% 47 | \contentsline {subsection}{常见的景}{114}{tcb@cnt@remark.3.4.18}% 48 | \contentsline {subsection}{典范与次典范拓扑}{116}{tcb@cnt@remark.3.4.26}% 49 | \contentsline {section}{\numberline {3.5}层化与 Grothendieck $+$构造}{117}{section.3.5}% 50 | \contentsline {section}{\numberline {3.6}Grothendieck 意象{}}{121}{section.3.6}% 51 | \contentsline {subsection}{层范畴中的极限与余极限}{122}{section.3.6}% 52 | \contentsline {subsection}{层范畴中的子对象分类子}{122}{tcb@cnt@remark.3.6.2}% 53 | \contentsline {subsection}{层范畴中的指数对象}{125}{tcb@cnt@remark.3.6.8}% 54 | \contentsline {subsection}{Grothendieck 意象{}}{126}{tcb@cnt@remark.3.6.10}% 55 | \contentsline {subsection}{位象型意象{}}{126}{tcb@cnt@remark.3.6.14}% 56 | \contentsline {section}{\numberline {3.7}Lawvere--Tierney 拓扑, 内蕴层化与局部化}{128}{section.3.7}% 57 | \contentsline {subsection}{Lawvere--Tierney 拓扑}{128}{section.3.7}% 58 | \contentsline {subsection}{层范畴的性质}{132}{tcb@cnt@remark.3.7.10}% 59 | \contentsline {subsubsection}{层范畴中的有限极限}{132}{tcb@cnt@remark.3.7.10}% 60 | \contentsline {subsubsection}{层范畴中的子对象}{132}{tcb@cnt@remark.3.7.12}% 61 | \contentsline {subsubsection}{层范畴中的指数对象}{134}{tcb@cnt@remark.3.7.16}% 62 | \contentsline {subsection}{层化与局部化}{134}{tcb@cnt@remark.3.7.17}% 63 | \contentsline {section}{\numberline {3.8}意象{}之间的态射}{134}{section.3.8}% 64 | \contentsline {subsection}{几何态射}{134}{section.3.8}% 65 | \contentsline {subsection}{逻辑态射}{137}{tcb@cnt@remark.3.8.9}% 66 | \contentsline {subsection}{嵌入与满射}{137}{tcb@cnt@remark.3.8.11}% 67 | \contentsline {subsection}{满--单分解}{138}{tcb@cnt@remark.3.8.15}% 68 | \contentsline {subsection}{群作用与张量--同态伴随}{140}{tcb@cnt@remark.3.8.17}% 69 | \contentsline {section}{\numberline {3.9}景之间的态射}{146}{section.3.9}% 70 | \contentsline {subsection}{预层意象{}的点}{148}{tcb@cnt@remark.3.9.4}% 71 | \contentsline {subsection}{景取值的平坦函子}{149}{tcb@cnt@remark.3.9.7}% 72 | \contentsline {subsection}{层意象{}的点}{149}{tcb@cnt@remark.3.9.9}% 73 | \contentsline {subsection}{景之间的态射}{150}{tcb@cnt@remark.3.9.12}% 74 | \contentsline {subsection}{比较原理}{150}{tcb@cnt@remark.3.9.14}% 75 | \contentsline {section}{\numberline {3.10}意象{}的几何性质}{152}{section.3.10}% 76 | \contentsline {subsection}{平展性}{152}{section.3.10}% 77 | \contentsline {subsection}{连通性}{152}{tcb@cnt@remark.3.10.1}% 78 | \contentsline {subsection}{开几何态射}{153}{Item.52}% 79 | \contentsline {subsection}{本质几何态射}{154}{tcb@cnt@remark.3.10.6}% 80 | \contentsline {subsection}{紧合几何态射}{155}{tcb@cnt@remark.3.10.11}% 81 | \contentsline {section}{\numberline {3.11}Giraud 定理}{155}{section.3.11}% 82 | \contentsline {paragraph}{第一步, 景的构造}{155}{tcb@cnt@remark.3.11.2}% 83 | \contentsline {paragraph}{第二步, 层条件的验证}{156}{tcb@cnt@remark.3.11.2}% 84 | \contentsline {paragraph}{第三步, 范畴等价的证明}{156}{tcb@cnt@remark.3.11.2}% 85 | \contentsline {section}{\numberline {3.12}等变层与拓扑群胚}{157}{section.3.12}% 86 | \contentsline {chapter}{\numberline {4}意象{}的内语言}{161}{chapter.4}% 87 | \contentsline {section}{\numberline {4.1}Mitchell--B\'enabou 语言}{162}{section.4.1}% 88 | \contentsline {subsection}{使用 Mitchell--B\'enabou 语言表达意象{}中的对象和态射}{165}{tcb@cnt@remark.4.1.11}% 89 | \contentsline {section}{\numberline {4.2}Kripke--Joyal 语义}{168}{section.4.2}% 90 | \contentsline {subsection}{层语义}{170}{Item.61}% 91 | \contentsline {section}{\numberline {4.3}模态与层化}{172}{section.4.3}% 92 | \contentsline {section}{\numberline {4.4}内位象}{176}{section.4.4}% 93 | \contentsline {chapter}{\numberline {5}语法景与分类意象{}}{177}{chapter.5}% 94 | \contentsline {section}{\numberline {5.1}语法范畴: 语法--语义对偶}{177}{section.5.1}% 95 | \contentsline {subsection}{命题理论的语法范畴}{182}{tcb@cnt@remark.5.1.11}% 96 | \contentsline {subsection}{类型论的语境范畴}{182}{tcb@cnt@remark.5.1.11}% 97 | \contentsline {subsection}{语法景}{182}{tcb@cnt@remark.5.1.12}% 98 | \contentsline {section}{\numberline {5.2}分类意象{}}{183}{section.5.2}% 99 | \contentsline {subsection}{$G$-旋子的分类意象{}}{184}{section.5.2}% 100 | \contentsline {subsection}{对象的分类意象}{186}{tcb@cnt@remark.5.2.6}% 101 | \contentsline {subsection}{子终对象的分类意象}{187}{tcb@cnt@remark.5.2.9}% 102 | \contentsline {subsection}{命题理论的分类意象{}}{187}{tcb@cnt@remark.5.2.10}% 103 | \contentsline {subsection}{群的分类意象{}}{188}{tcb@cnt@remark.5.2.14}% 104 | \contentsline {subsection}{环的分类意象{}}{189}{tcb@cnt@remark.5.2.16}% 105 | \contentsline {subsection}{向量空间的分类意象{}}{189}{tcb@cnt@remark.5.2.16}% 106 | \contentsline {subsection}{几何理论的分类意象{}}{189}{tcb@cnt@remark.5.2.16}% 107 | \contentsline {chapter}{\numberline {6}意象{}理论的应用}{191}{chapter.6}% 108 | \contentsline {section}{\numberline {6.1}非标准分析}{191}{section.6.1}% 109 | \contentsline {subsection}{基本概念}{191}{section.6.1}% 110 | \contentsline {subsection}{滤商}{191}{tcb@cnt@remark.6.1.2}% 111 | \contentsline {subsection}{超滤范畴}{191}{tcb@cnt@remark.6.1.2}% 112 | \contentsline {section}{\numberline {6.2}可计算性理论与有效意象}{192}{section.6.2}% 113 | \contentsline {subsection}{基础知识}{192}{section.6.2}% 114 | \contentsline {section}{\numberline {6.3}代数几何的函子观点}{194}{section.6.3}% 115 | \contentsline {subsection}{``小'' 意象{}与 ``大'' 意象{}}{196}{tcb@cnt@remark.6.3.7}% 116 | \contentsline {section}{\numberline {6.4}综合微分几何与光滑无穷小分析}{196}{section.6.4}% 117 | \contentsline {subsection}{综合微分几何的理论}{196}{section.6.4}% 118 | \contentsline {paragraph}{Kock--Lawvere 公理与导数}{197}{tcb@cnt@remark.6.4.5}% 119 | \contentsline {paragraph}{Weil 代数与无穷小几何对象}{198}{tcb@cnt@remark.6.4.7}% 120 | \contentsline {subsection}{综合微分几何的模型}{200}{tcb@cnt@remark.6.4.16}% 121 | \contentsline {subsubsection}{``代数'' 模型}{200}{tcb@cnt@remark.6.4.16}% 122 | \contentsline {subsubsection}{光滑代数}{202}{tcb@cnt@remark.6.4.22}% 123 | \contentsline {section}{\numberline {6.5}量子理论与 Bohr 意象}{204}{section.6.5}% 124 | \contentsline {subsection}{$C^*$-代数, 经典语境与 Bohr 景}{205}{section.6.5}% 125 | \contentsline {subsubsection}{偏序集上的层}{207}{tcb@cnt@remark.6.5.9}% 126 | \contentsline {subsection}{Bohr 意象}{208}{tcb@cnt@remark.6.5.13}% 127 | \contentsline {subsubsection}{Gelfand 对偶}{208}{tcb@cnt@remark.6.5.14}% 128 | \contentsline {subsection}{Bohr 意象中的命题}{209}{tcb@cnt@remark.6.5.19}% 129 | \contentsline {section}{\numberline {6.6}连续统假设的独立性}{209}{section.6.6}% 130 | \contentsline {subsection}{双重否定与稠密拓扑}{210}{section.6.6}% 131 | \contentsline {subsection}{意象{}中基数的比较}{211}{tcb@cnt@remark.6.6.5}% 132 | \contentsline {subsection}{连续统假设反例的构造}{212}{tcb@cnt@remark.6.6.8}% 133 | \contentsline {section}{\numberline {6.7}凝聚态数学}{215}{section.6.7}% 134 | \contentsline {chapter}{\numberline {A}范畴论基础}{217}{appendix.A}% 135 | \contentsline {section}{\numberline {A.1}$2$-范畴}{218}{section.A.1}% 136 | \contentsline {subsection}{$2$-范畴中的万有性质}{223}{tcb@cnt@remark.A.1.11}% 137 | \contentsline {subsection}{俯 $2$-范畴}{224}{tcb@cnt@remark.A.1.12}% 138 | \contentsline {section}{\numberline {A.2}伴随}{224}{section.A.2}% 139 | \contentsline {subsection}{伴随保持极限}{226}{tcb@cnt@remark.A.2.4}% 140 | \contentsline {subsection}{伴随的自然变换}{229}{tcb@cnt@remark.A.2.10}% 141 | \contentsline {subsection}{伴随三元组}{230}{tcb@cnt@remark.A.2.12}% 142 | \contentsline {subsection}{伴随函子的 Frobenius 互反律}{231}{tcb@cnt@remark.A.2.16}% 143 | \contentsline {section}{\numberline {A.3}自反子范畴与局部化}{232}{section.A.3}% 144 | \contentsline {subsection}{局部对象}{236}{tcb@cnt@remark.A.3.8}% 145 | \contentsline {subsection}{分式计算}{240}{tcb@cnt@remark.A.3.19}% 146 | \contentsline {section}{\numberline {A.4}预层范畴与米田嵌入}{243}{section.A.4}% 147 | \contentsline {subsection}{米田引理}{243}{section.A.4}% 148 | \contentsline {subsection}{可表函子的余极限}{244}{tcb@cnt@remark.A.4.4}% 149 | \contentsline {subsection}{自由余完备化, 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{subsection}{提纲}{310}{tcb@cnt@remark.B.2.15}% 178 | \contentsline {subsection}{教条}{311}{tcb@cnt@remark.B.2.19}% 179 | \contentsline {section}{\numberline {B.3}高阶逻辑}{313}{section.B.3}% 180 | \contentsline {subsection}{高阶语言的基本要件}{313}{section.B.3}% 181 | \contentsline {section}{\numberline {B.4}类型论}{316}{section.B.4}% 182 | \contentsline {subsection}{命题是类型: Curry--Howard 同构}{319}{tcb@cnt@remark.B.4.4}% 183 | \contentsline {subsection}{类型论的范畴语义}{319}{tcb@cnt@remark.B.4.4}% 184 | \contentsline {section}{\numberline {B.5}模态逻辑}{320}{section.B.5}% 185 | \contentsline {chapter}{术语和符号表}{323}{appendix*.11}% 186 | \contentsline {chapter}{参考文献}{327}{appendix*.12}% 187 | -------------------------------------------------------------------------------- /toposCommands.tex: -------------------------------------------------------------------------------- 1 | 2 | 3 | % 术语翻译 4 | 5 | \newcommand{\topos}{意象} 6 | \newcommand{\regular}{常理} 7 | \newcommand{\coherent}{贯理} 8 | \newcommand{\cohesive}{凝集} 9 | \newcommand{\cohesion}{凝集} 10 | 11 | % 专有名词 12 | 13 | \newcommand{\nlab}{$n$Lab} 14 | 15 | % 常用记号 16 | 17 | \newcommand{\op}{\text{op}} % 对偶范畴 18 | \newcommand{\yo}{\!\text{{\mincho よ}}} % 米田嵌入 19 | \newcommand{\Top}{\mathcal T\hspace{-3pt}opos} 20 | \newcommand{\interpretation}[1]{{[\![#1]\!]}} 21 | \newcommand{\upward}[1]{\uparrow{\!}{#1}} 22 | 23 | % sequent calculus 24 | \newcommand{\sqc}[2]{\dfrac{\quad #1 \quad}{\quad #2 \quad}} 25 | \newcommand{\sqqc}[2]{ 26 | \begin{array} 27 | {c} 28 | #1 \\ \hline \hline #2 29 | \end{array} 30 | } 31 | 32 | \newcommand{\internalprop}[1]{{\ulcorner {#1} \urcorner}} 33 | 34 | \newcommand{\todo}[1]{{\color{red} [\textbf{未完成: #1}]}} 35 | -------------------------------------------------------------------------------- /toposes-in-Mondovi.aux: -------------------------------------------------------------------------------- 1 | \relax 2 | \providecommand\hyper@newdestlabel[2]{} 3 | \providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} 4 | 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-------------------------------------------------------------------------------- https://raw.githubusercontent.com/SimplicialCat/topos/59a726f343732ec6c90a22ae3c1f4d5fc29fe663/toposes-in-Mondovi.pdf -------------------------------------------------------------------------------- /toposes-in-Mondovi.synctex.gz: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/SimplicialCat/topos/59a726f343732ec6c90a22ae3c1f4d5fc29fe663/toposes-in-Mondovi.synctex.gz -------------------------------------------------------------------------------- /toposes-in-Mondovi.tex: -------------------------------------------------------------------------------- 1 | \documentclass{article} 2 | \usepackage[UTF8]{ctex} 3 | 4 | \usepackage{amsthm} 5 | \usepackage{amsmath} 6 | \usepackage{amssymb} 7 | \usepackage{mathrsfs} 8 | \usepackage{hyperref} 9 | \usepackage{stmaryrd} 10 | 11 | \usepackage{tikz} 12 | \usetikzlibrary{cd} 13 | \usetikzlibrary{decorations.pathmorphing} 14 | 15 | \usepackage{enumerate} 16 | 17 | \newcommand{\philoquote}[2]{ 18 | ~\\ 19 | \begin{center} 20 | \begin{minipage}{0.7\linewidth} 21 | {\quad\sffamily #1}\\ 22 | \begin{flushright} 23 | \textsf{#2} 24 | \end{flushright} 25 | \end{minipage} 26 | \end{center} 27 | ~\\ 28 | } 29 | 30 | \usepackage{geometry} 31 | \geometry{ 32 | b5paper, 33 | left=15mm, 34 | right=15mm, 35 | top=20mm, 36 | bottom=20mm 37 | } 38 | 39 | \input{toposCommands} 40 | 41 | \input{colorboxes-printing} 42 | 43 | 44 | \begin{document} 45 | 46 | \section{Claudio Fontanari: 模空间} 47 | 48 | \section{Joseph Bernstein: 什么是群表示} 49 | % 9/9 15:20 50 | \begin{definition} 51 | {(商群胚)} 52 | 设群 $G$ 作用于集合 $X$, 定义\emph{商群胚} (quotient groupoid) $X/G$ 为 53 | \end{definition} 54 | 55 | \begin{prop} 56 | {} 57 | $$ 58 | \operatorname{Sh}(X/G) \simeq \operatorname{Sh}_G (X). 59 | $$ 60 | \end{prop} 61 | 62 | \section{Pivet: $2$-范畴上的层} 63 | % 9/10 13:45 64 | 65 | 66 | \section{Peter Haine: 由平展\topos{}重构概形} 67 | 68 | 69 | 70 | 71 | \section{景的态射与余态射} 72 | 73 | \section{Matthias Ritter (Hutzler): 综合代数几何} 74 | % 9/11 11:40 75 | 76 | 我们使用的综合代数几何的语言是用同伦类型论表述的 Zariski ($\infty$-){\topos{}}的内语言. 77 | 78 | \begin{definition} 79 | {(射影空间)} 80 | $$ 81 | \mathbb P^n := \sum_{L \subset R^{n+1}\,\text{子模}} \|L\simeq R^1 \|_{\text{prop}}. 82 | $$ 83 | \end{definition} 84 | 85 | \begin{definition} 86 | {(抽象直线的空间, 线丛, Picard 群)} 87 | 定义\emph{抽象直线的空间} (space of abstract lines) 88 | $$ 89 | BR^\times := \sum_{L: R\mathsf{Mod}} \| L\simeq R^1\|_{\text{prop}}. 90 | $$ 91 | 92 | 由于 $R$-模的张量积满足 $R^1\otimes R^1\simeq R^1$, 有运算 93 | $$ 94 | \otimes\colon BR^\times \times BR^\times \to BR^\times, 95 | $$ 96 | 且 $\otimes$ 构成 $BR^\times$ 上的 (高阶) 群结构, 单位为 $R^1$, 逆为 $L\mapsto L^\vee=\operatorname{Hom}(L,R^1)$. 97 | 98 | 定义空间 (类型) $X$ 上的\emph{线丛}为映射 $X\to BR^\times$. 99 | 定义 $X$ 的 \emph{Picard 群}为 100 | $$ 101 | \operatorname{Pic}(X) := \|X\to BR^\times\|_{\text{set}}. 102 | $$ 103 | \end{definition} 104 | 105 | \begin{example} 106 | {(重言线丛, $\mathcal O(d)$)} 107 | 射影空间 $\mathbb P^n$ 上的\emph{重言线丛} $\mathcal O(-1)$ 定义为 108 | $$ 109 | \mathcal O(-1) \colon \mathbb P^n \to BR^\times,\, L\mapsto L. 110 | $$ 111 | 定义 $\mathcal O(d)$ 为 $\mathcal O(-1)$ 的 $(-d)$ 次张量积. 112 | \end{example} 113 | 114 | \begin{prop} 115 | {} 116 | $$ 117 | \operatorname{Pic}(\mathbb P^n) \simeq\mathbb Z. 118 | $$ 119 | \end{prop} 120 | 121 | \section{Michael Shulman: \topos{}图表的内语言} 122 | 123 | \begin{definition} 124 | {(\topos{}的图表)} 125 | 定义一个\emph{\topos{}的图表}是一个 $2$-函子 $\mathcal M \to \Top$, 其中 $\mathcal M$ 是任意 $2$-范畴, $\Top$ 是\topos{}的 $2$-范畴. 126 | \end{definition} 127 | 128 | \begin{example} 129 | {} 130 | 以下结构均为\topos{}的图表的特例. 131 | \begin{itemize} 132 | \item $\mathcal S$-\topos{}, 也即几何态射 $f\colon \mathcal E \to \mathcal S$; 133 | \item 局部 $\mathcal S$-\topos{}, 也即几何态射 $f\colon \mathcal E \to \mathcal S$ 及其 ...%左伴随 $c\colon \mathcal S\to\mathcal E$, 满足单位 $cf\to 1$ 为同构; 134 | \item 完全连通 $\mathcal S$-\topos{}. 135 | \end{itemize} 136 | \end{example} 137 | \end{document} --------------------------------------------------------------------------------