From: Nathaniel Wesley Filardo Date: Tue, 21 Jun 2011 21:46:38 +0000 (-0400) Subject: Further progress? X-Git-Url: https://hydra-www.ietfng.org/gitweb/?a=commitdiff_plain;h=03572b942ec9b829918d3e879dfe8b1f4de33849;p=ctcheat Further progress? --- diff --git a/ctcheat.tex b/ctcheat.tex index b116791..03e4d20 100644 --- a/ctcheat.tex +++ b/ctcheat.tex @@ -1,4 +1,4 @@ -\documentclass[10pt,twocolumn,letterpaper]{amsart} +\documentclass[10pt,twocolumn,letterpaper]{article} \DeclareSymbolFont{AMSb}{U}{msb}{m}{n} \DeclareMathAlphabet{\mathbbm}{U}{bbm}{m}{n} \title{Category Theory Cheat Sheet} @@ -6,6 +6,7 @@ \usepackage{amsmath,amssymb,amsthm,latexsym} \usepackage{fancyhdr} +\usepackage[tiny,center,compact,sc]{titlesec} \usepackage[cm]{fullpage} \usepackage{pstricks} \usepackage{graphicx} @@ -22,8 +23,11 @@ \renewcommand{\baselinestretch}{0.9} -\newtheorem{thm}{Thm}[section] -\newtheorem{dfn}{Def}[section] +%\newtheorem{thm}{Thm}[section] +%\newtheorem{dfn}{Def}[section] + +\setlength{\parindent}{0pt} +\setlength{\parskip}{3pt} %Scalable bracket-like \newcommand{\paren}[1]{\left({#1}\right)} @@ -48,7 +52,7 @@ \DeclareMathOperator{\mm}{\mid\mid} -\newcommand{\defn}[1]{\label{dfn:#1}{\em #1}} +\newcommand{\defn}[1]{{\bf #1}} \begin{document} @@ -57,122 +61,111 @@ \section{Basics} - \begin{dfn}A \defn{category} (p4,\S1.3) is a structure with + A \defn{category} (p4,\S1.3) is a structure with \begin{itemize} \item Objects \& arrows (from \defn{domain} to \defn{codomain}). \item An associative arrow composition operator $\circ$. \item Identity arrows ($1_A$) on each object $A$, unit of $\circ$ \end{itemize} - \end{dfn} - \begin{dfn}A \defn{functor} (p8,D1.2) $F$ is a map between categories which + A \defn{functor} (p8,D1.2) $F$ is a map between categories which sends $A \to B$ to $FA \to FB$, sends $1_A$ to $1_{FA}$, and honors composition. - \end{dfn} - - \begin{dfn}$\mbox{Hom}_\mathbf{C}(A,B)$ denotes the class $(A \to B) \in \mathbf{C}$.\end{dfn} - \begin{dfn}The \defn{dual} (p15,i2) category $\mathbf{C}^{op}$ which + The \defn{dual} (p15,i2) category $\mathbf{C}^{op}$ which exchanges domains and codomains of arrows in $\mathbf{C}$. - \end{dfn} - \begin{thm}Any CT statement implies its dual (interchange dom/cod and reverse compositions).\end{thm} + Any CT statement implies its dual (interchange dom/cod and reverse compositions). \subsection{Categories over $\mathbf{C}$'s objects} - \begin{dfn}The \defn{arrow} (p16,i3) category $\mathbf{C}^\to$ has arrows + The \defn{arrow} (p16,i3) category $\mathbf{C}^\to$ has arrows for commutative squares in $\mathbf{C}$. There are two functors \[\xymatrix{ \mathbf{C} & \ar[l]_{\mathbf{dom}} C^\to \ar[r]^{\mathbf{cod}} & \mathbf{C}}\] - \end{dfn} - \begin{dfn}The \defn{slice} (p16,i4) category $\mathbf{C}/C$ has objects + The \defn{slice} (p16,i4) category $\mathbf{C}/C$ has objects of arrows in $\mathbf{C}$ with codomain $C$. Arrows are tops of commutative triangles. - \end{dfn} \subsection{Foundations} - \begin{thm}Categories may be described (p21) as + Categories may be described (p21) as \[\xymatrix{ C_2 \ar[r]^\circ & C_1 \ar@<2ex>[r]_{cod} \ar@<-2ex>[r]_{dom} & C_0 \ar[l]^i }\] - \end{thm} - \begin{dfn}A category is (p24-25,D1.11-12)\dots + A category is (p24-25,D1.11-12)\dots \begin{itemize} - \item \defn{small} if $C_0$ and $C_1$ is a set and \defn{large} otherwise. + \item \defn{small} if $C_0$ and $C_1$ are sets and \defn{large} otherwise. \item \defn{locally small} if $\forall_{X,Y \in C_0} \mbox{Hom}_C(X,Y) \subseteq C_1$ is a set. \end{itemize} - \end{dfn} \section{Special Kinds of Arrows} - \begin{dfn}$m$ is \defn{monic} (p25,D2.1) if $mi = mj \Rightarrow i = j$ in + $m$ is \defn{monic} (p25,D2.1) if $mi = mj \Rightarrow i = j$ in \[\xymatrix{C \ar@<1ex>[r]^{i} \ar@<-1ex>[r]_j & A \ar@{>->}[r]^m & B} \] - \end{dfn} - \begin{dfn}A \defn{subobject} (p77,D5.1) of $X$ is mono with cod $X$.\end{dfn} + A \defn{subobject} (p77,D5.1) of $X$ is mono with cod $X$. - \begin{dfn}$e$ is \defn{epic} (p25,D2.1) if it is monic in $\mathbf{C}^{op}$, + $e$ is \defn{epic} (p25,D2.1) if it is monic in $\mathbf{C}^{op}$, {\it i.e.,} if $ie = je \Rightarrow i = j$ in \[\xymatrix{A \ar@{->>}[r]^e & B \ar@<1ex>[r]^{i} \ar@<-1ex>[r]_j & C} \] - \end{dfn} - \begin{thm}(p27,P2.6) Every iso is both monic and epic.\end{thm} + (p27,P2.6) Every iso is both monic and epic. - \begin{dfn}A \defn{split mono} (\defn{epi}) has a left (right) inverse. (p28,D2.7)\end{dfn} + A \defn{split mono} (\defn{epi}) has a left (right) inverse. (p28,D2.7) - \begin{dfn}Given $e : A \to B$ and $s : B \to A$ s.t. $es = 1_A$, $e$ is a \defn{retraction} - of $s$ and $s$ is a \defn{section} (\defn{splitting}) of $e$. (p28,D2.7)\end{dfn} + Given $e : A \to B$ and $s : B \to A$ s.t. $es = 1_A$, $e$ is a \defn{retraction} + of $s$ and $s$ is a \defn{section} (\defn{splitting}) of $e$. (p28,D2.7) - \begin{thm}Functors preserve split monos and epis.\end{thm} + Functors preserve split monos and epis. - \begin{dfn}An \defn{point} (p32) of $C$ is any $c : 1 \to C$.\end{dfn} - - \begin{thm}Arrows in $\mathbf{Sets}$, but not $\mathbf{Pos}$, are pointwise.\end{thm} + A \defn{point} (p32) of $C$ is any $c : 1 \to C$. (Arrows in + $\mathbf{Sets}$, but not $\mathbf{Pos}$, are pointwise.) -\section{Universal Constructions} + A \defn{cone} (p89,D5.15) to a diagram $D(J)$ is a collection of arrows + $c_j : C \to D_j$ s.t. $\forall_{D_\alpha \in D(J)} c_j = D_\alpha \circ + c_i$. + A morphism between cones is an arrow $\phi : C \to C'$ s.t. + $\forall_{D_j \in D(J)} c_j^\prime \circ \phi = c_j$. Cones form a + category. - \begin{thm}Objects defined by universal constructions are unique up to isomorphism. - \end{thm} +\section{Universal Constructions} - \begin{dfn}$0$ is \defn{initial} iff - $\forall_C \exists! u : 0 \to C$. - \end{dfn} + Objects defined by UCs are unique up to isomorphism. - \begin{dfn}$1$ is \defn{terminal} iff - $\forall_C \exists! u : C \to 1 $. - \end{dfn} + $0$ is \defn{initial} iff $\forall_C \exists!_u 0 \to C$. + $1$ is \defn{terminal} iff $\forall_C \exists!_u C \to 1 $. - \begin{dfn}$(A \times B,\pi_1,\pi_2)$ is a \defn{product} iff + $(A \times B,\pi_1,\pi_2)$ is a \defn{product} iff \[\xymatrix{ {}\save[]+<-1cm,0cm>*\txt<8pc>{$\forall_{Z,z_1,z_2} \exists!_u$\\$u\pi_1 = z_1 ~\wedge~ u\pi_2 = z_2$}\restore & & Z \ar[dl]_{z_1} \ar[dr]^{z_2} \ar[d]^u & \\ & A & \ar[l]_{\pi_1} A \times B \ar[r]^{\pi_2} & B \\ }\] - \end{dfn} - \begin{dfn}$(E,e)$ is an \defn{equalizer} (p56,D3.13) of $f,g$ iff + $(E,e)$ is an \defn{equalizer} (p56,D3.13) of $f,g$ iff \[\forall_{Z,z . zf = zg} \exists!_u eu = z \quad \xymatrix{ Z \ar@{..>}[r]^u \ar@/_1pc/[rr]^{z} & E \ar[r]^e & A \ar@<1ex>[r]^f \ar@<-1ex>[r]_g & B \\ }\] - \end{dfn} - \begin{dfn}$(P,p_1,p_2)$ is a \defn{pullback} (p80,D5.4) of $f,g$ iff + $(P,p_1,p_2)$ is a \defn{pullback} (p80,D5.4) of $f,g$ iff \[\xymatrix{ {}\save[]+<-1cm,0cm>*\txt<8pc>{$\forall_{Z,z_1,z_2 . fz_1 = gz_2}\exists!_u$\\$z_1 = p_1u ~\wedge~ z_2 = p_2u$}\restore & Z \ar[dr]_{z_2} \ar@/^1pc/[rr]_{z_1} \ar[r]_u & P \ar[d]^{p_1} \ar[r]_{p_2} & B \ar[d]^g \\ & & A \ar[r]^f & C }\] $P$ may be denoted $A \times_C B$ when $f,g$ are clear. - \end{dfn} + + A \defn{limit} (p90,D5.16) of a diagram $D(J)$ is a + terminal object in the category $\mathbf{Cone}(D(J))$. Written: + $c_i : (\varprojlim_{j} D_j) \to D_i$. \section{Properties of UCs} - \begin{thm}Equalizers are monic.\end{thm} + Equalizers are monic. - \begin{thm}(p81,P5.5) $(Z,u)$ in a pullback is an equalizer of $fp_1$ and $gp_2$. + (p81,P5.5) $(Z,u)$ in a pullback is an equalizer of $fp_1$ and $gp_2$. If $(E,e)$ is an equalizer of same, then $E,p_1e,p_2e$ is a pullback of $f,g$. - \end{thm} - \begin{thm}(p84,L5.8) In the commuting diagram + (p84,L5.8) In the commuting diagram \[\xymatrix{ F \ar[r]_{f'} \ar[d]^{h''} & E \ar[r]_{g'} \ar[d]^{h'} & D \ar[d]^{h} \\ A \ar[r]^f & B \ar[r]^g & C }\] @@ -180,37 +173,79 @@ \item If $FEBA$ and $EDCB$ are pullbacks, so is $FDCA$. \item If $FDCA$ and $EDCB$ are pullbacks, so is $FEBA$. \end{enumerate} - \end{thm} - \begin{thm}Pullbacks preserve commutative triangles.\end{thm} + (p84,C5.9) Pullbacks preserve commutative triangles. + + UCs reduce to limits (p91,E5.17-20): + \begin{tabular}{cccc} + terminals & products & equalizers & pullbacks \\ + % + & $\xymatrix@C5pt{x & y}$ + & $\xymatrix{x \ar@<1ex>[r]^{\alpha} \ar@<-1ex>[r]_{\beta} & y}$ + & $\xymatrix@C5pt@R5pt{& x \ar[d] \\ y \ar[r] & z}$ + \end{tabular} + +\section{Exponentials} + + (p107,D6.1) In a category with binary products, given two objects $B$ and $C$, + their \defn{exponential} is an object $C^B$ and arrow $\epsilon : C^B \times B \to C$ + s.t. + \[\xymatrix{ + {}\save[]+<-1cm,0cm>*\txt<8pc>{$\forall_{A,f : A \times B \to C}\exists!_{\tilde f : A \to C^B}$\\ + $\epsilon \circ (\tilde f \times 1_B) = f$}\restore + & C^B & C^B \times B \ar[r]^\epsilon & C \\ + & A \ar[u]^{\tilde f} & A \times B \ar[u]^{\tilde f \times 1_B} \ar[ur]_f + }\] + The arrows $f$ and $\tilde f$ are ``exponential transposes.'' + + Exponential transposition is self inverse (p108). This implies + \[ \mbox{Hom}_{\bf C}(A \times B, C) \simeq \mbox{Hom}_{\bf C}(A, C^B) \] + +\section{Properties of Functors} + + A functor $F : C \to C'$ \defn{preserves limits of type $J$} if + \[ \forall_{D : J \to C}\forall_{\varprojlim_j D_j} F(\varprojlim_j D_j) \simeq \varprojlim_j F(D_j).\] + A functor is \defn{continuous} if it preserves all limits. (p94,D5.24) + + A functor $F : C \to C'$ \defn{creates limits of type $J$} if $\forall_{D : J \to C}$ + and all limits $L = \varprojlim_j FD_j$ (i.e., bundle $p_j : L \to FD_j$ in $C'$), + $\exists! (\bar{p_j} : \bar{L} \to D_j) \in C'$ with $F(\bar L) = L$, $F(\bar{p_j}) = p_j$, + and $\bar L = \varprojlim_j D_j$. \section{Special Functors} - \begin{dfn}The \defn{covariant representable functor} (p44) is + The \defn{covariant representable functors} (p44) are \[\mbox{Hom}(A,\text{---}) : \mathbf{C} \to \mathbf{Sets}\] - \end{dfn} + These functors preserve all limits (p94,P5.25). - \begin{thm}(p85,P5.10) Pullback defines a functor + (p85,P5.10) Pullback defines a functor \[ h^* : (A \stackrel{\alpha}{\to} C) \in \mathbf{C}/C \mapsto (C' \times_C A \stackrel{\alpha'}{\to} C') \in \mathbf{C}/C' \] where $\alpha'$ is the pullback of $\alpha$ along $h$. - \end{thm} \section{Glossary} - \begin{dfn}A category is \defn{finitely presented} (p75) if it is the + A category is \defn{cartesian closed} (p108,D6.2) if it has all finite + products and exponentials. + + A \defn{diagram} (p89,D5.15) is a functor $D : J \to C$ from + some indexing category $J$. + + A category is \defn{finitely presented} (p75) if it is the free category over a finite graph quotiented by a finite set of equations. - \end{dfn} - \begin{dfn} A structure is \defn{free} over $S$ if its elements are ``generated'' from $S$ and no ``nontrivial'' equations exist. - \end{dfn} - \begin{dfn}Subobject $m$'s \defn{local membership relation}: + $\mbox{Hom}_\mathbf{C}(A,B)$ denotes the class $(A \to B) \in \mathbf{C}$. + + Subobject $m$'s \defn{local membership relation}: \[ \forall_{m : M \rightarrowtail X} - \brak{ z \in_X M \Leftrightarrow \exists_{f:Z \to M} z = mf} \] - \end{dfn} + \brak{ z \in_X M \Leftrightarrow \exists_{f:Z \to M} z = mf} \] + + An \defn{$\omega$-complete Partial Order} ($\omega$CPO) is a Poset which + has all {\em co}limits of type $(\mathbb{N},\le)$. (All countably + infinite ascending chains have a top.) (p101,E5.33) \end{document}