-\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}
\usepackage{amsmath,amssymb,amsthm,latexsym}
\usepackage{fancyhdr}
+\usepackage[tiny,center,compact,sc]{titlesec}
\usepackage[cm]{fullpage}
\usepackage{pstricks}
\usepackage{graphicx}
\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)}
\DeclareMathOperator{\mm}{\mid\mid}
-\newcommand{\defn}[1]{\label{dfn:#1}{\em #1}}
+\newcommand{\defn}[1]{{\bf #1}}
\begin{document}
\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
\[\xymatrix{ F \ar[r]_{f'} \ar[d]^{h''} & E \ar[r]_{g'} \ar[d]^{h'} & D \ar[d]^{h} \\
A \ar[r]^f & B \ar[r]^g & C
}\]
\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}
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