From 120e73d3e9d648e2a79f1609a7d91f3f5b92739f Mon Sep 17 00:00:00 2001 From: Nathaniel W Filardo Date: Wed, 11 May 2011 16:04:07 -0400 Subject: [PATCH] Initial revision --- ctcheat.tex | 217 ++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 217 insertions(+) create mode 100644 ctcheat.tex diff --git a/ctcheat.tex b/ctcheat.tex new file mode 100644 index 0000000..b116791 --- /dev/null +++ b/ctcheat.tex @@ -0,0 +1,217 @@ +\documentclass[10pt,twocolumn,letterpaper]{amsart} +\DeclareSymbolFont{AMSb}{U}{msb}{m}{n} +\DeclareMathAlphabet{\mathbbm}{U}{bbm}{m}{n} +\title{Category Theory Cheat Sheet} +%\author{Nathaniel Wesley Filardo} + +\usepackage{amsmath,amssymb,amsthm,latexsym} +\usepackage{fancyhdr} +\usepackage[cm]{fullpage} +\usepackage{pstricks} +\usepackage{graphicx} +\usepackage{verbatim} +\usepackage{bm} +\usepackage{ifthen} +\usepackage{epsfig} +\usepackage[all]{xypic} +\usepackage{textcomp} +\usepackage{url} +\usepackage{multirow} +\usepackage{hyperref} +\usepackage{breakurl} + +\renewcommand{\baselinestretch}{0.9} + +\newtheorem{thm}{Thm}[section] +\newtheorem{dfn}{Def}[section] + +%Scalable bracket-like +\newcommand{\paren}[1]{\left({#1}\right)} +\newcommand{\brak}[1]{\left[{#1}\right]} +\newcommand{\abs}[1]{\left\lvert{#1}\right\rvert} +\newcommand{\ang}[1]{\left\langle{#1}\right\rangle} +\newcommand{\set}[1]{\left\{{#1}\right\}} + +%Mathematics +\newcommand{\condexp}[1]{\ifthenelse{\equal{#1}{false}}{}{^{#1}}} +\newcommand{\dd}[3][false]{\frac{d\condexp{#1}{#2}}{d{#3}\condexp{#1}}} +\newcommand{\pd}[3][false]{\frac{\partial\condexp{#1}{#2}}{\partial{#3}\condexp{#1}}} + +\newcommand{\ifrac}[2]{{#1}/{#2}} + +%Quantum Mechanics +\newcommand{\ket}[1]{\left\lvert{#1}\right\rangle} +\newcommand{\bra}[1]{\left\langle{#1}\right\rvert} +\newcommand{\braket}[2]{\left\langle{#1}\middle\vert{#2}\right\rangle} +\newcommand{\Braket}[3]{\left\langle{#1}\middle\vert{#2}\middle\vert{#3}\right\rangle} +\newcommand{\dyad}[2]{\left\lvert{#1}\middle\rangle\middle\langle{#2}\right\rvert} + +\DeclareMathOperator{\mm}{\mid\mid} + +\newcommand{\defn}[1]{\label{dfn:#1}{\em #1}} + +\begin{document} + +%\maketitle +\bibliographystyle{plainurl} + +\section{Basics} + + \begin{dfn}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 + 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 + 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} + + \subsection{Categories over $\mathbf{C}$'s objects} + + \begin{dfn}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 + 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 + \[\xymatrix{ C_2 \ar[r]^\circ & C_1 \ar@<2ex>[r]_{cod} \ar@<-2ex>[r]_{dom} & C_0 \ar[l]^i }\] + \end{thm} + + \begin{dfn}A category is (p24-25,D1.11-12)\dots + \begin{itemize} + \item \defn{small} if $C_0$ and $C_1$ is a set 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 + \[\xymatrix{C \ar@<1ex>[r]^{i} \ar@<-1ex>[r]_j & A \ar@{>->}[r]^m & B} \] + \end{dfn} + + \begin{dfn}A \defn{subobject} (p77,D5.1) of $X$ is mono with cod $X$.\end{dfn} + + \begin{dfn}$e$ 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} + + \begin{dfn}A \defn{split mono} (\defn{epi}) has a left (right) inverse. (p28,D2.7)\end{dfn} + + \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} + + \begin{thm}Functors preserve split monos and epis.\end{thm} + + \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} + +\section{Universal Constructions} + + \begin{thm}Objects defined by universal constructions are unique up to isomorphism. + \end{thm} + + \begin{dfn}$0$ is \defn{initial} iff + $\forall_C \exists! u : 0 \to C$. + \end{dfn} + + \begin{dfn}$1$ is \defn{terminal} iff + $\forall_C \exists! u : C \to 1 $. + \end{dfn} + + \begin{dfn}$(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 + \[\forall_{Z,z . zf = zg} \exists!_u eu = z \quad + \xymatrix{ + Z \ar@{..>}[r]^u \ar@/_1pc/[rr]^{z} & E \ar[r]^e & A \ar@<1ex>[r]^f \ar@<-1ex>[r]_g & B \\ + }\] + \end{dfn} + + \begin{dfn}$(P,p_1,p_2)$ is a \defn{pullback} (p80,D5.4) of $f,g$ iff + \[\xymatrix{ + {}\save[]+<-1cm,0cm>*\txt<8pc>{$\forall_{Z,z_1,z_2 . fz_1 = gz_2}\exists!_u$\\$z_1 = p_1u ~\wedge~ z_2 = p_2u$}\restore + & Z \ar[dr]_{z_2} \ar@/^1pc/[rr]_{z_1} \ar[r]_u & P \ar[d]^{p_1} \ar[r]_{p_2} & B \ar[d]^g \\ + & & A \ar[r]^f & C + }\] + $P$ may be denoted $A \times_C B$ when $f,g$ are clear. + \end{dfn} + +\section{Properties of UCs} + + \begin{thm}Equalizers are monic.\end{thm} + + \begin{thm}(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 + \[\xymatrix{ F \ar[r]_{f'} \ar[d]^{h''} & E \ar[r]_{g'} \ar[d]^{h'} & D \ar[d]^{h} \\ + A \ar[r]^f & B \ar[r]^g & C + }\] + \begin{enumerate} + \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} + +\section{Special Functors} + + \begin{dfn}The \defn{covariant representable functor} (p44) is + \[\mbox{Hom}(A,\text{---}) : \mathbf{C} \to \mathbf{Sets}\] + \end{dfn} + + \begin{thm}(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 + 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}: + \[ \forall_{m : M \rightarrowtail X} + \brak{ z \in_X M \Leftrightarrow \exists_{f:Z \to M} z = mf} \] + \end{dfn} + +\end{document} + +% vim:ts=2:expandtab -- 2.50.1