+% Header <<<
+\RequirePackage[l2tabu,orthodox]{nag}
+\RequirePackage{fixltx2e}
+
\documentclass[10pt,twocolumn,letterpaper]{article}
\DeclareSymbolFont{AMSb}{U}{msb}{m}{n}
\DeclareMathAlphabet{\mathbbm}{U}{bbm}{m}{n}
\title{Category Theory Cheat Sheet}
%\author{Nathaniel Wesley Filardo}
+\usepackage{xparse}
\usepackage{amsmath,amssymb,amsthm,latexsym}
\usepackage{fancyhdr}
-\usepackage[tiny,center,compact,sc]{titlesec}
+\usepackage{titlesec} % [tiny,center,compact,sc]
\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{breakurl}
\usepackage{enumitem}
\usepackage{etoolbox}
\usepackage{hyperref}
+\hypersetup{
+ colorlinks,
+ linkcolor={red!50!black},
+ citecolor={blue!50!black},
+ urlcolor={blue!80!black}
+}
+\usepackage{makeidx}\makeindex
+
%\setlist{nolistsep}
%http://tex.stackexchange.com/questions/126750/how-can-i-number-paragraphs-without-higher-level-counters
\DeclareMathOperator{\mm}{\mid\mid}
\newcommand{\natto}{\overset{\cdot}{\to}}
-\newcommand{\defn}[2][]{{\ifstrempty{#1}{\label{defn:#2}}{\label{defn:#1}}{\bf #2}}}
+\newcommand{\defnref}[3][]{\ifstrempty{#1}{\ref{defn:#2}}{\ref{defn:#1}}}
+\DeclareDocumentCommand{\defn}{ O{} D<>{} m }{%
+ {\ifstrempty{#1}%
+ {\label{defn:#3}}%
+ {\label{defn:#1}}%
+ \ifstrempty{#2}%
+ {\index{#3|defnref[#1]{#3}}\bfseries #3}%
+ {\index{#2#3|defnref[#1]{#3}}\bfseries #3}}%
+}
\newcommand\xrdefnhelper[1]{defn:#1}
\newcommand{\xrdefn}[1]{\ref{\forcsvlist{\xrdefnhelper}{#1}}}
-
+\newcommand{\hrdefn}[2][]{\ifstrempty{#1}{\hyperref[defn:#2]{#2}[\ref{defn:#2}]}
+ {\hyperref[defn:#1]{#2}[\ref{defn:#1}]}}
\begin{document}
-
-Unless otherwise notated, references are to Ji\v{r}\'i Ad\'amek, Horst
-Herrlich, George E. Strecker's \textit{Abstract and Concrete Categories: The
-Joy of Cats}. Notation follows theirs with some contamination from Awodey's
-\textit{Category Theory} and Pierce's \textit{Basic Category Theory for
-Computer Scientists}.
+%>>>
+% Intro <<<
+Unless otherwise notated, references are to \textit{Abstract and Concrete
+Categories: The Joy of Cats}, \cite{adamek:joy}. Notation follows theirs
+with some contamination from Awodey's \textit{Category Theory},
+\cite{awodey:ct}, and Pierce's \textit{Basic Category Theory for Computer
+Scientists}, \cite{pierce:basicct}.
Entries within each section are roughly sorted by definition, alphabetically.
-\section{Basics}
+%>>>
+\section{Basics} % <<<
\paragraph{}
%
\item \defn{thin} if $\forall_{A,B} \mbox{hom}(A,B) \simeq \set{*}$. (\S3.26.2)
\end{itemize}
-\section{Derived Categories}
+% >>>
+\section{Derived Categories} % <<<
\paragraph{}
%
- The \defn[conecat]{cone} category over a given diagram, $\mathbf{Cone}(D(J))$, has
- as objects cones to that diagram and a morphism between cones is an arrow
- $\phi : C \to C'$ s.t. $\forall_{D_j \in D(J)} c_j^\prime \circ \phi =
- c_j$. \xrdefn{cone}
+ The \defn{arrow} (Awodey:p16,i3) category $\mathbf{C}^\to$ has arrows for
+ commutative squares in $\mathbf{C}$. There are two functors
+ $\mathbf{cod}, \mathbf{dom} : \mathbf{C}^\to \to \mathbf{C}$.
\paragraph{}
%
- The \defn{dual} (\S3.5;Awodey:p15,i2) category $\mathbf{A}^\text{op}$
- which exchanges domains and codomains of arrows in $\mathbf{A}$. Any
- statement implies its dual.
+ The \defn[conecat]{cone} category over a given diagram,
+ $\mathbf{Cone}(D(J))$, has as objects \hrdefn[cone]{cones} to that diagram
+ and a morphism between cones is an arrow $\phi : C \to C'$ s.t.
+ $\forall_{D_j \in D(J)} c_j^\prime \circ \phi = c_j$.
\paragraph{}
%
- The \defn{arrow} (Awodey:p16,i3) category $\mathbf{C}^\to$ has arrows for
- commutative squares in $\mathbf{C}$. There are two functors
- $\mathbf{cod}, \mathbf{dom} : \mathbf{C}^\to \to \mathbf{C}$.
+ The \defn{dual} (\S3.5;Awodey:p15,i2) category $\mathbf{A}^\text{op}$
+ which exchanges domains and codomains of arrows in $\mathbf{A}$. Any
+ purely-categorical statement implies its dual.
\paragraph{}
%
arrows in $\mathbf{C}$ with codomain $C$. Arrows are tops of commutative
triangles.
-\section{Object Properties}
+% >>>
+\section{Object Properties} % <<<
\paragraph{}
%
& A & \ar[l]_{\pi_1} A \times B \ar[r]^{\pi_2} & B \\
}\]
+ \paragraph{}
+ %
+ The \defn{product category} $\mathbf{C} \times \mathbf{D}$ of two
+ categories $\mathbf{C}$ and $\mathbf{D}$ consists of objects which are
+ each an ordered pair of an object from $\mathbf{C}$ and one from
+ $\mathbf{D}$; morphisms are, similarly, pairs of morphisms from
+ $\mathbf{C}$ and $\mathbf{D}$. This sense of $\times$ is itself the
+ trivial \hrdefn{bifunctor}.
+
\paragraph{}
%
$(P,p_1,p_2)$ is a \defn{pullback} (Awodey:p80,D5.4) of $f,g$ iff (UMP)
An object that is both initial and terminal is called a \defn{zero}.
(\S7.7)
-\section{Arrow Properties}
+% >>>
+\section{Arrow Properties} % <<<
\paragraph{}
$(Q,q)$ is a \defn{coequalizer} (\S7.51) of $f,g$ iff (UMP) $qf = qg$ and
\paragraph{}
%
Let $G: \mathbf{A} \to \mathbf{B}$ and $B \in \mathbf{B}$. A
- \defn[gstrarr]{$G$-structured arrow with domain $B$} is a pair $(f : B \to GA, A)$.
- (\S8.30) It is
+ \defn[gstrarr]<@G-structured arrow with domain B> {$G$-structured arrow
+ with domain $B$} is a pair $(f : B \to GA, A)$. (\S8.30) It is
+ %
\begin{itemize}
+ %
\item \defn{generating} if $\forall_{r,s : A \to A'} . Gr \circ f = Gs
\circ f \implies r = s$
+ %
\item \defn{extremally generating} if it is generating and $\forall_{m :
A' \to A, m ~\text{mono}, (g,A')} . f = Gm \circ g \implies m ~\text{iso}$.
- \item \defn{$G$-universal for $B$} if $\forall_{(f', A')} .
+ %
+ \item \defn[gunivarr]<@G-universal for B>{$G$-universal for $B$} if
+ $\forall_{(f', A')} .
+ %
\exists!_{\check f} . f' = G{\check f} \circ f$. That is,
\[\xymatrix{
B \ar[r]^f \ar@/_1.25pc/[rr]^{f'}
\end{itemize}
%(XXX stopped around \S7.60; there's more to be said)
-\section{Exponentials}
+% >>>
+\section{Exponentials} % <<<
\paragraph{}
%
\paragraph{}
%
Exponential transposition is self inverse (Awodey:p108). This implies
- \[ \mbox{hom}_{\bf C}(A \times B, C) \simeq \mbox{hom}_{\bf C}(A, C^B) \]
+ \[ \mbox{hom}_{\mathbf{C}}(A \times B, C) \simeq \mbox{hom}_{\mathbf{C}}(A, C^B) \]
+
+ \paragraph{}
+ %
+ The \defn{exponential category} $\mathbf{D}^\mathbf{C}$ has as objects
+ \hrdefn[functor]{functors} from $\mathbf{C}$ to $\mathbf{D}$ and as
+ morphisms the \hrdefn[nattrans]{natural transformations} between these
+ functors.
\paragraph{}
%
A category is \defn{cartesian closed} (Awodey:p108,D6.2) if it has all
finite products and exponentials.
-\section{Functors}
+% >>>
+\section{Functors} % <<<
+% Basics <<<
\paragraph{}
%
\paragraph{}
%
- A \defn{(covariant) functor} $F$ (\S3.17;Awodey:D1.2) assigns to each
- $\mathbf{A}$-object a $\mathbf{B}$-object and to each
- $\mathbf{A}$-morphism a $\mathbf{B}$-morphism s.t. composition and
- identites are {\em preserved}.
+ A \defn{covariant functor} (or just \defn{functor}) $F$
+ (\S3.17;Awodey:D1.2) assigns to each $\mathbf{A}$-object a
+ $\mathbf{B}$-object and to each $\mathbf{A}$-morphism a
+ $\mathbf{B}$-morphism s.t. composition and identites are {\em preserved}.
\paragraph{}
%
- A \defn{contravariant functor} $F$ (\S3.20.5) is a (covariant) functor
+ A \defn[contrafunc]{contravariant functor} $F$ (\S3.20.5) is a (covariant) functor
$\mathbf{A}^\text{op} \to \mathbf{B}$.
\paragraph{}
and $\bar L = \varprojlim_j D_j$.
\end{itemize}
+ \paragraph{}
+ %
+ A (covariant) \defn{bifunctor} is a functor from a \hrdefn{product
+ category} such that each partial application is {\em also} a functor.
+ (See \cite{hinze:f} and bifunctors.tex for more.) A \defn{profunctor} is a
+ bifunctor which is \hrdefn[contrafunc]{contravariant} in one argument and
+ covariant in the other.
+
\paragraph{}
%
A functor $F$ is (\S3.27, \S3.33)
\paragraph{}
%
- A \defn{natural transformation} $\tau : F \natto G$ assigns each
+ All functors \defn{preserve} (in $\mathbf{A}$ implies in $\mathbf{B}$)
+ isomorphisms (\S3.21), sections (\S7.22), and retractions (\S7.28).
+
+ \paragraph{}
+ %
+ Some functors \defn{reflect} (in $\mathbf{B}$ implies in $\mathbf{A}$) useful properties:
+ \begin{itemize}
+ \item Full, faithful functors reflect sections (\S7.23) and retractions (\S7.29).
+ \item Faithful functors reflect monos (\S7.37.2) and epis (\S7.44).
+ \end{itemize}
+
+% >>>
+\subsection{Transformations} % <<<
+
+ \paragraph{}
+ %
+ A \defn[nattrans]{natural transformation} $\tau : F \natto G$ assigns each
$A \in \mathbf{A}$ to $\tau_A : FA \to GA$ s.t.
$\forall_{f : A \to A'} . G f \circ \tau_A = \tau_{A'} \circ F f$
(\S6.1;Awodey:D7.6).
{}\save[]+<-1cm,0cm>*\txt<8pc>{$\forall_{A,B,f \in C}$\\$Gf\circ \tau_A = \tau_B\circ Ff$}\restore
& FA \ar[r]^{\tau_A} \ar[d]_{Ff} & GA \ar[d]^{Gf} \\
& FB \ar[r]^{\tau_B} & GB} \]
+ %
+ More generally, given any functor from a \hrdefn{product category}, we may
+ say that it is natural in the $i$-th position if, for all ways of fixing
+ the other positions, the resulting partial applications form natural
+ transformations.
\paragraph{}
%
$(H\tau)_A = H(\tau_A)$ and $\tau H : FH \natto GH$ defined by $(\tau H)_A
= \tau_{HA}$.
- \paragraph{}
- %
- All functors \defn{preserve} (in $\mathbf{A}$ implies in $\mathbf{B}$)
- isomorphisms (\S3.21), sections (\S7.22), and retractions (\S7.28).
+% XXX Not yet
+% \paragraph{}
+% %
+% A \defn[exttrans]{extranatural transformation} is one where
+
+% >>>
+\subsection{Special Functors} % <<<
\paragraph{}
%
- Some functors \defn{reflect} (in $\mathbf{B}$ implies in $\mathbf{A}$) useful properties:
- \begin{itemize}
- \item Full, faithful functors reflect sections (\S7.23) and retractions (\S7.29).
- \item Faithful functors reflect monos (\S7.37.2) and epis (\S7.44).
- \end{itemize}
-
-\subsection{Special Functors}
+ For every category $\mathbf{C}$ and object $D \in \mathbf{D}$ there is
+ a unique \defn{constant functor} $\mathbf{!}_D$ which sends every
+ $C$ to $D$ and every $f$ to $1_D$.
\paragraph{}
%
\mapsto (C' \times_C A \stackrel{\alpha'}{\to} C') \in \mathbf{C}/C' \]
where $\alpha'$ is the pullback of $\alpha$ along $h$. (Awodey:P5.10)
-
-\section{Cones and Sources}
+% >>>
+% >>>
+\section{Cones and Sources} % <<<
\paragraph{}
%
A \defn{cone} (Awodey:D5.15) to a diagram $D(J)$ is a collection of arrows
$c_j : C \to D_j$ s.t. $\forall_{D_\alpha \in D(J)} c_j = D_\alpha \circ
- c_i$.
+ c_i$. (Cones are also \hrdefn[nattrans]{natural
+ transformations} from the \hrdefn{constant functor} to the inclusion
+ functor of the diagram $D$. \cite{milewski:limits}) (Cones are
+ \hrdefn[source]{sources} subject to commutation diagrams implied by the
+ diagram.)
\paragraph{}
%
A \defn{mono-source} (\S10.5) is $(A,\set{f_i})$ s.t. \\ $\forall r,s: B \to A.
\brak{\forall_{i\in I} . f_i \circ r = f_i \circ s} \Rightarrow r = s$.
-\section{Concrete Categories}
+% >>>
+% \section{Ends} % <<<
+% XXX not yet
+% \paragraph{}
+% %
+%
+%
+% >>>
+\section{Concrete Categories} % <<<
\paragraph{}
%
For this section, $\mathbf{A}$ is a \defn{concrete category} over
- $\mathbf{X}$ with \defn{forgetful} functor $U : \mathbf{A} \to \mathbf{X}$
- faithful, denoted $(\mathbf{A}, U)$. (\S5.1.1)
+ $\mathbf{X}$ with \defn{forgetful} \hrdefn{functor} $U : \mathbf{A} \to
+ \mathbf{X}$ \hrdefn{faithful}, denoted $(\mathbf{A}, U)$. (\S5.1.1)
\paragraph{}
%
- When $\mathbf{A} = \mathbf{X}$, $\mathbf{Alg}(U)$ has as objects
- $U$-\defn{algebra}s $(X \in \mathbf{X}, h : UX \to X)$ and morphisms
- $f : (X,h) \to (X',h')$ s.t. $f \circ h = h' \circ T(f)$. (\S5.37)
+ When $\mathbf{A} = \mathbf{X}$, $\mathbf{Alg}(U)$ has
+ \hrdefn[falg]{$U$-algebras} as objects and algebra homomorphisms as
+ morphisms.
\paragraph{}
%
\paragraph{}
%
- $(UA \overset{f}{\to} UB) \in \mathbf{X}$ \defn[Amporphism]{is an $\mathbf{A}$-morphism}
- if $f$ has a unique $U$-preimage in $\mathbf{A}$. (\S5.3, \S6.22)
+ $(UA \overset{f}{\to} UB) \in \mathbf{X}$ \defn[Amporphism]{is an
+ $\mathbf{A}$-morphism} if $f$ has an {\em unique} $U$-preimage in
+ $\mathbf{A}$. (\S5.3, \S6.22)
%An object $A\in\mathbf{A}$ is
%\dots\! if $\forall_{B \in \mathbf{A}}$, \dots is an $\mathbf{A}$ arrow.
\paragraph{}
%
A \defn{free object} $A \in \mathbf{A}$ is one with a ($U$-structured)
- universal arrow $(u,UA)$ in $B$. (\S8.22+\S8.30)
+ universal arrow $(u,UA)$ in $B$. (\S8.22+\S8.30) \xrdefn{gunivarr}
%$f \in \mathbf{A}$ is \defn{initial} if $\forall_{C \in \mathbf{A}}$ $UC
%\overset{f \circ g}{\to} UB$ is an $\mathbf{A}$-morphism implies that $UC
%\overset{g}{\to} UA$ is an $\mathbf{A}$-morphism.
-\section{Adjoints and Adjoint Situations}
+% >>>
+\section{Adjoints and Adjoint Situations} % <<<
\subsection{Joy Approach}
%
A functor $G : \mathbf{A} \to \mathbf{B}$ is \defn{adjoint} if
$\forall_{B \in \mathbf{B}}$ there exists a $G$-structured universal
- arrow with domain $B$. (\S18.1)
+ arrow with domain $B$. (\S18.1) \xrdefn{gunivarr}
\paragraph{}
%
- Adjoints compose (\S8.5), preserve mono sources (\S8.6), and preserve
- limits (\S8.9)
+ Adjoints compose (\S8.5), preserve \hrdefn[mono-source]{mono-sources}
+ (\S8.6), and preserve \hrdefn[limit]{limits} (\S8.9)
\paragraph{}
%
\paragraph{}
%
- An \defn{adjunction} (Awodey:D9.1) of $F : C \to D$ and $G : D \to C$ is a natural
- transformation $\eta : I_C \stackrel{\cdot}{\to} (G\circ F)$ s.t.
+ An \defn{adjunction} (Awodey:D9.1) of $F : C \to D$ and $G : D \to C$ is a
+ \hrdefn[nattrans]{natural transformation} $\eta : I_C
+ \stackrel{\cdot}{\to} (G\circ F)$ s.t.
+ %
\[\xymatrix{
{}\save[]+<-1cm,0cm>*\txt<8pc>{$\forall_{f:X \to GY}\exists!_{f^\#:FX\to Y}$\\
$f = Gf^\# \circ \eta_X$}\restore
& FX\ar@{..>}[d]^{f^\#} & X \ar[dr]^f \ar[r]^{\eta_X} & GFX \ar@{..>}[d]^{Gf^\#} \\
& Y & & GY
}\]
+ %
Equivalently (Awodey:D9.7), a natural {\em isomorphism}
- \[ \phi : \mbox{Hom}_D(FC,D) \simeq \mbox{Hom}_C(C,GD), \quad \eta_X = \phi(1_{FX}) \]
+ %
+ \[ \phi : \mbox{Hom}_D(FC,D) \simeq \mbox{Hom}_C(C,GD),
+ \quad \eta_X = \phi(1_{FX}) \]
\subsection{Moving Right Along}
& TX &
}\]
+% >>>
\appendix
-\section{Miscellaneous Terminology}
+\section{Miscellaneous Terminology} % <<<
+
+ \paragraph{}
+ %
+ Given an \hrdefn{endofunctor} $F$ on $\mathbf{C}$, a
+ \defn[falg]<@F-algebra>{$F$-algebra} is a pair of a \defn{carrier} $X \in
+ \mathbf{C}$ and interpretation morphism $h : FX \to X \in \mathbf{C}$. A
+ \defn{algebra homomorphism} is a morphism $f$ such that $f : (X,h) \to
+ (X',h')$ s.t. $f \circ h = h' \circ T(f)$. (\S5.37)
+
\paragraph{}
%
\paragraph{}
%
- An \defn[wCPO]{$\omega$-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.) (Awodey:p101,E5.33)
+ An \defn[wCPO]<@w-complete Partial Order>{$\omega$-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.)
+ (Awodey:p101,E5.33)
-\section{Miscellaneous Useful Properties}
+% >>>
+\section{Miscellaneous Useful Properties} % <<<
\paragraph{}
%
%
Objects defined by UCs are unique up to isomorphism.
-\section{Examples To Jog Your Memory}
+% >>>
+\section{Examples To Jog Your Memory} % <<<
\subsection{$\mathbf{Set}$}
\paragraph{}
%
- Mono-epics are not isos ($(\mathbf{N},+,0) \to (\mathbf{Z},+,0)$).
- (Pierce:\S1.6.3)
+ \hrdefn[bimorphism]{Bimorphisms} are not isos: ($(\mathbf{N},+,0) \to
+ (\mathbf{Z},+,0)$). (Pierce:\S1.6.3)
\paragraph{}
%
F(X^*)$ (that is, a concatenation of symbols from $GFX$) and re-imposes
structure to obtain $\epsilon_{FX} y \in FX$.
+% >>>
+\section{Bootstrapping Category Theory} % <<<
+
+ \paragraph{}
+ %
+ \defn[catcat]<@Cat>{$\mathbf{Cat}$} is the category which has locally
+ small categories as objects and \hrdefn[functor]{functors} as morphisms.
+ (It is not, itself, locally small, and so is not an object in itself.)
+ $\mathbf{Cat}$ is \hrdefn{cartesian closed} (see \hrdefn{product category}
+ and \hrdefn{exponential category}). Its initial object is the empty
+ category and its terminal object is the category of a single object and
+ its identity morphism.
+
+% >>>
+% Footer <<<
+\printindex
+
+\bibliographystyle{alphaurl}
+\bibliography{ctcheat}
+
\end{document}
+
+% vim: ai:expandtab:ts=4:foldmethod=marker:foldmarker=<<<,>>>
+% >>>
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