From: Nathaniel Wesley Filardo Date: Tue, 9 Sep 2014 21:16:54 +0000 (-0400) Subject: Unify cheatsheets, add .gitignore X-Git-Url: https://hydra-www.ietfng.org/gitweb/?a=commitdiff_plain;h=aa866de9a53e1d1366f129e196ddb830598d9c9d;p=ctcheat Unify cheatsheets, add .gitignore --- diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..48de77e --- /dev/null +++ b/.gitignore @@ -0,0 +1,7 @@ +*.aux +*.dvi +*.fdb_latexmk +*.fls +*.log +*.out +*.pdf diff --git a/ctcheat-joy.tex b/ctcheat-joy.tex deleted file mode 100644 index 260845e..0000000 --- a/ctcheat-joy.tex +++ /dev/null @@ -1,344 +0,0 @@ -\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{amsmath,amssymb,amsthm,latexsym} -\usepackage{fancyhdr} -\usepackage[tiny,center,compact,sc]{titlesec} -\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} -\usepackage{enumitem} -\setlist{nolistsep} - -\renewcommand{\baselinestretch}{0.9} - -%\newtheorem{thm}{Thm}[section] -%\newtheorem{dfn}{Def}[section] - -\setlength{\parindent}{0pt} -\setlength{\parskip}{2pt} - -% http://www.latex-community.org/forum/viewtopic.php?f=46&t=3837&start=0#p15112 -\makeatletter -\g@addto@macro\normalsize{% -\setlength\abovedisplayskip{0pt}% -\setlength\abovedisplayshortskip{0pt}% -\setlength\belowdisplayskip{0pt}% -\setlength\belowdisplayshortskip{0pt}% -} -\makeatother - -% http://comments.gmane.org/gmane.comp.tex.xy-pic/223 -% fixes {>->} having the tail overlap the item. -\newdir{ >}{{}*!/-2.6667\jot/\dir{>}} - - -%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{\natto}{\overset{\cdot}{\to}} - -\newcommand{\defn}[1]{{\bf #1}} - -\begin{document} - -References are to Ji\v{r}\'i Ad\'amek, Horst Herrlich, George -E. Strecker's {\em Abstract and Concrete Categories: The Joy of Cats}. -Notation follows theirs with some contamination from Awodey and Pierce's -texts. - -Entries within each section are roughly sorted by definition, alphabetically. - -\section{Basics} - - A \defn{category} $\mathbf{C}$ (\S3.1) is a quadruple - $(\mathcal{O},\mbox{hom},id,\circ)$ with - \begin{itemize} - \item A collection of objects $\mathcal{O}$ - \item For each pair of objects $A,B$, a (disjoint) collection of arrows - from \defn{domain} $A$ to \defn{codomain} $B$, - $\mbox{hom}(A,B)$ (also written $\mathbf{C}(A,B)$). - \item An associative arrow composition operator $\circ$. - \item Identity arrows ($id_A$) on each object $A$, unit of $\circ$ - \end{itemize} - - The \defn{dual} (\S3.5) category $\mathbf{A}^\text{op}$ which - exchanges domains and codomains of arrows in $\mathbf{A}$. - - A predicate $P$ is \defn{essentially unique} (\S7.3) if it is unique up to - isomorphism: - \begin{itemize} - \item If both $PA$ and $PB$, then $A \simeq B$ - \item If $PA$ and $A \simeq B$, then $PB$. - \end{itemize} - - - A category is$\dots$ - \begin{itemize} - \item \defn{balanced} if all bi are iso (\S7.49.2) - \item \defn{discrete} if all morphisms are identities. (\S3.26.1) - \item \defn{thin} if $\forall_{A,B} \mbox{hom}(A,B) \simeq \set{*}$. (\S3.26.2) - \end{itemize} - -\section{Kinds of Objects} - - $C$ is a \defn{coseparator} if $\forall_{f,g : B \to A} . f \ne g - \Rightarrow \exists_{h : A \to C} . h \circ f \ne h \circ g$. (\S7.17) - (Contrast monomorphism.) - - An object $0$ is \defn{initial} if $\forall_B . \exists! f_B : 0 \to B$. - Initial objects are essentially unique. (\S7.1) - - $S$ is a \defn{separator} if $\forall_{f,g : A \to B} . f \ne g - \Rightarrow \exists_{h : S \to A} . f \circ h \ne g \circ h$. (\S7.10) - (Contrast epimorphism.) - - $S$ is a separator iff $\mbox{hom}(S,-)$ is faithful. (\S7.12) - - A set of objects $\mathcal{T}$ is a \defn{separating set} if - $\forall_{f,g : A \to B} . f \ne g \Rightarrow \exists{S \in \mathcal{T}, - h : S \to A} . f \circ h \ne g \circ h$. (\S7.14) - - An object $1$ is \defn{terminal} if $\forall_A . \exists! f_A : A \to 1$. - Terminal objects are essentially unique. (\S7.4) - - An object that is both initial and terminal is called a \defn{zero}. - (\S7.7) - -\section{Kinds of Arrows} - - $e$ is an \defn{epimorphism} (\S7.39) (the dual of a monomorphism) if - $ie = je \Rightarrow i = j$ in - \[\xymatrix{A \ar@{->>}[r]^e & B \ar@<1ex>[r]^{i} \ar@<-1ex>[r]_j & C} \] - If $f$ and $g$ are epis, then so is $g \circ f$; if $g \circ f$ is epi, - then so is $g$. (\S7.41) Epis generalize \defn{surjection} - - $(E,e)$ is an \defn{equalizer} (\S7.51) of $f,g$ iff $fe = ge$ and - \[\forall_{Z,z . zf = zg} \exists!_u eu = z \quad - \xymatrix{ - Z \ar@{..>}[r]^u \ar@/_1pc/[rr]^{z} & E \ar[r]^e & A \ar@<1ex>[r]^f \ar@<-1ex>[r]_g & B \\ - }\] - Equalizers are essentially unique. (\S7.53) - - A mono $m$ is a \defn{extremal} (\S7.61) if $e$ epic and - $m = f \circ e$ implies that $e$ iso. - - Let $G: \mathbf{A} \to \mathbf{B}$ and $B \in \mathbf{B}$. A - \defn{$G$-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 $\forall_{(f', A')} . - \exists!_{\check f} . f' = G{\check f} \circ f$. That is, - \[\xymatrix{ - B \ar[r]^f \ar@/_1.25pc/[rr]^{f'} - & GA \ar@{.>}[r]^{G{\check f}} - & GA' - & A \ar@{.>}[r]^{\check f} - & A' - }\] - \end{itemize} - - $f : A \to B$ is an \defn{isomorphism} if $\exists!_g . f \circ g = id_B - ~\wedge~ g \circ f = id_A$. (\S3.8; ! in \S3.11) - - $f$ is a \defn{monomorphism} (\S7.32) if $mi = mj \Rightarrow i = j$ in - \[\xymatrix{C \ar@<1ex>[r]^{i} \ar@<-1ex>[r]_j & A \ar@{{ >}->}[r]^m & B} \] - If $f$ and $g$ are monos, then so is $g \circ f$; if $g \circ f$ is mono, - then so is $f$. (\S7.34) Monos generalize \defn{injection} in - $\mathbf{Set}$. - - $f$ is a \defn{regular monomorphism} (\S7.56) if it is an equalizer of - some pair of morphisms. - - $f : A \to B$ is a \defn{retraction} if $\exists_g . f \circ g = 1_B$. - If $f$ and $g$ are retractions, then so is $g \circ f$; if $g \circ f$ - is a retraction, then so is $g$. (\S7.27) - - $f : A \to B$ is a \defn{section} if $\exists_g . g \circ f = 1_A$. - (\S7.19) - If $f$ and $g$ are sections, then so is $g \circ f$; - if $g \circ f$ is a section, then so is $f$. (\S7.21) - - Several morphism properties combine in useful ways: - \begin{itemize} - \item mono, epi $\Rightarrow$ \defn{bimorphism} (\S7.49) - \item section $\Rightarrow$ regular mono (\S7.35, \S7.59.1) - \item regular mono $\Rightarrow$ extremal mono (\S7.59.2, \S7.63) - \item retraction $\Rightarrow$ epi (\S7.42) - \item mono, retraction $\Leftrightarrow$ isomorphism (\S7.36) - \item section, epi $\Leftrightarrow$ isomorphism (\S7.43) - \end{itemize} - %(XXX stopped around \S7.60; there's more to be said) - -\section{Functors} - - By default, functors $F,G : \mathbf{A} \to \mathbf{B}$. - - A \defn{(covariant) functor} $F$ (\S3.17) 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{contravariant functor} $F$ (\S3.20.5) is a (covariant) functor - $\mathbf{A}^\text{op} \to \mathbf{B}$. - - A \defn{endofunctor} has $\mathbf{A} = \mathbf{B}$. $F \circ F$ may be - denoted $F^2$, etc. (\S3.23; ftn 15) - - Functors compose. (\S3.23) - - A functor $F$ is (\S3.27, \S3.33) - \begin{itemize} - \item \defn{amnestic} if $f$ is an identity iff $Ff$ is an identity. - \item an \defn{equivalence} if it is full, faithful, and - isomorphism-dense. - \item an \defn{embedding} if it is injective on morphisms. - \item \defn{faithful} if $\forall_{A,A'} . F\vert_{\mathbf{A}(A,A')} - \subseteq \mathbf{B}(FA, FA')$ is injective. - \item \defn{full} if $\forall_{A,A'} . F\vert_{\mathbf{A}(A,A')}$ surjective. - \item \defn{isomorphism-dense} if $\forall_B . \exists_A . F(A) \simeq B$. - \end{itemize} - - A \defn{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) - - There is special notation for functors applied to natural transformations - and vice-versa (\S6.2): $(F\tau)_A = F(\tau_A)$ and $(\tau F)_A = - \tau_{FA}$. - - All functors \defn{preserve} (in $\mathbf{A}$ implies in $\mathbf{B}$)$\dots$ - \begin{itemize} - \item isomorphisms. (\S3.21) - \item sections (\S7.22) - \item retractions (\S7.28) - \end{itemize} - - Representable functors preserve monos. (\S7.37.1) - - 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 (7.44). - \end{itemize} - -\section{Sources and Sinks} - - A \defn{source} in category $\mathbf{A}$ indexed by $I$ is a pair $(A, - \set{f_i : A \to A_i}_{i \in I})$. This source has domain $A$ and - codomain $\set{A_i}_{i\in I}$. (\S10.1) - - Given $(A,\set{f_i}_{i \in I})$ and - $\{(A_i,\set{g_{ij}}_{j \in J_i})\}_{i \in I}$ - all sources, their \defn{composite} is $(A, \set{g_{ij} \circ f_i}_{i\in I, - j\in J_i})$. (\S10.3) - - 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} - - 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) - - 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) - - If $\mathbf{X}$ is $\mathbf{Set}$, $\mathbf{A}$ is a \defn{construct}. - (\S5.1.2) - - $(UA \overset{f}{\to} UB) \in \mathbf{X}$ \defn{is an $\mathbf{A}$-morphism} - if $f$ has a 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. - %\begin{itemize} - % \item \defn{discrete}, $(UA \to UB)$ (\S8.1) - % \item \defn{indiscrete}, $(UB \to UA)$ (\S8.3) - %\end{itemize} - - A \defn{free object} $A \in \mathbf{A}$ is one with a ($U$-structured) - universal arrow $(u,UA)$ in $B$. (\S8.22+\S8.30) - - %$f \in \mathbf{A}$ 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} - - 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) - - Adjoints compose (\S8.5), preserve mono sources (\S8.6), and preserve - limits (\S8.9) - - Given adjoint $G$ with $\eta_B : B \to G(A_B)$ the $G$-structured - universal arrow with domain $B$, $\exists!_F$ such that $FB = A_B$ and - $\eta : id_B \natto G \circ F$ is natural; further, there is a unique, - natural $\epsilon : F \circ G \natto id_A$ with $G\epsilon \circ \eta G = - id_G$ and $\epsilon F \circ F \eta = id_F$. (\S19.1) - - $(\eta,\epsilon) : F \dashv G : \mathbf{A} \to \mathbf{B}$ is a - \defn{adjoint situation} if the above relationships hold. (\S19.7) - - A \defn{monad} (\S20.1) on $\mathbf{X}$ is $(T : \mathbf{X} \to \mathbf{X}, - \eta : id_{\mathbf{X}} \natto T, \mu : T^2 \natto T)$ s.t. - \[\forall_X \quad - \xymatrix@R=10pt{ - T^3X \ar[r]^{T(\mu_X)} \ar[d]^{\mu_{TX}} & T^2X \ar[d]^{\mu_{X}} \\ - T^2X \ar[r]^{\mu_X} & TX - } \quad \xymatrix@R=10pt{ - TX \ar[r]^{T(\eta_X)} \ar[dr]_{id_{TX}} & T^2X \ar[d]^{\mu_X} & TX \ar[l]_{\eta_{TX}} \ar[dl]^{id_{TX}} \\ - & TX & - }\] - - -\pagebreak\appendix\section{Examples To Jog Your Memory} - -\subsection{$\mathbf{Mon}$} - - $(\set{*},\cdot,id_*)$ is a zero (both initial and terminal). - - Mono-epics are not isos ($(\mathbf{N},+,0) \to (\mathbf{Z},+,0)$). - (Pierce:\S1.6.3) - -\end{document} diff --git a/ctcheat.tex b/ctcheat.tex index 03c4820..379bb06 100644 --- a/ctcheat.tex +++ b/ctcheat.tex @@ -20,6 +20,8 @@ \usepackage{multirow} \usepackage{hyperref} \usepackage{breakurl} +\usepackage{enumitem} +%\setlist{nolistsep} \renewcommand{\baselinestretch}{0.9} @@ -27,7 +29,22 @@ %\newtheorem{dfn}{Def}[section] \setlength{\parindent}{0pt} -\setlength{\parskip}{3pt} +\setlength{\parskip}{5pt} + +% http://www.latex-community.org/forum/viewtopic.php?f=46&t=3837&start=0#p15112 +\makeatletter +\g@addto@macro\normalsize{% +\setlength\abovedisplayskip{0pt}% +\setlength\abovedisplayshortskip{0pt}% +\setlength\belowdisplayskip{0pt}% +\setlength\belowdisplayshortskip{0pt}% +} +\makeatother + +% http://comments.gmane.org/gmane.comp.tex.xy-pic/223 +% fixes {>->} having the tail overlap the item. +\newdir{ >}{{}*!/-2.6667\jot/\dir{>}} + %Scalable bracket-like \newcommand{\paren}[1]{\left({#1}\right)} @@ -51,230 +68,475 @@ \newcommand{\dyad}[2]{\left\lvert{#1}\middle\rangle\middle\langle{#2}\right\rvert} \DeclareMathOperator{\mm}{\mid\mid} +\newcommand{\natto}{\overset{\cdot}{\to}} \newcommand{\defn}[1]{{\bf #1}} \begin{document} -%\maketitle -\bibliographystyle{plainurl} +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}. + +Entries within each section are roughly sorted by definition, alphabetically. \section{Basics} - A \defn{category} (p4,\S1.3) is a structure with + A \defn{category} $\mathbf{C}$ (\S3.1) is a quadruple + $(\mathcal{O},\mbox{hom},id,\circ)$ with \begin{itemize} - \item Objects \& arrows (from \defn{domain} to \defn{codomain}). + \item A collection of objects $\mathcal{O}$ + \item For each pair of objects $A,B$, a (disjoint) collection of arrows + from \defn{domain} $A$ to \defn{codomain} $B$, + $\mbox{hom}(A,B)$ (also written $\mathbf{C}(A,B)$). \item An associative arrow composition operator $\circ$. - \item Identity arrows ($1_A$) on each object $A$, unit of $\circ$ + \item Identity arrows ($id_A$) on each object $A$, unit of $\circ$ \end{itemize} - 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. - - The \defn{dual} (p15,i2) category $\mathbf{C}^{op}$ which - exchanges domains and codomains of arrows in $\mathbf{C}$. - - Any CT statement implies its dual (interchange dom/cod and reverse compositions). - - \subsection{Categories over $\mathbf{C}$'s objects} - - The \defn{arrow} (p16,i3) category $\mathbf{C}^\to$ has arrows - for commutative squares in $\mathbf{C}$. There are two functors - \[\xymatrix{ \mathbf{C} & \ar[l]_{\mathbf{dom}} C^\to \ar[r]^{\mathbf{cod}} & \mathbf{C}}\] - - 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. - - \subsection{Foundations} - - Categories may be described (p21) as + Categories may be described (Awodey:p21) as \[\xymatrix{ C_2 \ar[r]^\circ & C_1 \ar@<2ex>[r]_{cod} \ar@<-2ex>[r]_{dom} & C_0 \ar[l]^i }\] - A category is (p24-25,D1.11-12)\dots + A category is (Awodey:p24-25,D1.11-12)\dots \begin{itemize} \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. + \item \defn{locally small} if $\forall_{X,Y \in C_0} \mbox{hom}_C(X,Y) \subseteq C_1$ is a set. \end{itemize} -\section{Special Kinds of Arrows} + A predicate $P$ is \defn{essentially unique} (\S7.3) if it is unique up to + isomorphism: + \begin{itemize} + \item If both $PA$ and $PB$, then $A \simeq B$ + \item If $PA$ and $A \simeq B$, then $PB$. + \end{itemize} - $m$ is \defn{monic} (p25,D2.1) if $mi = mj \Rightarrow i = j$ in - \[\xymatrix{C \ar@<1ex>[r]^{i} \ar@<-1ex>[r]_j & A \ar@{>->}[r]^m & B} \] + $\mathbf{B}$ is a \defn{subcategory} of $\mathbf{A}$ if it has + subcollections of objects and morphisms with identical composition and + identity (\S4.1.1). $\mathbf{B}$ is additionally \dots + \begin{itemize} + \item \defn{full} if it has all morphisms from $\mathbf{A}$. (\S4.1.2) + \item \defn{reflective} if each $B$ has an $\mathbf{A}$-reflection. (\S4.16.2) + \end{itemize} - A \defn{subobject} (p77,D5.1) of $X$ is mono with cod $X$. + A category is$\dots$ + \begin{itemize} + \item \defn{balanced} if all bi are iso (\S7.49.2) + \item \defn{discrete} if all morphisms are identities. (\S3.26.1) + \item \defn{thin} if $\forall_{A,B} \mbox{hom}(A,B) \simeq \set{*}$. (\S3.26.2) + \end{itemize} - $e$ is \defn{epic} (p25,D2.1) if it is monic in $\mathbf{C}^{op}$, - {\it i.e.,} if $ie = je \Rightarrow i = j$ in - \[\xymatrix{A \ar@{->>}[r]^e & B \ar@<1ex>[r]^{i} \ar@<-1ex>[r]_j & C} \] +\section{Object Properties} - (p27,P2.6) Every iso is both monic and epic. + $C$ is a \defn{coseparator} if $\forall_{f,g : B \to A} . f \ne g + \Rightarrow \exists_{h : A \to C} . h \circ f \ne h \circ g$. (\S7.17) + (Contrast monomorphism.) - A \defn{split mono} (\defn{epi}) has a left (right) inverse. (p28,D2.7) + An object $0$ is \defn{initial} if $\forall_B . \exists! f_B : 0 \to B$. + (\S7.1) - 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) + A \defn{limit} (Awodey:D5.16) of a diagram $D(J)$ is a + terminal object in the category $\mathbf{Cone}(D(J))$. Written: + $c_i : (\varprojlim_{j} D_j) \to D_i$. - Functors preserve split monos and epis. + $(A \times B,\pi_1,\pi_2)$ is a \defn{product} iff (UMP) + \[\xymatrix{ + {}\save[]+<-1cm,0cm>*\txt<8pc>{$\forall_{Z,z_1,z_2} \exists!_u$\\$u\pi_1 = z_1 ~\wedge~ u\pi_2 = z_2$}\restore + & & Z \ar[dl]_{z_1} \ar[dr]^{z_2} \ar@{..>}[d]^u & \\ + & A & \ar[l]_{\pi_1} A \times B \ar[r]^{\pi_2} & B \\ + }\] - A \defn{point} (p32) of $C$ is any $c : 1 \to C$. (Arrows in - $\mathbf{Sets}$, but not $\mathbf{Pos}$, are pointwise.) + $(P,p_1,p_2)$ is a \defn{pullback} (Awodey:p80,D5.4) of $f,g$ iff (UMP) + \[\xymatrix{ + {}\save[]+<-1cm,0cm>*\txt<8pc>{$\forall_{Z,z_1,z_2 . fz_1 = gz_2}\exists!_u$\\$z_1 = p_1u ~\wedge~ z_2 = p_2u$}\restore + & Z \ar[dr]_{z_2} \ar@/^1pc/[rr]_{z_1} \ar@{..>}[r]_u & P \ar[d]^{p_1} \ar[r]_{p_2} & B \ar[d]^g \\ + & & A \ar[r]^f & C + }\] + $P$ may be denoted $A \times_C B$ when $f,g$ are clear. - 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. + $S$ is a \defn{separator} if $\forall_{f,g : A \to B} . f \ne g + \Rightarrow \exists_{h : S \to A} . f \circ h \ne g \circ h$. (\S7.10) + (Contrast epimorphism.) + $S$ is a separator iff $\mbox{hom}(S,-)$ is faithful. (\S7.12) -\section{Universal Constructions} + A set of objects $\mathcal{T}$ is a \defn{separating set} if + $\forall_{f,g : A \to B} . f \ne g \Rightarrow \exists{S \in \mathcal{T}, + h : S \to A} . f \circ h \ne g \circ h$. (\S7.14) - Objects defined by UCs are unique up to isomorphism. + An object $1$ is \defn{terminal} if $\forall_A . \exists! f_A : A \to 1$. + (\S7.4) - $0$ is \defn{initial} iff $\forall_C \exists!_u 0 \to C$. - $1$ is \defn{terminal} iff $\forall_C \exists!_u C \to 1 $. + An object that is both initial and terminal is called a \defn{zero}. + (\S7.7) - $(A \times B,\pi_1,\pi_2)$ is a \defn{product} iff - \[\xymatrix{ - {}\save[]+<-1cm,0cm>*\txt<8pc>{$\forall_{Z,z_1,z_2} \exists!_u$\\$u\pi_1 = z_1 ~\wedge~ u\pi_2 = z_2$}\restore - & & Z \ar[dl]_{z_1} \ar[dr]^{z_2} \ar[d]^u & \\ - & A & \ar[l]_{\pi_1} A \times B \ar[r]^{\pi_2} & B \\ - }\] +\section{Arrow Properties} - $(E,e)$ is an \defn{equalizer} (p56,D3.13) of $f,g$ iff + $e$ is an \defn{epimorphism} (\S7.39) (the dual of a monomorphism) + (equiv: is \defn{epic} (Awodey:D2.1)) if + % + \[\xymatrix{\forall_{i,j} . ie = je \Rightarrow i = j & A \ar@{->>}[r]^e & B \ar@<1ex>[r]^{i} \ar@<-1ex>[r]_j & C} \] + % + If $f$ and $g$ are epis, then so is $g \circ f$; if $g \circ f$ is epi, + then so is $g$. (\S7.41) Epis generalize \defn{surjection} in + $\mathbf{Set}$. + + $(E,e)$ is an \defn{equalizer} (\S7.51) of $f,g$ iff (UMP) $fe = ge$ and \[\forall_{Z,z . zf = zg} \exists!_u eu = z \quad \xymatrix{ Z \ar@{..>}[r]^u \ar@/_1pc/[rr]^{z} & E \ar[r]^e & A \ar@<1ex>[r]^f \ar@<-1ex>[r]_g & B \\ }\] + Equalizers are essentially unique (\S7.53) and monic. % XXX Cite? - $(P,p_1,p_2)$ 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. + A mono $m$ is a \defn{extremal} (\S7.61) if $e$ epic and + $m = f \circ e$ implies that $e$ iso. - A \defn{limit} (p90,D5.16) of a diagram $D(J)$ is a - terminal object in the category $\mathbf{Cone}(D(J))$. Written: - $c_i : (\varprojlim_{j} D_j) \to D_i$. + Let $G: \mathbf{A} \to \mathbf{B}$ and $B \in \mathbf{B}$. A + \defn{$G$-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 $\forall_{(f', A')} . + \exists!_{\check f} . f' = G{\check f} \circ f$. That is, + \[\xymatrix{ + B \ar[r]^f \ar@/_1.25pc/[rr]^{f'} + & GA \ar@{.>}[r]^{G{\check f}} + & GA' + & A \ar@{.>}[r]^{\check f} + & A' + }\] + \end{itemize} + When $G$ is a subcategory inclusion, a $G$-structured universal arrow is + a \defn{reflection} (\S4.16). -\section{Properties of UCs} + $f : A \to B$ is an \defn{isomorphism} if $\exists!_g . f \circ g = id_B + ~\wedge~ g \circ f = id_A$. (\S3.8; ! in \S3.11). Every isomorphism + is both monic and epic (Awodey:P2.6). - Equalizers are monic. + $f$ is a \defn{monomorphism} (\S7.32) (equiv: is \defn{monic} + (Awodey:D2.1)) if + \[\xymatrix{\forall_{i,j} . mi = mj \Rightarrow i = j & C \ar@<1ex>[r]^{i} \ar@<-1ex>[r]_j & A \ar@{{ >}->}[r]^m & B} \] + If $f$ and $g$ are monos, then so is $g \circ f$; if $g \circ f$ is mono, + then so is $f$. (\S7.34) Monos generalize \defn{injection} in + $\mathbf{Set}$. Objects with monomorphisms to $X$ are called + \defn{subobjects} of $X$ (Awodey:D5.1). - (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 $f,g$. + A \defn{point} (Awodey:p32) of $C$ is any $c : 1 \to C$. - (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} + $f$ is a \defn{regular monomorphism} (\S7.56) if it is an equalizer of + some pair of morphisms. - (p84,C5.9) Pullbacks preserve commutative triangles. + $f : A \to B$ is a \defn{retraction} if $\exists_g . f \circ g = 1_B$ + (\S7.24) aka \defn{split epi} (Awodey:D2.7). If $f$ and $g$ are + retractions, then so is $g \circ f$; if $g \circ f$ is a retraction, then + so is $g$. (\S7.27) - 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} + $f : A \to B$ is a \defn{section} if $\exists_g . g \circ f = 1_A$. + (\S7.19) aka \defn{split mono} (Awodey:D2.7). + If $f$ and $g$ are sections, then so is $g \circ f$; + if $g \circ f$ is a section, then so is $f$. (\S7.21) + + Several morphism properties combine in useful ways: + \begin{itemize} + \item mono, epi $\Rightarrow$ \defn{bimorphism} (\S7.49) + \item section $\Rightarrow$ regular mono (\S7.35, \S7.59.1) + \item regular mono $\Rightarrow$ extremal mono (\S7.59.2, \S7.63) + \item retraction $\Rightarrow$ epi (\S7.42) + \item mono, retraction $\Leftrightarrow$ isomorphism (\S7.36) + \item section, epi $\Leftrightarrow$ isomorphism (\S7.43) + \end{itemize} + %(XXX stopped around \S7.60; there's more to be said) \section{Exponentials} - (p107,D6.1) In a category with binary products, given two objects $B$ and $C$, + (Awodey: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. \[\xymatrix{ {}\save[]+<-1cm,0cm>*\txt<8pc>{$\forall_{A,f : A \times B \to C}\exists!_{\tilde f : A \to C^B}$\\ $\epsilon \circ (\tilde f \times 1_B) = f$}\restore & C^B & C^B \times B \ar[r]^\epsilon & C \\ - & A \ar[u]^{\tilde f} & A \times B \ar[u]^{\tilde f \times 1_B} \ar[ur]_f + & A \ar@{..>}[u]^{\tilde f} & A \times B \ar@{..>}[u]^{\tilde f \times 1_B} \ar[ur]_f }\] The arrows $f$ and $\tilde f$ are ``exponential transposes.'' - Exponential transposition is self inverse (p108). This implies - \[ \mbox{Hom}_{\bf C}(A \times B, C) \simeq \mbox{Hom}_{\bf C}(A, C^B) \] + Exponential transposition is self inverse (Awodey:p108). This implies + \[ \mbox{hom}_{\bf C}(A \times B, C) \simeq \mbox{hom}_{\bf C}(A, C^B) \] + + A category is \defn{cartesian closed} (Awodey:p108,D6.2) if it has all + finite products and exponentials. \section{Functors} - A functor $F : C \to D$\dots - \begin{itemize} - \item is \defn{faithful} (D7.1) if the induced - \[ F_{A,B} : \mbox{Hom}_C(A,B) \to \mbox{Hom}_D(FA,FB) \] - is injective. + Default notation here: functors $F,G : \mathbf{A} \to \mathbf{B}$. + + 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{contravariant functor} $F$ (\S3.20.5) is a (covariant) functor + $\mathbf{A}^\text{op} \to \mathbf{B}$. - \item is \defn{full} if $F_{A,B}$ is surjective. + A \defn{diagram} (Awodey:D5.15) is a functor $D : J \to C$ from some + indexing category $J$. + A \defn{endofunctor} has $\mathbf{A} = \mathbf{B}$. $F \circ F$ may be + denoted $F^2$, etc. (\S3.23; ftn 15) + + Functors compose. (\S3.23) + + % XXX Cite + A functor $F : C \to D$\dots + \begin{itemize} \item \defn{preserves limits of type $J$} if \[ \forall_{D : J \to C}\forall_{\varprojlim_j D_j} F(\varprojlim_j D_j) \simeq \varprojlim_j F(D_j).\] - \item is \defn{continuous} if it preserves all limits. (p94,D5.24) - \item \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 $\bar L = \varprojlim_j D_j$. \end{itemize} - A \defn{natural transformation} (p134,D7.6) from $F : C \to D$ to $G : C \to D$ ($F - \stackrel{\cdot}{\to} D$) is a family of $D$-arrows $\paren{\eta_X}_{X \in C_0}$ s.t. + A functor $F$ is (\S3.27, \S3.33) + \begin{itemize} + \item \defn{amnestic} if $f$ is an identity iff $Ff$ is an identity. + \item \defn{continuous} if it preserves all limits. (Awodey:D5.24) + \item an \defn{equivalence} if it is full, faithful, and + isomorphism-dense. + \item an \defn{embedding} if it is injective on morphisms. + \item \defn{faithful} if $\forall_{A,A'} . F\vert_{\mathbf{A}(A,A')} + \subseteq \mathbf{B}(FA, FA')$ is injective. + \item \defn{full} if $\forall_{A,A'} . F\vert_{\mathbf{A}(A,A')}$ surjective. + \item \defn{isomorphism-dense} if $\forall_B . \exists_A . F(A) \simeq B$. + \end{itemize} + + A \defn{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). + That is, + % \[\xymatrix{ - {}\save[]+<-1cm,0cm>*\txt<8pc>{$\forall_{A,B,f \in C}$\\$Gf\circ \eta_A = \eta_B\circ Ff$}\restore - & FA \ar[r]^{\eta_A} \ar[d]_{Ff} & GA \ar[d]^{Gf} \\ - & FB \ar[r]^{\eta_B} & GB} \] + {}\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} \] + + There is special notation for functors ($H$) applied to natural + transformations and vice-versa (\S6.3): $H\tau : HF \natto HG$ defined by + $(H\tau)_A = H(\tau_A)$ and $\tau H : FH \natto GH$ defined by $(\tau H)_A + = \tau_{HA}$. + + All functors \defn{preserve} (in $\mathbf{A}$ implies in $\mathbf{B}$) + isomorphisms (\S3.21), sections (\S7.22), and retractions (\S7.28). + + 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} + + The \defn{covariant representable functor} (Awodey:p44) at $A \in + \mathbf{C}$ is defined by $\mbox{Hom}(A,\text{---}) : \mathbf{C} \to + \mathbf{Sets}$. These functors are continuous (Awodey:P5.25). + + Representable functors preserve monos. (\S7.37.1) + + 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 $h$. (Awodey:P5.10) + + +\section{Cones and Sources} + + A \defn{cone} (Awodey:D5.15) to a diagram $D(J)$ is a collection of arrows + $c_j : C \to D_j$ s.t. $\forall_{D_\alpha \in D(J)} c_j = D_\alpha \circ + c_i$. + + A \defn{source} in category $\mathbf{A}$ indexed by $I$ is a pair $(A, + \set{f_i : A \to A_i}_{i \in I})$. This source has domain $A$ and + codomain $\set{A_i}_{i\in I}$. (\S10.1) + + Given $(A,\set{f_i}_{i \in I})$ and + $\{(A_i,\set{g_{ij}}_{j \in J_i})\}_{i \in I}$ + all sources, their \defn{composite} is $(A, \set{g_{ij} \circ f_i}_{i\in I, + j\in J_i})$. (\S10.3) + + 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} + + 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) + + 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) + + If $\mathbf{X}$ is $\mathbf{Set}$, $\mathbf{A}$ is a \defn{construct}. + (\S5.1.2) + + $(UA \overset{f}{\to} UB) \in \mathbf{X}$ \defn{is an $\mathbf{A}$-morphism} + if $f$ has a 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. + %\begin{itemize} + % \item \defn{discrete}, $(UA \to UB)$ (\S8.1) + % \item \defn{indiscrete}, $(UB \to UA)$ (\S8.3) + %\end{itemize} + + A \defn{free object} $A \in \mathbf{A}$ is one with a ($U$-structured) + universal arrow $(u,UA)$ in $B$. (\S8.22+\S8.30) + + %$f \in \mathbf{A}$ 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{Derived Categories} + + The \defn{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. $\forall_{D_j \in D(J)} c_j^\prime \circ \phi = + c_j$. + + 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{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{slice} (Awodey:p16,i4) category $\mathbf{C}/C$ has objects of + arrows in $\mathbf{C}$ with codomain $C$. Arrows are tops of commutative + triangles. + +\section{Adjoints and Adjoint Situations} - An \defn{adjunction} (p180,D9.1) of $F : C \to D$ and $G : D \to C$ is a natural +\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) + + Adjoints compose (\S8.5), preserve mono sources (\S8.6), and preserve + limits (\S8.9) + + Given adjoint $G$ with $\eta_B : B \to G(A_B)$ the $G$-structured + universal arrow with domain $B$, $\exists!_F$ such that $FB = A_B$ and + $\eta : id_B \natto G \circ F$ is natural; further, there is a unique, + natural $\epsilon : F \circ G \natto id_A$ with $G\epsilon \circ \eta G = + id_G$ and $\epsilon F \circ F \eta = id_F$. (\S19.1) + + $(\eta,\epsilon) : F \dashv G : \mathbf{A} \to \mathbf{B}$ is a + \defn{adjoint situation} if the above relationships hold. (\S19.7) + +\subsection{Awodey Approach} + + 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. \[\xymatrix{ {}\save[]+<-1cm,0cm>*\txt<8pc>{$\forall_{f:X \to GY}\exists!_{f^\#:FX\to Y}$\\ $f = Gf^\# \circ \eta_X$}\restore - & X \ar[dr]^f \ar[r]^{\eta_X} & GFX \ar[d]^{Gf^\#} \\ - & & GY + & FX\ar@{..>}[d]^{f^\#} & X \ar[dr]^f \ar[r]^{\eta_X} & GFX \ar@{..>}[d]^{Gf^\#} \\ + & Y & & GY }\] - Equivalently (p189,D9.7), a natural {\em isomorphism} + Equivalently (Awodey:D9.7), a natural {\em isomorphism} \[ \phi : \mbox{Hom}_D(FC,D) \simeq \mbox{Hom}_C(C,GD), \quad \eta_X = \phi(1_{FX}) \] -\section{Special Functors} +\subsection{Moving Right Along} + + A \defn{monad} (\S20.1) on $\mathbf{X}$ is $(T : \mathbf{X} \to \mathbf{X}, + \eta : id_{\mathbf{X}} \natto T, \mu : T^2 \natto T)$ s.t. + \[\forall_X \quad + \xymatrix@R=10pt{ + T^3X \ar[r]^{T(\mu_X)} \ar[d]^{\mu_{TX}} & T^2X \ar[d]^{\mu_{X}} \\ + T^2X \ar[r]^{\mu_X} & TX + } \quad \xymatrix@R=10pt{ + TX \ar[r]^{T(\eta_X)} \ar[dr]_{id_{TX}} & T^2X \ar[d]^{\mu_X} & TX \ar[l]_{\eta_{TX}} \ar[dl]^{id_{TX}} \\ + & TX & + }\] - The \defn{covariant representable functors} (p44) are - \[\mbox{Hom}(A,\text{---}) : \mathbf{C} \to \mathbf{Sets}\] - These functors preserve all limits (p94,P5.25). +\appendix +\section{Miscellaneous Terminology} - (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 $h$. + A category is \defn{finitely presented} (Awodey:p75) if it is the free + category over a finite graph quotiented by a finite set of equations. + + The \defn{local membership relation} for generalized element $z : Z \to C$ + and subobject $M$ (i.e., with monic $m : M \to C$), $z \in_X M$, holds iff + $\exists_{f:Z \to M} . z = mf$. + + 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.) (Awodey:p101,E5.33) + +\section{Miscellaneous Useful Properties} + + (Awodey: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} -\section{Glossary} + (Awodey:p84,C5.9) Pullbacks preserve commutative triangles. - A category is \defn{cartesian closed} (p108,D6.2) if it has all finite - products and exponentials. + Universal Constructions (or Universal Mapping Properties, UMP) reduce to + limits (Awodey:p91,E5.17-20): + % + \begin{tabular}{|c|c|c|c|} + \hline + terminals & products & equalizers & pullbacks \\ + % + \hline + & $\xymatrix@C5pt{x & y}$ + & $\xymatrix{x \ar@<1ex>[r]^{\alpha} \ar@<-1ex>[r]_{\beta} & y}$ + & $\xymatrix@C5pt@R5pt{& x \ar[d] \\ y \ar[r] & z}$\\ + \hline + \end{tabular} - A \defn{diagram} (p89,D5.15) is a functor $D : J \to C$ from - some indexing category $J$. + Objects defined by UCs are unique up to isomorphism. - A category is \defn{finitely presented} (p75) if it is the - free category over a finite graph quotiented by a finite set of equations. +\section{Examples To Jog Your Memory} - A structure is \defn{free} over $S$ if its elements are ``generated'' - from $S$ and no ``nontrivial'' equations exist. +\subsection{$\mathbf{Mon}$} - $\mbox{Hom}_\mathbf{C}(A,B)$ denotes the class $(A \to B) \in \mathbf{C}$. + Mono-epics are not isos ($(\mathbf{N},+,0) \to (\mathbf{Z},+,0)$). + (Pierce:\S1.6.3) - Subobject $m$'s \defn{local membership relation}: - \[ \forall_{m : M \rightarrowtail X} - \brak{ z \in_X M \Leftrightarrow \exists_{f:Z \to M} z = mf} \] + $(\set{*},\cdot,*)$ is a (the) zero. - 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) + Each monoid $M$ has only one point, $1 \to M$. -\end{document} +\subsection{Adjoint Situations and Monads} + + Consider $(\eta, \epsilon) : F \dashv G : \mathbf{Mon} \to \mathbf{Set}$. + $\eta_X : X \to GFX$ is insertion of generators: $\forall x \in X . \eta_X x = x$. + $\epsilon_Y : FGY \to Y$ is the re-introduction of structure; + if $FGY = ((GY)^*, \cdot, \varepsilon)$ and $Y = (GY, +, 0)$ then + \[ \epsilon_Y \varepsilon = 0 + \quad \epsilon_Y (y \cdot z) = y + z + \quad \epsilon_Y (y \in GY) = y + \] -% vim:ts=2:expandtab + Further, $T = G \circ F$ is a monad. Generically, $\mu$... + \begin{align*} + \mu_X (TTX) &= (G \epsilon F)_X (TTX) = (G \epsilon_{FX}) (GFGFX) \\ + &= G ((\epsilon_{FX})(FGFX)) = GFX + \end{align*} + So here $\mu$ is the $G$-image of a function which takes $y \in FGFX = + F(X^*)$ (that is, a concatenation of symbols from $GFX$) and re-imposes + structure to obtain $\epsilon_{FX} y \in FX$. + +\end{document}