From 1db16292642c61b6ff429b7a7f5cfcfdea73cb9d Mon Sep 17 00:00:00 2001 From: Nathaniel Wesley Filardo Date: Thu, 12 Mar 2015 01:12:26 -0400 Subject: [PATCH] Misc improvements --- adjoints-diag.tex | 4 +- bifunctors.tex | 38 ++++--- ctcheat.tex | 284 ++++++++++++++++++++++++++++++++++------------ 3 files changed, 231 insertions(+), 95 deletions(-) diff --git a/adjoints-diag.tex b/adjoints-diag.tex index 8ce4dc7..8ee732e 100644 --- a/adjoints-diag.tex +++ b/adjoints-diag.tex @@ -59,12 +59,12 @@ \section{Getting Started} -The ``diagonal functor'' $\Delta : \mathbf{C} \to \mathbf{C}^2$ is +The ``diagonal functor'' $\Delta : \mathcal{C} \to \mathcal{C}^2$ is charmingly degenerate: $\Delta X = (X,X), \Delta (f : A \to B) = (f,f) : (A,A) \to (B,B)$. (In the ${}^2$ category, morphisms are such that $(f,g)(a,b) = (f a,g b)$.) For notational clarity, we'll use $A \times B$ for products within -$\mathbf{C}$ and $(A,B)$ for objects in $\mathbf{C}^2$. +$\mathcal{C}$ and $(A,B)$ for objects in $\mathcal{C}^2$. Define $\Pi_1 : \mathcal{C}^2 \to \mathcal{C}$ to be the first projector, and $\Pi_2$ the second. If $\mathcal{C}$ has diff --git a/bifunctors.tex b/bifunctors.tex index 7bb2f1b..8ebefd1 100644 --- a/bifunctors.tex +++ b/bifunctors.tex @@ -33,13 +33,13 @@ \begin{document} -\section{Bifunctors} - In the paper {\bf Functional Pearl: F for Functor} from ICPF '12, the concept of a \defn{bifunctor} is introduced quickly and somewhat confusingly. Herein, as Neil Gaiman wrote in Good Omens, ``the text will be slowed down to allow the sleight of hand to be followed.'' +\section{Bifunctors} + A \defn{bifunctor} is a two-argument object, here denoted $\textrm{---} \otimes \textrm{---} \in \mathcal{E}^{\mathcal{C} \times \mathcal{D}}$, which @@ -47,25 +47,27 @@ which \begin{itemize} % \item Sends an object $C \times D \in \mathcal{C} \times \mathcal{D}$ to -an object $C \otimes D \in \mathcal{E}$. + an object $C \otimes D \in \mathcal{E}$ and a morphism $f \times g \in + \Hom{\mathcal{C} \times \mathcal{D}}{C \times D}{C' \times D'}$ to a + morphism $f \otimes g \in \Hom{\mathcal{E}}{C \otimes D}{C' \otimes D'}$. % - \item Sends a morphism $f \times g \in \Hom{\mathcal{C} \times -\mathcal{D}}{C \times D}{C' \times D'}$ to a morphism -$f \otimes g \in \Hom{\mathcal{E}}{C \otimes D}{C' \otimes D'}$. -% - \item Preserves identities: $id_C \otimes id_D = id_{C \otimes D}$. -% - \item Preserves composition: $(f' \circ f) \otimes (g' \circ g) = (f' -\otimes g') \circ (f \otimes g)$. + \item Preserves identities ($id_C \otimes id_D = id_{C \otimes D}$) and + composition: $(f' \circ f) \otimes (g' \circ g) = (f' \otimes g') \circ (f + \otimes g)$. % \item Has a $\mathcal{C}$-object-indexed collection of {\em functors} -obtained by partial application on the left: a $L^\otimes_C = \paren{C \otimes -\textrm{---}} \in \mathcal{E}^\mathcal{D}$ for each object $C \in -\mathcal{C}$. -% - \item Has a $\mathcal{D}$-object-indexed collection of {\em functors} -obtained by partial application on the right: a $R^\otimes_D \paren{\textrm{---} \otimes -D} \in \mathcal{E}^\mathcal{C}$ for each object $D \in \mathcal{D}$. + obtained by partial application on the left: a $L^\otimes_C = \paren{C + \otimes \textrm{---}} \in \mathcal{E}^\mathcal{D}$ for each object $C \in + \mathcal{C}$, and a $\mathcal{D}$-object-indexed collection from the + right: a $R^\otimes_D \paren{\textrm{---} \otimes D} \in + \mathcal{E}^\mathcal{C}$ for each object $D \in \mathcal{D}$.% + % + \footnote{Of course, there are also functor families indexed by arrows, + which might be designated $\tilde{L}^\otimes_f = \paren{f \otimes + \textrm{---}} \in \mathcal{E}^\mathcal{D}$ for each $f \in \mathcal{C}$. + However, these bring no new degrees of freedom to the table, as + $\tilde{L}^\otimes_f(D) = f \otimes D = R^\otimes_D f$ and + $\tilde{L}^\otimes_f(g) = f \otimes g$.} % \end{itemize} diff --git a/ctcheat.tex b/ctcheat.tex index f58f345..ac70c5d 100644 --- a/ctcheat.tex +++ b/ctcheat.tex @@ -1,27 +1,38 @@ +% 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 @@ -79,21 +90,32 @@ \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{} % @@ -151,26 +173,27 @@ Entries within each section are roughly sorted by definition, alphabetically. \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{} % @@ -178,7 +201,8 @@ Entries within each section are roughly sorted by definition, alphabetically. arrows in $\mathbf{C}$ with codomain $C$. Arrows are tops of commutative triangles. -\section{Object Properties} +% >>> +\section{Object Properties} % <<< \paragraph{} % @@ -207,6 +231,15 @@ Entries within each section are roughly sorted by definition, alphabetically. & 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) @@ -240,7 +273,8 @@ Entries within each section are roughly sorted by definition, alphabetically. 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 @@ -277,14 +311,20 @@ Entries within each section are roughly sorted by definition, alphabetically. \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'} @@ -348,7 +388,8 @@ Entries within each section are roughly sorted by definition, alphabetically. \end{itemize} %(XXX stopped around \S7.60; there's more to be said) -\section{Exponentials} +% >>> +\section{Exponentials} % <<< \paragraph{} % @@ -366,14 +407,23 @@ Entries within each section are roughly sorted by definition, alphabetically. \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{} % @@ -381,14 +431,14 @@ Entries within each section are roughly sorted by definition, alphabetically. \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{} @@ -419,6 +469,14 @@ Entries within each section are roughly sorted by definition, alphabetically. 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) @@ -436,7 +494,23 @@ Entries within each section are roughly sorted by definition, alphabetically. \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). @@ -446,6 +520,11 @@ Entries within each section are roughly sorted by definition, alphabetically. {}\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{} % @@ -454,20 +533,19 @@ Entries within each section are roughly sorted by definition, alphabetically. $(H\tau)_A = H(\tau_A)$ 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{} % @@ -486,14 +564,19 @@ Entries within each section are roughly sorted by definition, alphabetically. \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{} % @@ -513,19 +596,27 @@ Entries within each section are roughly sorted by definition, alphabetically. 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{} % @@ -534,8 +625,9 @@ Entries within each section are roughly sorted by definition, alphabetically. \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. @@ -547,13 +639,14 @@ Entries within each section are roughly sorted by definition, alphabetically. \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} @@ -561,12 +654,12 @@ Entries within each section are roughly sorted by definition, alphabetically. % 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{} % @@ -585,16 +678,21 @@ Entries within each section are roughly sorted by definition, alphabetically. \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} @@ -611,8 +709,18 @@ Entries within each section are roughly sorted by definition, alphabetically. & 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{} % @@ -627,11 +735,13 @@ Entries within each section are roughly sorted by definition, alphabetically. \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{} % @@ -668,7 +778,8 @@ Entries within each section are roughly sorted by definition, alphabetically. % Objects defined by UCs are unique up to isomorphism. -\section{Examples To Jog Your Memory} +% >>> +\section{Examples To Jog Your Memory} % <<< \subsection{$\mathbf{Set}$} @@ -687,8 +798,8 @@ Entries within each section are roughly sorted by definition, alphabetically. \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{} % @@ -722,4 +833,27 @@ Entries within each section are roughly sorted by definition, alphabetically. 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=<<<,>>> +% >>> -- 2.50.1