From: Nathaniel Wesley Filardo Date: Tue, 17 Apr 2012 18:43:21 +0000 (-0400) Subject: More improvements to ctcheat-joy X-Git-Url: https://hydra-www.ietfng.org/gitweb/?a=commitdiff_plain;h=b627f48438c58b3d3b7fe0b87a13fd941bbb6193;p=ctcheat More improvements to ctcheat-joy --- diff --git a/ctcheat-joy.tex b/ctcheat-joy.tex index f27e8c0..32ed9b3 100644 --- a/ctcheat-joy.tex +++ b/ctcheat-joy.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}{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)} @@ -51,24 +68,28 @@ \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} -Notation and references are to Ji\v{r}\'i Ad\'amek, Horst Herrlich, George +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} (\S3.1) is a quadruple + 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 object $A,B$, a (disjoint) collection of arrows - $\mbox{hom}(A,B)$ (from \defn{domain} to \defn{codomain}). + \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} @@ -83,15 +104,11 @@ Entries within each section are roughly sorted by definition, alphabetically. \item If $PA$ and $A \simeq B$, then $PB$. \end{itemize} - In a concrete category $(\mathbf{A}, U)$ over $\mathbf{X}$, $(UA - \overset{f}{\to} UB) \in \mathbf{X}$ \defn{is an $\mathbf{A}$-morphism} if - $f$ has a unique $U$-preimage in $\mathbf{A}$. (\S5.3, \S6.22) - \section{Predicates on Categories} A category is$\dots$ \begin{itemize} - \item \defn{balanced} if all bimorphisms are isomorphisms (\S7.49.2) + \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} @@ -105,19 +122,14 @@ Entries within each section are roughly sorted by definition, alphabetically. $C$ 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) - - An object $A$ in a concrete category $\mathbf{A}$ over $\mathbf{X}$ 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} + (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) @@ -133,11 +145,11 @@ Entries within each section are roughly sorted by definition, alphabetically. \section{Kinds of Arrows} - $e$ is an \defn{epimorphism} (\S7.39) if it is monic in $\mathbf{C}^{op}$, - {\it i.e.,} if $ie = je \Rightarrow i = j$ in + $e$ 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) + 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 @@ -149,25 +161,33 @@ Entries within each section are roughly sorted by definition, alphabetically. A mono $m$ is a \defn{extremal} (\S7.61) if $e$ epic and $m = f \circ e$ implies that $e$ iso. - Let $G$ be a functor $\mathbf{A} \to \mathbf{B}$ and $B$ a - $\mathbf{B}$-object. A \defn{$G$-structured arrow with domain $B$} - is a pair $(f : B \to GA, A)$. (\S8.30) It is + 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 $\forall_{(f', A')} . - \exists!_{\check f} . f' = G{\check f} \circ f$. + \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 - \[\xymatrix{C \ar@<1ex>[r]^{i} \ar@<-1ex>[r]_j & A \ar@{>->}[r]^m & B} \] + \[\xymatrix{C \ar@<1ex>[r]^{i} \ar@<-1ex>[r]_j & A \ar@{{ >}->}[r]^m & B} \] If $f$ and $g$ are monos, then so is $g \circ f$; if $g \circ f$ is mono, - then so is $f$. (\S7.34) + 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. @@ -181,65 +201,120 @@ Entries within each section are roughly sorted by definition, alphabetically. 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 section, retraction $\Leftrightarrow$ isomorphism (\S7.26) \item mono, retraction $\Leftrightarrow$ isomorphism (\S7.36) \item section, epi $\Leftrightarrow$ isomorphism (\S7.43) - \item section $\Rightarrow$ regular mono (\S7.35; regular \S7.59.1) - \item regular mono $\Rightarrow$ extremal mono (\S7.59.2; extremal \S7.63) \end{itemize} - (XXX stopped around 7.60; there's more to be said) - -\subsection{Arrows in Concrete Categoies} - - For this section, $\mathbf{A}$ is a concrete category over $\mathbf{X}$. - - $f \in \mathbf{A}$ 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. + %(XXX stopped around \S7.60; there's more to be said) \section{Functors} - A \defn{(covariant) functor} $F : \mathbf{A} \to \mathbf{B}$ (\S3.17) - assigns to each $\mathbf{A}$-object a $\mathbf{B}$-object and to each + 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} $F : \mathbf{A} \to \mathbf{B}$ (\S3.20.5) - is a functor from $\mathbf{A}^\text{op} \to \mathbf{B}$. + 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 is (\S3.27) + 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{embedding} if it is injective on morphisms. - \item \defn{faithful} if $\forall_{A,A'}$ the restriction $F\vert_{\mbox{hom}_A(A,A')} - \subseteq \mbox{hom}(FA, FA')$ is injective. - \item \defn{full} if said restrictions are surjective. - \end{itemize} - - A functor $F : \mathbf{A} \to \mathbf{B}$ is (\S3.33) - \begin{itemize} \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} - All functors preserve$\dots$ + 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) + + 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} - All full, faithful functors reflect sections (\S7.23) and retractions - (\S7.29). + 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{Concrete Categories} + + For this section, $\mathbf{A}$ is a concrete category over $\mathbf{X}$ + with forgetful functor $U : \mathbf{A} \to \mathbf{X}$, denoted $(\mathbf{A}, U)$. + + 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}. + + $(UA \overset{f}{\to} UB) \in \mathbf{X}$ \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) - All representable functors preserve monos. (\S7.37.1) + $(\eta,\epsilon) : F \dashv G : \mathbf{A} \to \mathbf{B}$ is a + \defn{adjoint situation} if the above relationships hold. (\S19.7) - Faithful functors reflect monos (\S7.37.2) and epis (7.44). + A \defn{monad} on $\mathbf{X}$ is $(T : \mathbf{X} \to \mathbf{X}, + \eta : id_{\mathbf{X}} \natto, \mu : T^2 \natto T)$ s.t. + \[\xymatrix@R=10pt{ + T^3 \ar[r]^{T\mu} \ar[d]^{\mu T} & T^2 \ar[d]^\mu \\ + T^2 \ar[r]^{\mu} & T + } \quad \xymatrix@R=10pt{ + T \ar[r]^{T\eta} \ar[dr]_{id} & T^2 \ar[d]^\mu & T \ar[l]_{\eta T} \ar[dl]^{id} \\ + & T & + }\] \end{document}