From: Nathaniel Wesley Filardo Date: Tue, 27 Mar 2012 18:49:24 +0000 (-0400) Subject: Improvements to ctcheat-joy? X-Git-Url: https://hydra-www.ietfng.org/gitweb/?a=commitdiff_plain;h=bb03ed77c7cfcb01f4e2289958aa254431c22101;p=ctcheat Improvements to ctcheat-joy? --- diff --git a/ctcheat-joy.tex b/ctcheat-joy.tex index 029cfe4..f27e8c0 100644 --- a/ctcheat-joy.tex +++ b/ctcheat-joy.tex @@ -59,6 +59,8 @@ Notation and references are to Ji\v{r}\'i Ad\'amek, Horst Herrlich, George E. Strecker's {\em Abstract and Concrete Categories: The Joy of Cats}. +Entries within each section are roughly sorted by definition, alphabetically. + \section{Basics} A \defn{category} (\S3.1) is a quadruple @@ -74,16 +76,82 @@ E. Strecker's {\em Abstract and Concrete Categories: The Joy of Cats}. The \defn{dual} (\S3.5) category $\mathbf{A}^\text{op}$ which exchanges domains and codomains of arrows in $\mathbf{A}$. -\section{Special Relations on Categories} + 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} + + 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{Special Kinds of Arrows} +\section{Predicates on Categories} - $f : A \to B$ is an \defn{isomorphism} if $\exists!_g . f \circ g = id_B - ~\wedge~ g \circ f = id_A$. (\S3.8; ! in \S3.11) + A category is$\dots$ + \begin{itemize} + \item \defn{balanced} if all bimorphisms are isomorphisms (\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} + + A \defn{concrete category} over $\mathbf{X}$ is a pair $(\mathbf{A},U : + \mathbf{A} \to \mathbf{X}$) with $U$ faithful. (\S5.1.1) + + A \defn{construct} is a concrete category over $\mathbf{Set}$. (\S5.1.2) + +\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) + + 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} + + 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) + + $S$ 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) if it is monic in $\mathbf{C}^{op}$, + {\it i.e.,} if $ie = je \Rightarrow i = j$ in + \[\xymatrix{A \ar@{->>}[r]^e & B \ar@<1ex>[r]^{i} \ar@<-1ex>[r]_j & C} \] + If $f$ and $g$ are epis, then so is $g \circ f$; if $g \circ f$ is epi, + then so is $g$. (\S7.41) + + $(E,e)$ 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 $G$ 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 $(f : B \to GA, A)$. It is + is a pair $(f : B \to GA, A)$. (\S8.30) It is \begin{itemize} \item \defn{generating} if $\forall_{r,s : A \to A'} . Gr \circ f = Gs \circ f \implies r = s$ @@ -93,6 +161,47 @@ E. Strecker's {\em Abstract and Concrete Categories: The Joy of Cats}. \exists!_{\check f} . f' = G{\check f} \circ f$. \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) + + $f$ 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 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. + \section{Functors} A \defn{(covariant) functor} $F : \mathbf{A} \to \mathbf{B}$ (\S3.17) @@ -103,22 +212,34 @@ E. Strecker's {\em Abstract and Concrete Categories: The Joy of Cats}. A \defn{contravariant functor} $F : \mathbf{A} \to \mathbf{B}$ (\S3.20.5) is a functor from $\mathbf{A}^\text{op} \to \mathbf{B}$. - All functors preserve isomorphisms. (\S3.21) - A functor is (\S3.27) \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. - \item \defn{amnestic} if $f$ is an identity iff $Ff$ is an identity. \end{itemize} A functor $F : \mathbf{A} \to \mathbf{B}$ is (\S3.33) \begin{itemize} - \item \defn{isomorphism-dense} if $\forall_B . \exists_A . F(A) \simeq B$. \item an \defn{equivalence} if it is full, faithful, and isomorphism-dense. + \item \defn{isomorphism-dense} if $\forall_B . \exists_A . F(A) \simeq B$. \end{itemize} + All functors preserve$\dots$ + \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). + + All representable functors preserve monos. (\S7.37.1) + + Faithful functors reflect monos (\S7.37.2) and epis (7.44). + \end{document}