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+\usepackage{enumitem}
+\setlist{nolistsep}
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+\setlength{\parskip}{2pt}
+
+% http://www.latex-community.org/forum/viewtopic.php?f=46&t=3837&start=0#p15112
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+\g@addto@macro\normalsize{%
+\setlength\abovedisplayskip{0pt}%
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+\makeatother
+
+% http://comments.gmane.org/gmane.comp.tex.xy-pic/223
+% fixes {>->} having the tail overlap the item.
+\newdir{ >}{{}*!/-2.6667\jot/\dir{>}}
+
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\newcommand{\paren}[1]{\left({#1}\right)}
\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}
\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}
$C$ 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)
\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
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.
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}
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