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-\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}
-
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-\usepackage{fancyhdr}
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-%\newtheorem{thm}{Thm}[section]
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-\setlength{\parindent}{0pt}
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-% http://www.latex-community.org/forum/viewtopic.php?f=46&t=3837&start=0#p15112
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-%Mathematics
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-%Quantum Mechanics
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-
-\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}
\usepackage{multirow}
\usepackage{hyperref}
\usepackage{breakurl}
+\usepackage{enumitem}
+%\setlist{nolistsep}
\renewcommand{\baselinestretch}{0.9}
%\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)}
\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}
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