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+%http://tex.stackexchange.com/questions/126750/how-can-i-number-paragraphs-without-higher-level-counters
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-\newcommand{\defn}[1]{{\bf #1}}
+\newcommand{\defn}[2][]{{\ifstrempty{#1}{\label{defn:#2}}{\label{defn:#1}}{\bf #2}}}
+\newcommand\xrdefnhelper[1]{defn:#1}
+\newcommand{\xrdefn}[1]{\ref{\forcsvlist{\xrdefnhelper}{#1}}}
\begin{document}
\section{Basics}
+ \paragraph{}
+ %
A \defn{category} $\mathbf{C}$ (\S3.1) is a quadruple
$(\mathcal{O},\mbox{hom},id,\circ)$ with
\begin{itemize}
\item Identity arrows ($id_A$) on each object $A$, unit of $\circ$
\end{itemize}
+ \paragraph{}
+ %
Categories may be described (Awodey:p21) as
\[\xymatrix{ C_2 \ar[r]^\circ & C_1 \ar@<2ex>[r]_{cod} \ar@<-2ex>[r]_{dom} & C_0 \ar[l]^i }\]
+ \paragraph{}
+ %
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.
\end{itemize}
+ \paragraph{}
+ %
A predicate $P$ is \defn{essentially unique} (\S7.3) if it is unique up to
isomorphism:
\begin{itemize}
\item If $PA$ and $A \simeq B$, then $PB$.
\end{itemize}
+ \paragraph{}
+ %
$\mathbf{B}$ 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[fullcat]{full} if it has all morphisms from $\mathbf{A}$
+ between objects in $\mathbf{B}$. (\S4.1.2)
\item \defn{reflective} if each $B$ has an $\mathbf{A}$-reflection. (\S4.16.2)
+ \xrdefn{reflection}
\end{itemize}
+ \paragraph{}
+ %
A category is$\dots$
\begin{itemize}
\item \defn{balanced} if all bi are iso (\S7.49.2)
\item \defn{thin} if $\forall_{A,B} \mbox{hom}(A,B) \simeq \set{*}$. (\S3.26.2)
\end{itemize}
+\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}
+
+ \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.
+
+ \paragraph{}
+ %
+ The \defn{arrow} (Awodey:p16,i3) category $\mathbf{C}^\to$ has arrows for
+ commutative squares in $\mathbf{C}$. There are two functors
+ $\mathbf{cod}, \mathbf{dom} : \mathbf{C}^\to \to \mathbf{C}$.
+
+ \paragraph{}
+ %
+ 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{Object Properties}
+ \paragraph{}
+ %
$C$ 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.)
+ \paragraph{}
+ %
An object $0$ is \defn{initial} if $\forall_B . \exists! f_B : 0 \to B$.
(\S7.1)
- A \defn{limit} (Awodey:D5.16) of a diagram $D(J)$ is a
- terminal object in the category $\mathbf{Cone}(D(J))$. Written:
- $c_i : (\varprojlim_{j} D_j) \to D_i$.
-
+ \paragraph{}
+ %
+ A \defn{limit} (Awodey:D5.16) of a diagram $D(J)$ is a terminal object in
+ the category $\mathbf{Cone}(D(J))$. Written: $c_i : (\varprojlim_{j} D_j)
+ \to D_i$. A \defn{colimit} (Awodey:\S5.6) is an initial object in the
+ category of cocones; $c_i : D_i \to (\varinjlim_j D_j)$. \xrdefn{cone}
+
+ \paragraph{}
+ %
$(A \times B,\pi_1,\pi_2)$ is a \defn{product} iff (UMP)
\[\xymatrix{
{}\save[]+<-1cm,0cm>*\txt<8pc>{$\forall_{Z,z_1,z_2} \exists!_u$\\$u\pi_1 = z_1 ~\wedge~ u\pi_2 = z_2$}\restore
& A & \ar[l]_{\pi_1} A \times B \ar[r]^{\pi_2} & B \\
}\]
+ \paragraph{}
+ %
$(P,p_1,p_2)$ is a \defn{pullback} (Awodey:p80,D5.4) of $f,g$ iff (UMP)
\[\xymatrix{
{}\save[]+<-1cm,0cm>*\txt<8pc>{$\forall_{Z,z_1,z_2 . fz_1 = gz_2}\exists!_u$\\$z_1 = p_1u ~\wedge~ z_2 = p_2u$}\restore
}\]
$P$ may be denoted $A \times_C B$ when $f,g$ are clear.
+ \paragraph{}
+ %
$S$ 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)
+ \paragraph{}
+ %
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)
+ \paragraph{}
+ %
An object $1$ is \defn{terminal} if $\forall_A . \exists! f_A : A \to 1$.
(\S7.4)
+ \paragraph{}
+ %
An object that is both initial and terminal is called a \defn{zero}.
(\S7.7)
\section{Arrow Properties}
+ \paragraph{}
+ %
$e$ is an \defn{epimorphism} (\S7.39) (the dual of a monomorphism)
(equiv: is \defn{epic} (Awodey:D2.1)) if
%
then so is $g$. (\S7.41) Epis generalize \defn{surjection} in
$\mathbf{Set}$.
+ \paragraph{}
+ %
$(E,e)$ is an \defn{equalizer} (\S7.51) of $f,g$ iff (UMP) $fe = ge$ and
\[\forall_{Z,z . zf = zg} \exists!_u eu = z \quad
\xymatrix{
}\]
Equalizers are essentially unique (\S7.53) and monic. % XXX Cite?
+ \paragraph{}
+ %
A mono $m$ is a \defn{extremal} (\S7.61) if $e$ epic and
$m = f \circ e$ implies that $e$ iso.
+ \paragraph{}
+ %
Let $G: \mathbf{A} \to \mathbf{B}$ and $B \in \mathbf{B}$. A
- \defn{$G$-structured arrow with domain $B$} is a pair $(f : B \to GA, A)$.
+ \defn[gstrarr]{$G$-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
When $G$ is a subcategory inclusion, a $G$-structured universal arrow is
a \defn{reflection} (\S4.16).
+ \paragraph{}
+ %
$f : A \to B$ 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).
+ \paragraph{}
+ %
$f$ is a \defn{monomorphism} (\S7.32) (equiv: is \defn{monic}
(Awodey:D2.1)) if
\[\xymatrix{\forall_{i,j} . mi = mj \Rightarrow i = j & C \ar@<1ex>[r]^{i} \ar@<-1ex>[r]_j & A \ar@{{ >}->}[r]^m & B} \]
$\mathbf{Set}$. Objects with monomorphisms to $X$ are called
\defn{subobjects} of $X$ (Awodey:D5.1).
+ \paragraph{}
+ %
A \defn{point} (Awodey:p32) of $C$ is any $c : 1 \to C$.
+ \paragraph{}
+ %
$f$ is a \defn{regular monomorphism} (\S7.56) if it is an equalizer of
some pair of morphisms.
+ \paragraph{}
+ %
$f : A \to B$ 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)
+ \paragraph{}
+ %
$f : A \to B$ 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)
+ \paragraph{}
+ %
Several morphism properties combine in useful ways:
\begin{itemize}
\item mono, epi $\Rightarrow$ \defn{bimorphism} (\S7.49)
\section{Exponentials}
+ \paragraph{}
+ %
(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.
}\]
The arrows $f$ and $\tilde f$ are ``exponential transposes.''
+ \paragraph{}
+ %
Exponential transposition is self inverse (Awodey:p108). This implies
\[ \mbox{hom}_{\bf C}(A \times B, C) \simeq \mbox{hom}_{\bf C}(A, C^B) \]
+ \paragraph{}
+ %
A category is \defn{cartesian closed} (Awodey:p108,D6.2) if it has all
finite products and exponentials.
\section{Functors}
+ \paragraph{}
+ %
Default notation here: functors $F,G : \mathbf{A} \to \mathbf{B}$.
+ \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}.
+ \paragraph{}
+ %
A \defn{contravariant functor} $F$ (\S3.20.5) is a (covariant) functor
$\mathbf{A}^\text{op} \to \mathbf{B}$.
+ \paragraph{}
+ %
A \defn{diagram} (Awodey:D5.15) is a functor $D : J \to C$ from some
indexing category $J$.
+ \paragraph{}
+ %
A \defn{endofunctor} has $\mathbf{A} = \mathbf{B}$. $F \circ F$ may be
denoted $F^2$, etc. (\S3.23; ftn 15)
+ \paragraph{}
+ %
Functors compose. (\S3.23)
% XXX Cite
+ \paragraph{}
+ %
A functor $F : C \to D$\dots
\begin{itemize}
- \item \defn{preserves limits of type $J$} if
+ \item \defn[fpresvlim]{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 \defn{creates limits of type $J$} if $\forall_{D : J \to C}$
+ \item \defn[fcreatlim]{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}
+ \paragraph{}
+ %
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 $\forall_{A,A'} . F\vert_{\mathbf{A}(A,A')}
\subseteq \mathbf{B}(FA, FA')$ is injective.
- \item \defn{full} if $\forall_{A,A'} . F\vert_{\mathbf{A}(A,A')}$ surjective.
+ \item \defn[fullfunc]{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}
+ \paragraph{}
+ %
A \defn{natural transformation} $\tau : F \natto G$ assigns each
$A \in \mathbf{A}$ to $\tau_A : FA \to GA$ s.t.
$\forall_{f : A \to A'} . G f \circ \tau_A = \tau_{A'} \circ F f$
& FA \ar[r]^{\tau_A} \ar[d]_{Ff} & GA \ar[d]^{Gf} \\
& FB \ar[r]^{\tau_B} & GB} \]
+ \paragraph{}
+ %
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 $(\tau H)_A
= \tau_{HA}$.
+ \paragraph{}
+ %
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).
\subsection{Special Functors}
+ \paragraph{}
+ %
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).
+ \paragraph{}
+ %
Representable functors preserve monos. (\S7.37.1)
+ \paragraph{}
+ %
Pullback defines a functor
\[ h^* : (A \stackrel{\alpha}{\to} C) \in \mathbf{C}/C
\mapsto (C' \times_C A \stackrel{\alpha'}{\to} C') \in \mathbf{C}/C' \]
\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. $\forall_{D_\alpha \in D(J)} c_j = D_\alpha \circ
c_i$.
+ \paragraph{}
+ %
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)
+ \paragraph{}
+ %
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)
+ \paragraph{}
+ %
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}
+ \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)
+ \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)
+ \paragraph{}
+ %
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}
+ \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)
%An object $A\in\mathbf{A}$ is
% \item \defn{indiscrete}, $(UB \to UA)$ (\S8.3)
%\end{itemize}
+ \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)
%\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}
\subsection{Joy Approach}
+ \paragraph{}
+ %
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)
+ \paragraph{}
+ %
Adjoints compose (\S8.5), preserve mono sources (\S8.6), and preserve
limits (\S8.9)
+ \paragraph{}
+ %
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)
+ \paragraph{}
+ %
$(\eta,\epsilon) : F \dashv G : \mathbf{A} \to \mathbf{B}$ is a
\defn{adjoint situation} if the above relationships hold. (\S19.7)
\subsection{Awodey Approach}
+ \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.
\[\xymatrix{
\subsection{Moving Right Along}
+ \paragraph{}
+ %
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
\appendix
\section{Miscellaneous Terminology}
+ \paragraph{}
+ %
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.
+ \paragraph{}
+ %
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
+ \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)
\section{Miscellaneous Useful Properties}
+ \paragraph{}
+ %
(Awodey:p84,L5.8) In the commuting diagram
\[\xymatrix{ F \ar[r]_{f'} \ar[d]^{h''} & E \ar[r]_{g'} \ar[d]^{h'} & D \ar[d]^{h} \\
A \ar[r]^f & B \ar[r]^g & C
\item If $FDCA$ and $EDCB$ are pullbacks, so is $FEBA$.
\end{enumerate}
+ \paragraph{}
+ %
(Awodey:p84,C5.9) Pullbacks preserve commutative triangles.
+ \paragraph{}
+ %
Universal Constructions (or Universal Mapping Properties, UMP) reduce to
limits (Awodey:p91,E5.17-20):
%
\hline
\end{tabular}
+ \paragraph{}
+ %
Objects defined by UCs are unique up to isomorphism.
\section{Examples To Jog Your Memory}
\subsection{$\mathbf{Mon}$}
+ \paragraph{}
+ %
Mono-epics are not isos ($(\mathbf{N},+,0) \to (\mathbf{Z},+,0)$).
(Pierce:\S1.6.3)
+ \paragraph{}
+ %
$(\set{*},\cdot,*)$ is a (the) zero.
+ \paragraph{}
+ %
Each monoid $M$ has only one point, $1 \to M$.
\subsection{Adjoint Situations and Monads}
+ \paragraph{}
+ %
Consider $(\eta, \epsilon) : F \dashv G : \mathbf{Mon} \to \mathbf{Set}$.
$\eta_X : X \to GFX$ is insertion of generators: $\forall x \in X . \eta_X x = x$.
$\epsilon_Y : FGY \to Y$ is the re-introduction of structure;
\quad \epsilon_Y (y \in GY) = y
\]
+ \paragraph{}
+ %
Further, $T = G \circ F$ is a monad. Generically, $\mu$...
\begin{align*}
\mu_X (TTX) &= (G \epsilon F)_X (TTX) = (G \epsilon_{FX}) (GFGFX) \\
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