--- /dev/null
+@book{adamek:joy,
+ title = {Abstract and Concrete Categories: The Joy of Cats},
+ url = {http://www.prgmea.com/pdf/abstract/9.pdf},
+ urldate = {2014-03-20},
+ author = {{Jirı Adámek} and {Horst Herrlich} and Strecker, George E.},
+ year = {2004},
+}
+
+
+@book{awodey:ct,
+ address = {Oxford; New York},
+ title = {Category Theory},
+ isbn = {9780199237180},
+ language = {English},
+ publisher = {Oxford University Press},
+ author = {Awodey, Steve},
+ month = aug,
+ year = {2010}
+}
+
+@book{pierce:basicct,
+ address = {Cambridge, Massachusetts},
+ series = {Foundations of Computing Series},
+ title = {Basic Category Theory for Computer Scientists},
+ isbn = {9780262660716},
+ language = {English},
+ publisher = {{MIT} Press},
+ author = {Pierce, Benjamin C.},
+ month = aug,
+ year = {1991}
+}
+
+@article{hinze:f,
+ title = {Functional {Pearl}: {F} for {Functor}},
+ url = {http://www.cs.ox.ac.uk/people/daniel.james/functor/functor.pdf},
+ journal = {ICFP},
+ author = {Ralf Hinze and Jennifer Hackett and Daniel W. H. James},
+ year = {2012},
+}
+
+@misc{milewski:limits,
+ title = {Understanding Limits},
+ url = {http://bartoszmilewski.com/2014/05/08/understanding-limits-2/},
+ journal = {Bartosz Milewski's Programming Cafe},
+ author = {Bartosz Milewski},
+}
+
+@misc{milewski:ends,
+ title = {Natural Transformations and Ends},
+ url = {http://bartoszmilewski.com/2014/07/15/natural-transformations-and-ends/},
+ journal = {Bartosz Milewski's Programming Cafe},
+ author = {Bartosz Milewski},
+}
Entries within each section are roughly sorted by definition, alphabetically.
+Quantifiers are written perhaps unusually in this document, as $Q_{\phi}$,
+where $Q$ is $\forall$, $\exists$, $\bigcup$, etc. and $\phi$ is a list of
+variables or an expression whose free variables are quantified over.
+Constrained quantification may be written as $v_1 : \tau_1, v_2 : \tau_2 .
+\phi(v_1,v_2)$ to indicate ``the pairs of values $v_1$ ($\in \tau_1$) and
+$v_2$ ($\in \tau_2$) such that $\phi(v_1,v_2)$ holds''. Strings of
+quantifiers are represented $Q_{\phi} Q'_{\phi'}$ etc. There is not
+necessarily a dot between quantifiers or between the quantifiers and
+quantified formula.
+
%>>>
\section{Basics} % <<<
\paragraph{}
%
- $C$ is a \defn{coseparator} if $\forall_{f,g : B \to A} . f \ne g
+ $C$ 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{}
+% %
+% The \defn{end} of a diagonal profunctor $S : \mathbf{A}^\text{op} \times
+% \mathbf{A} \to \mathbf{B}$ is the object
+
\paragraph{}
%
- An object $0$ is \defn{initial} if $\forall_B . \exists! f_B : 0 \to B$.
+ An object $0$ is \defn{initial} if $\forall_B \exists! f_B : 0 \to B$.
(\S7.1)
\paragraph{}
\paragraph{}
%
- $S$ is a \defn{separator} if $\forall_{f,g : A \to B} . f \ne g
+ $S$ 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},
+ $\forall_{f,g : A \to B} f \ne g \Rightarrow \exists{S \in \mathcal{T},
h : S \to A} . f \circ h \ne g \circ h$. (\S7.14)
\paragraph{}
%
- An object $1$ is \defn{terminal} if $\forall_A . \exists! f_A : A \to 1$.
+ An object $1$ is \defn{terminal} if $\forall_A \exists! f_A : A \to 1$.
(\S7.4)
\paragraph{}
$e$ is an \defn{epimorphism} (\S7.39) (the dual of a monomorphism)
(equiv: is \defn{epic} (Awodey:D2.1)) if
%
- \[\xymatrix{\forall_{i,j} . ie = je \Rightarrow i = j & A \ar@{->>}[r]^e & B \ar@<1ex>[r]^{i} \ar@<-1ex>[r]_j & C} \]
+ \[\xymatrix{\forall_{i,j} ie = je \Rightarrow i = j & A \ar@{->>}[r]^e & B \ar@<1ex>[r]^{i} \ar@<-1ex>[r]_j & C} \]
%
If $f$ and $g$ are epis, then so is $g \circ f$; if $g \circ f$ is epi,
then so is $g$. (\S7.41)
%
\begin{itemize}
%
- \item \defn{generating} if $\forall_{r,s : A \to A'} . Gr \circ f = Gs
+ \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 $\forall_{m :
- A' \to A, m ~\text{mono}, (g,A')} . f = Gm \circ g \implies m ~\text{iso}$.
+ A' \to A, m ~\text{mono}, (g,A')} f = Gm \circ g \implies m ~\text{iso}$.
%
\item \defn[gunivarr]<@G-universal for B>{$G$-universal for $B$} if
- $\forall_{(f', A')} .
+ $\forall_{(f', A')}
%
- \exists!_{\check f} . f' = G{\check f} \circ f$. That is,
+ \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}}
%
$f$ is a \defn{monomorphism} (\S7.32) (equiv: is \defn{monic}
(Awodey:D2.1)) if
- \[\xymatrix{\forall_{i,j} . mi = mj \Rightarrow i = j & C \ar@<1ex>[r]^{i} \ar@<-1ex>[r]_j & A \ar@{{ >}->}[r]^m & B} \]
+ \[\xymatrix{\forall_{i,j} mi = mj \Rightarrow i = j & C \ar@<1ex>[r]^{i} \ar@<-1ex>[r]_j & A \ar@{{ >}->}[r]^m & B} \]
If $f$ and $g$ are monos, then so is $g \circ f$; if $g \circ f$ is mono,
then so is $f$. (\S7.34) Objects with monomorphisms to $X$ are called
\defn{subobjects} of $X$ (Awodey:D5.1).
\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')}
+ \item \defn{faithful} if $\forall_{A,A'} F\vert_{\mathbf{A}(A,A')}
\subseteq \mathbf{B}(FA, FA')$ is injective.
- \item \defn[fullfunc]{full} if $\forall_{A,A'} . F\vert_{\mathbf{A}(A,A')}$ surjective.
- \item \defn{isomorphism-dense} if $\forall_B . \exists_A . F(A) \simeq B$.
+ \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[nattrans]{natural transformation} $\tau : F \natto G$ assigns each
$A \in \mathbf{A}$ to $\tau_A : FA \to GA$ s.t.
- $\forall_{f : A \to A'} . G f \circ \tau_A = \tau_{A'} \circ F f$
+ $\forall_{f : A \to A'} G f \circ \tau_A = \tau_{A'} \circ F f$
(\S6.1;Awodey:D7.6).
That is,
%
% \paragraph{}
% %
% A \defn[exttrans]{extranatural transformation} is one where
+%
+% \paragraph{}
+% %
+% A \defn[dinat]{dinatural transform} is
% >>>
\subsection{Special Functors} % <<<
\paragraph{}
%
- A \defn{mono-source} (\S10.5) is $(A,\set{f_i})$ s.t. \\ $\forall r,s: B \to A.
- \brak{\forall_{i\in I} . f_i \circ r = f_i \circ s} \Rightarrow r = s$.
+ 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{Ends} % <<<
-% XXX not yet
-% \paragraph{}
-% %
-%
-%
% >>>
\section{Concrete Categories} % <<<
\paragraph{}
Coequalizers correspond to equivalence classes (\S7.69.1): Let $\sim$ be
- {\em the smallest} eq. rel. s.t. $\forall_{a \in A} . f(a) \sim g(a)$;
+ {\em the smallest} eq. rel. s.t. $\forall_{a \in A} f(a) \sim g(a)$;
then $(Q,q) = (B/\sim, b \mapsto \brak{b}_\sim)$ is a coequalizer of $f$
and $g$.
\paragraph{}
%
Consider $(\eta, \epsilon) : F \dashv G : \mathbf{Mon} \to \mathbf{Set}$.
- $\eta_X : X \to GFX$ is insertion of generators: $\forall x \in X . \eta_X x = x$.
+ $\eta_X : X \to GFX$ 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
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