Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
Last updates: 11 August 2012
Morphisms
A groupoid is a category such that all morphisms are isomorphisms.
An isomorphism is a morphism
$f:X\to Y$
such that there exists a morphism $g:Y\to X$
such that
$f\circ g={\mathrm{id}}_{Y}$
and
$g\circ f={\mathrm{id}}_{X}$.
Let $f:X\to Y$
be a morphism.
A section, or right inverse, of $f$
is a morphism
$g:Y\to X$
such that
$f\circ g={\mathrm{id}}_{Y}$.
A retraction, or left inverse, of $f$
is a morphism
$g:Y\to X$
such that
$g\circ f={\mathrm{id}}_{X}$.
A monomorphism is a morphism
$f:X\to Y$
with the left cancellation property, i.e. a morphism
$f:X\to Y$
such that if
${g}_{1}:Z\to X$${g}_{2}:Z\to X$
and
$f\circ {g}_{1}=f\circ {g}_{2}$
then
${g}_{1}={g}_{2}$.
A epimorphism is a morphism
$f:X\to Y$
with the right cancellation property, i.e. a morphism
$f:X\to Y$
such that if
${g}_{1}:Y\to Z$${g}_{2}:Y\to Z$
and
${g}_{1}\circ f={g}_{2}\circ f$
then
${g}_{1}={g}_{2}$.
Let $X$ be an object in $\mathcal{C}$.
A subobject of $X$ is an isomorphism class of a monomorphism
$f:Y\to X$.
A quotient of $X$ is an isomorphism class of an epimorphism
$f:X\to Q$.
Limits, initial and terminal objects, sums and products, kernels and cokernels, fiber products and coproducts
The fiber product, or pullback, of
is
${X}_{0}{\times}_{Y}{X}_{1}$
given by
i.e., it is given by
such that if
then there exists a unique $Z\u21e2{X}_{0}{\times}_{Y}{X}_{1}$ such that
commutes.
The fiber coproduct, or pushout, of
is
such that if
then there exists a unique
${Y}_{0}{\bigsqcup}_{X}{Y}_{1}\u21e2Z$
such that
commutes.
The product of ${X}_{0}$
and ${X}_{1}$ is
${X}_{0}\times {X}_{1}$ given by
(In the category Set this is product.)
The coproduct of ${Y}_{0}$
and ${Y}_{1}$ is
${Y}_{0}\bigsqcup {Y}_{1}$
given by
(In the category Set this is disjoint union.)
The kernel of ${X}_{0}\underset{g}{\overset{f}{\rightrightarrows}}{X}_{1}$
is $K$ given by
The cokernel of ${X}_{0}\underset{g}{\overset{f}{\rightrightarrows}}{X}_{1}$
is $L$ given by
HW: Show that, in a small category,
$${X}_{0}{\times}_{Y}{X}_{1}=\left\{\right(x,y)\in {X}_{0}\times {X}_{1}\phantom{\rule{0.3em}{0ex}}|\phantom{\rule{0.3em}{0ex}}f\left(x\right)=g\left(y\right)\}\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{Y}_{0}{\bigsqcup}_{X}{Y}_{1}=\frac{{Y}_{0}\bigsqcup {Y}_{1}}{\u27e8f\left(x\right)=g\left(x\right)\u27e9}.$$
Let $\mathcal{C}$ be an additive category and let
$f:X\to Y$ be a morphism.
The kernel of $f$ is the fiber product of
The cokernel of $f$ is the fiber coproduct of
An inductive system indexed by $I$ is a functor
$\alpha :I\to \mathcal{C}$.
An projective system indexed by $I$ is a functor
$\alpha :{I}^{\mathrm{op}}\to \mathcal{C}$.
The inductive limit of
$\alpha :I\to \mathcal{C}$
is $\underrightarrow{\mathrm{lim}}\alpha $ given by
for $i,j\in I$ and $s:i\to j$.
The projective limit of
$\beta :{I}^{\mathrm{op}}\to \mathcal{C}$
is $\underleftarrow{\mathrm{lim}}\beta $ given by
for $i,j\in I$ and $s:i\to j$.
Fiber products and coproducts are inductive and projective limits corresponding
to the category
$I=\phantom{\rule{1em}{0ex}}\u2022\leftarrow \u2022\to \u2022$.
Products and sums are inductive and projective limits corresponding
to the category
$I=\phantom{\rule{1em}{0ex}}\u2022\phantom{\rule{1em}{0ex}}\u2022\phantom{\rule{1em}{0ex}}\u2022\phantom{\rule{.5em}{0ex}}\cdots \phantom{\rule{1em}{0ex}}$ (the discrete category with set $I$).
Cokernels and kernels are inductive and projective limits corresponding
to the category
$I=\phantom{\rule{1em}{0ex}}\u2022\rightrightarrows \u2022$.
Pairing of states as an (n-1) functor$C$$C$$C$$C$${V}_{x}$${V}_{y}$${V}_{x\prime}$${V}_{y\prime}$${e}_{1}(x)$$\Downarrow $${e}_{1}(y)$${\overline{e}}_{2}(x\prime )$$\Downarrow $${\overline{e}}_{2}(y\prime )$$\mathrm{tra}({\gamma}_{1})$$\Downarrow \mathrm{tra}(\Sigma )$$\mathrm{tra}({\gamma}_{2})$$X$
References
[KS]
M. Kashiwara and P. Schapira, Categories and sheaves,
Grundlehren der Mathematischen Wissenschaften 332
Springer-Verlag, Berlin, 2006. x+497 pp. ISBN: 978-3-540-27949-5; 3-540-27949-0
MR2182076.