## 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 $𝒞$.

• 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 ${X}_{0}{×}_{Y}{X}_{1}$ given by
i.e., it is given by
such that if
then there exists a unique $Z⇢{X}_{0}{×}_{Y}{X}_{1}$ such that
commutes.

• The fiber coproduct, or pushout, of
is
such that if
then there exists a unique ${Y}_{0}{\bigsqcup }_{X}{Y}_{1}⇢Z$ such that
commutes.
• The product of ${X}_{0}$ and ${X}_{1}$ is ${X}_{0}×{X}_{1}$ 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 ${X}_{0}\underset{g}{\overset{f}{⇉}}{X}_{1}$ is $K$ given by
• The cokernel of ${X}_{0}\underset{g}{\overset{f}{⇉}}{X}_{1}$ is $L$ given by

HW: Show that, in a small category, $X0 ×Y X1 = {(x,y) ∈X0× X1 | f(x)=g(y)} and Y0 ⊔X Y1 = Y0 ⊔ Y1 ⟨ f(x) =g(x) ⟩ .$

Let $𝒞$ 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 $\alpha :I\to 𝒞$.
• An projective system indexed by $I$ is a functor $\alpha :{I}^{\mathrm{op}}\to 𝒞$.
• The inductive limit of $\alpha :I\to 𝒞$ is $\underset{\to }{\mathrm{lim}}\alpha$ given by
for $i,j\in I$ and $s:i\to j$.
• The projective limit of $\beta :{I}^{\mathrm{op}}\to 𝒞$ is $\underset{←}{\mathrm{lim}}\beta$ 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 $I=\phantom{\rule{1em}{0ex}}•←•\to •$.
Products and sums are inductive and projective limits corresponding to the category $I=\phantom{\rule{1em}{0ex}}•\phantom{\rule{1em}{0ex}}•\phantom{\rule{1em}{0ex}}•\phantom{\rule{.5em}{0ex}}\cdots \phantom{\rule{1em}{0ex}}$ (the discrete category with set $I$).
Cokernels and kernels are inductive and projective limits corresponding to the category $I=\phantom{\rule{1em}{0ex}}•⇉•$.

## Notes and References

These notes are distilled from [KS, Chapt. 1-2].

Pairing of states as an (n-1) functor $C$ $C$ $C$ $C$ ${V}_{x}$ ${V}_{y}$ ${V}_{x\prime }$ ${V}_{y\prime }$ ${e}_{1}\left(x\right)$ $⇓$ ${e}_{1}\left(y\right)$ ${\overline{e}}_{2}\left(x\prime \right)$ $⇓$ ${\overline{e}}_{2}\left(y\prime \right)$ $\mathrm{tra}\left({\gamma }_{1}\right)$ $⇓\mathrm{tra}\left(\Sigma \right)$ $\mathrm{tra}\left({\gamma }_{2}\right)$ $X$