1.2 Categories
Definition 1.2.1 (Category).label A category $\mathfrak{C}$ is a collection of objects $\ob{\frak{C}}$, such that for any $X,Y,Z\in\text{Ob}(\frak{C})$ there exists collections of morphisms $\mor{}{X }{Y },\mor{}{Y }{Z }$ and a composition law:
satisfying:
- (CAT1)
For any $X,Y,Z,W\in \ob{\frak{C}}$, $\mor{}{X }{Y }$ and $\mor{}{Z }{W }$ are disjoint or equal
- (CAT2)
For any $X,Y\in\ob{\frak{C}}$, there exists $\text{Id}_{X}\in \mor{}{X}{X},\text{Id}_{Y}\in \mor{}{Y }{Y }$ such that for each $f\in \mor{}{X }{Y }$, $f\circ \text{Id}_{X}=f=\text{Id}_{Y}\circ f$
- (CAT3)
For any $X,Y,Z,W\in \ob{\frak{C}}$, $f\in \mor{}{X }{Y },g\in \mor{}{Y }{Z }$ and $h\in \mor{}{Z }{W }$, $(h\circ g)\circ f=h\circ (g\circ f)$
Definition 1.2.2 (Isomorphism).label Let $\frak{C}$ be a category and $X,Y\in \ob{\frak{C}}$ then $X$ and $Y$ are isomorphic, denoted by $X\iso Y$, if there are morphisms $f\in \mor{}{X }{Y },g\in \mor{ }{Y }{X }$ such that $f\circ g=\text{Id}_{Y}$ and $g\circ f=\text{Id}_{X}$.
Definition 1.2.3 (Universal Objects).label An initial (resp. final) object in a category $\frak{C}$ is an object $A\in \ob{\frak{C}}$ such that for each $X\in \ob{\frak{C}}$ the set $\mor{}{A }{X }$ (resp. $\mor{}{X }{A }$) has a single element. Initial and Final objects if exists in a category is unique up to unique isomorphism.
An extremal object is an object that is initial and/or final. A zero object is an object that is initial and final.
Proof. Let $A,A'$ be initial objects of a category $\frak{C}$ then there are unique morphisms $f\in \mor{}{A }{A'},g\in \mor{}{A' }{A }$ so $g\circ f\in \mor{}{A }{A }$ hence $g\circ f=\text{Id}_{A}$ and similarly $f\circ g=\text{Id}_{A}'$ showing $A\iso A'$. Since $\mor{}{A }{A'}$ has a single element, this isomorphism is unique. By Definition 1.2.4 we have also shown the property for final objects.$\square$
Definition 1.2.4 (Opposite Category).label Let $\frak{C}$ be a category then the oppposite category $\frak{C}^{\text{op}}$ is a category with the same objects with $\mor{\frak{C}^{\text{op}}}{X}{Y}\define \mor{\frak{C}}{Y}{X}$ and composition law
Then $\parens{\cd}^{\text{op}}$ is a strictly involutive contravariant endofunctor on $\text{CAT}_{U}$ as in Definition 1.2.5 and Definition 1.2.7, in particular $\parens{\frak{C}^{\text{op}}}^{\text{op}}=\frak{C}$ and $A\in \ob{\frak{C}}$ is intial (final) if and only if $A\in \ob{\frak{C}^{\text{op}}}$ is final (intial).
Proof. Observe
- (CAT1)
By construction
- (CAT2)
By construction
- (CAT3)
Let $X,Y,Z,W\in \ob{\frak{C}^{\text{op}}}$, $f\in \mor{\frak{C}^{\text{op}}}{X }{Y },g\in \mor{\frak{C}^{\text{op}}}{Y }{Z }$ and $h\in \mor{\frak{C}^{\text{op}}}{Z }{W }$ then
\begin{align*}(h\circ^{\text{op}}g)\circ^{\text{op}}f=(g\circ h)\circ^{\text{op}}f=f\circ(g\circ h)=(f\circ g)\circ h=h\circ^{\text{op}}(g\circ^{\text{op}}f)\end{align*}showing $\frak{C}^{\text{op}}$ is a category. The rest follows by construction.
$\square$
Definition 1.2.5 (The 2-category of $U$-small categories).label Let $U$ be a Grothendieck universe then define $\text{CAT}_{U}$ to be the $2$-category whose:
Objects are $U$-small categories, i.e. categories $\frak{C}$ such that $\ob{\frak{C}}\in U$ and $\mor{\frak{C}}{X }{Y }$ for all $X,Y\in\ob{\frak{C}}$.
$1$-morphisms are functors between $U$-small categories.
$2$-morphisms are natural transformations between such functors.
Remark 1.2.6 (Axiom of Universes).label As in Grothendieck’s original proposal we adopt axiom of universes which states:
For every ZFC-set $s$, there exists a universe $U$ such that $s\in U$.
which means every set is small if your universe is large enough.
Definition 1.2.7 (Functor).label Let $\frak{C},\frak{D}$ be categories. A covariant (resp. contravariant) functor is a rule which
associates each object $X\in \ob{\frak{C}}$ with an object $F(X)\in \ob{\frak{D}}$
associates each morphism $f\in \mor{\frak{C}}{X}{Y}$ with a morphism $F(f)\in \mor{\frak{D}}{F(X)}{F(Y)}$ (resp. $\mor{\frak{D}}{F(Y)}{F(X)}$) such that:
- (F1)
$F(\text{Id}_{X})=\text{Id}_{F(X)}$ for each $X\in \ob{\frak{X}}$
- (F2)
$F(g\circ f)=F(g)\circ F(f)$ (resp. $=F(f)\circ F(g)$) for each morphisms $f\in \mor{\frak{C}}{X}{Y},g\in \mor{\frak{C}}{Y}{Z}$
We denote the collection of functors $\text{Fun}_{\text{cov}}\parens{\frak{C},\frak{D}}$ (resp. $\text{Fun}_{\text{contra}}\parens{\frak{C},\frak{D}}$). Then there is a canonical bijection $\text{Fun}_{\text{contra}}\parens{\frak{C},\frak{D}}\to \text{Fun}_{\text{cov}}\parens{\frak{C}^{\text{op}},\frak{D}}$.
We say that $F$ is an endofunctor if $\frak{C}=\frak{D}$. In which case we say that $F$ is involutive if $F\circ F$ is naturally isomorphic to $\indi{\frak{C}}$ and strictly involutive if $F\circ F=\indi{\frak{C}}$.
Proof. Consider the map
to be continued$\square$
Definition 1.2.8 (Natural Transformations).label Let $F,G:\frak{C}\to\frak{D}$ be covariant functors. A natural transformation or morphism of functors $\al$ from $F$ to $G$ is a rule which associates each object $X\in \ob{\frak{C}}$ with a morphism $\al_{X}:F(X)\to G(X)$ such that for each morphism $f\in \mor{\frak{C}}{X }{Y }$ we have a commutative diagram:
If each $\al_{X}$ is an isomorphism then $F$ and $G$ are naturally equivalent or isomorphic, denoted by $F\iso G$. In which case
A similar definition can be made for contravariant functors by observation in Definition 1.2.7.
Definition 1.2.9 (Equivalence of Categories).label Let $\frak{C},\frak{D}$ be categories then we say they are equivalent (resp. antiequivalent) if there are covariant (resp. contravariant) functors $F:\frak{C}\to \frak{D}$ and $G:\frak{D}\to \frak{C}$ such that $G\circ F\iso \indi{\frak{C}}$ and $F\circ G\iso \indi{\frak{D}}$.
Theorem 1.2.10 (Equivalence).label Let $F:\frak{C}\to \frak{D}$ be a covariant (resp. contravariant) functor. There exists a functor $G:\frak{D}\to \frak{C}$ of the same variance such that $(F,G)$ is an equivalence (resp. antiequivalence) of categories if and only if:
$F$ is full i.e. surjective on morphisms
$F$ is faithful i.e. injective on morphisms
$F$ is essentially surjective i.e. surjective on objects up to isomorphism
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