# Category

$ \newcommand{\inv}{^{-1}} \newcommand{\op}{^\mathrm{op}} % rm \newcommand{\act}{{\mathrm{act}}} \newcommand{\Obj}{{\mathrm{Obj}}} \newcommand{\Hom}{{\mathrm{Hom}}} \newcommand{\id}{{\mathrm{id}}} % cat \newcommand{\catf}[1]{{\mathbf{#1}}} \newcommand{\cA}{\catf{A}} \newcommand{\cB}{\catf{B}} \newcommand{\cC}{\catf{C}} \newcommand{\cX}{\catf{X}} \newcommand{\cY}{\catf{Y}} \newcommand{\cZ}{\catf{Z}} \newcommand{\cSet}{\catf{Set}} % sets \newcommand{\cRel}{\catf{Rel}} % relations \newcommand{\cPos}{\catf{Pos}} % posets \newcommand{\cMon}{\catf{Mon}} % monoids \newcommand{\cGrp}{\catf{Grp}} % groups \newcommand{\cAb}{\catf{Ab}} % Abelian groups \newcommand{\cCat}{\catf{Cat}} % categories \newcommand{\cGrpd}{\catf{Grpd}} % groupoids \newcommand{\cVect}{\catf{Vect}} % vector spaces $

## Category

A **category** $\cC = (\Obj, \Hom, \id, \circ)$ consists of

- $\Obj$:
a collection of
**objects**$A, B, C, \dots$ - $\Hom(A, B)$:
a collection of
**morphisms**(arrows) $f, g, h, \dots$ from $A$ to $B$ for each pair $(A, B)$ of objects - $\id_A$:
an
**identity**morphism for each object $A$ - $\circ$:
**composition**$\Hom(B, C) \times \Hom(A, B) \to \Hom(A, C)$

subject to the following conditions:

- composition is associative: $h \circ (g \circ f) = (h \circ g) \circ f$.
- identity and composition are compatible: $f = f \circ \id_A = \id_B \circ f$.

$A \in \cC$ means $A \in \Obj_\cC$ and $f \in \cC$ means $f \in \Hom_\cC(A, B)$ for some $A$ and $B$.

A morphism $f$ from $A$ to $B$ is also denoted by $f: A \to B$ or $A \xrightarrow{f} B$.

Roughly speaking, a category consists of a collection of objects and a collection of arrows between them,

and we have an identity arrow for each object and we know how to compose two arrows to get a new one.

Indeed, the subject might better have been called

abstract function theory, or, perhaps even better:archery[Awodey, 2006].

### Examples

- The first example is the category $\cSet$ of
**sets**with**functions**between sets as morphisms.

The identity $\id_A$ is the identity function $\id_A: a \mapsto a$ and the composition is the function composition. - More generally, we have the category $\cRel$ of
**sets**with**binary relations**as morphisms. - Further, we have categories of
**structured sets**and**structure-preserving functions**between them, such as:- $\cPos$: posets and monotonic functions
- $\cMon$: monoids and monoid homomorphisms
- $\cGrp$: groups and group homomorphisms
- $\cAb$: Abelian groups and group homomorphisms
- $\cVect$: vector spaces and linear functions

- A
**monoid**is a**single-object**category where morphisms are its elements. - A
**proset**is a**single-morphism**category where objects are its elements. - A
**groupoid**is a category in which all morphisms are isomorphisms.

A group is a single-object groupoid.

### Size

**small**: $\Obj$ is a set and $\Hom(A, B)$ is a set.**locally small**: $\Hom(A, B)$ is a set.**large**: otherwise.

### Isomorphism

An **inverse** of a morphism $f: A \to B$ is $f\inv: B \to A$ that $f\inv \circ f = \id_A$ and $f \circ f\inv = \id_B$.

A morphism which has an inverse is called an **isomorphism**.

Two objects $A$ and $B$ are isomorphic if there is an insomorphism between them, denoted by $A \cong B$.

## Functor

For two categories $\cA$ and $\cB$, a **functor** $F: \cA \to \cB$ is a function
sending $\cA$-objects to $\cB$-objects $F(A) = FA$ and
$\cA$-morphisms to $\cB$-morphisms
$F(A \xrightarrow{f} A’) = FA \xrightarrow{Ff} FA’$,
in such a way that

- $F$ preserves composition: $F(g \circ_\cA f) = Fg \circ_\cB Ff$.
- $F$ preserves identity morphisms: $F(\id_A) = \id_{FA}$.

A category is a *generalized* **monoid** and a functor is a *generalized* **monoid homomorphism**.

A category is a *generalized* **proset** and a functor is a *generalized* **monotonic function**.

### Forgetful functor

For ordinary categories, an **essentially $k$-surjective** functor $F: \cA \to \cB$ is

**essentially surjective**: $F_\Obj$ is surjective up to isomorphism.**full**: $F_{\Hom(A, A’)}$ is surjective.**faithful**: $F_{\Hom(A, A’)}$ is injective.- essentially $k$-surjective for all $k \geq 3$.

Then, a **forgetful functor** is defined as follows:

- ($k \geq 0$) a fully faithful and essentially surjective functor forgets
**nothing**. - ($k \geq 1$) a fully faithful functor forgets
**only properties**. - ($k \geq 2$) a faithful functor forgets
**at most structure**. - ($k \geq 3$) a functor may forget
**everything**.

A fully faithful and essentially surjective functor realizes an **equivalence of categories**.

## Natural Transformation

For functors $F, G: \cA \to \cB$, a **natural transformation** $\tau$ from $F$ to $G$ is a collection of $\cB$-morphisms indexed by $\cA$-objects $\tau_A: FA \to GA$ in such a way that the following diagram commutes: