Categories

Category

A category is a collection of objects and morphisms such that

1. Each morphism has a domain and codomain object
2. Each object has a identity morphism $1_X:X\to X$
3. For morphisms $f:X\to Y$ and $g:Y\to Z$, there is a morphism $gf:X\to Z$.

We have the additional requirement that composition is associative, i.e., $f(gh)=(fg)h$.

A common alternative name for morphisms are arrows or maps.

Some examples of Categories:

1. $\text{Set}$ has sets as objects and functions as morphisms
2. $\text{Top}$ has topological spaces as objects and continuous functions as morphisms
3. $\text{Group}$ has groups as objects and group homomorphisms as morphisms

These are concrete categories, i.e., categories whose objects have underlying sets and morphisms are functions. An example of an abstract category is $\text{Htpy}$ which has topological spaces as objects but homotopy classes of continuous maps as morphisms.

The term “collection” in the definition is to avoid Russell’s paradox: let

$$R=\{x\mid x\notin x\}$$

then $R\in R$ if and only if $R\notin R$. However, it is ultimately not important as a primary focus in category theory is functors rather than categories themselves.

Small Category

A category is small if it has a only a set’s worth of arrows.

There is a bijective correspondence between the objects of a category and their identity morphisms. Hence if $C$ is a small category, then there are functions that send a morphism to its domain and its codomain and object to its identity.

$$\text{mor}C\stackrel{dom}{\to }\text{ob}C,\text{mor}C\stackrel{cod}{\to }\text{ob}C,\text{ob}C\stackrel{id}{\to }\text{mor}C$$

None of our examples are small, but they are locally small.

Locally Small

A category is locally small if between any pair of objects there is a set’s worth of arrows.

The set of morphisms from $X$ to $Y$ is typically denoted $C(X,Y)$ or $\text{Hom}(X,Y)$.

The most important type of morphism in any category is the isomorphism.

Isomorphism

An isomorphism in a category is a morphism $f:X\to Y$ for which there is another morphism $g:Y\to X$ such that $gf=1_X$ and $fg=1_Y$. In such a case, $X$ and $Y$ are isomorphic and denoted $X\cong Y$.

In our three examples above:

1. The isomorphisms in $\text{Set}$ are bijections
2. The isomorphisms in $\text{Top}$ are homeomorphisms
3. The isomorphisms in $\text{Group}$ are bijective homomorphisms.

A category where every morphism is an isomorphism is called a groupoid. Given any category $C$, the subcategory containing all of the objects and only those morphisms that are isomorphisms is the maximal groupoid.

Exercises

1. Show that a morphism can have at most one inverse isomorphism.

Let $f:X\to Y$ be an isomorphism and $g,h:Y\to X$ are inverse isomorphisms. Then

$$gf=1_X=hf\text{ and }fg=1_Y=fh.$$

Notice that $gfh=1_{X}h=h$ and $gfh=g{1}_Y=g$. Therefore, $g=h$.

2. Show that the collection of isomorphisms in a category $C$ defines a subcategory.

First notice that for $1_X:X\to X$, $1_X{1}_X=1_X$, i.e., $1_X$ is an isomorphism. Next let $f:X\to Y$ and $g:Y\to Z$ be isomorphism. Then there are morphisms $f^{-1}:Y\to X$ and $g^{-1}:Z\to Y$ such that $f{f}^{-1}=1_Y$, $f^{-1}f=1_X$, $g{g}^{-1}=1_Z$ and $g^{-1}g=1_Y$. Notice that

$$gf{f}^{-1}{g}^{-1}=g{1}_Y{g}^{-1}=g{g}^{-1}=1_Z$$

and

$$f^{-1}{g}^{-1}gf=f^{-1}{1}_{Y}f=f^{-1}f=1_X.$$

Thus, $gf:X\to Z$ is an isomorphism.

3. For any category $C$ and $c\in C$ show that there is a category $c/C$ whose objects are morphism $f:c\to x$ with domain $c$ and in which a morphism from $f:c\to x$ to $g:c\to y$ is a map $h:x\to y$ so that $g=hf$.

Let $f:c\to x$. Notice that the map $1_x:x\to x$ satisfies $f=1_{x}f$. Thus, each morphism has an identity morphism. Next let $h:x\to y$ and $h^{\prime }:y\to z$ be morphisms from $f:c\to x$ to $g:c\to y$ and from $g$ to $g^{\prime }:c\to z$ respectively. Then $g=hf$ and $g^{\prime }=h^{\prime }g$. Notice that

$$g^{\prime }=h^{\prime }g=h^{\prime }hf.$$

Therefore, $h^{\prime }h$ is a morphism.