Category

Definition

A category $C$ consists of

• A collection $C_0$, whose elements are called the objects of $C$ and are usually denoted by uppercase letters $X,Y,Z,\dots$
• A collection $C_1$, whose elements are called the morphisms or arrows of $C$ and are usually denoted by lowercase letter $f,g,h,\dots$ such that
• Each morphism has assigned two objects, called source and target, or domain and codomain. We denote the source and target of the morphism $f$ by $s(f)$ and $t(f)$, respectively. If the morphism $f$ has source $X$ and target $Y$, we also write $f:X\to Y$, or more graphically, $X\stackrel{f}{\to }Y$
• Each object $X$ has a distinguished morphism ${\mathrm{id}}_X:X\to X$, called identity morphism
• For each pair of morphism $f,g$ such that $t(f)=s(g)$ there exists a specified morphism $g\circ f$, called the composite morphism, such that $s(g\circ f)=s(f)$ and $t(g\circ f)=t(g)$. More graphically:

$$f:X\to Y,g:Y\to Z⇝g\circ f:X\to Z$$

These structures need to satisfy the following axioms

• Unitality: for every morphism $f:X\to Y$, the compositions $f\circ {\mathrm{id}}_X$ and ${\mathrm{id}}_Y\circ f$ are both equal to $f$
• Associativity: for $f:X\to Y,g:Y\to Z$, and $h:Z\to W$, the composition $h\circ (g\circ f)$ and $(h\circ g)\circ f$ are equal

Categories as operations

Another way to think about categories is

A category is a collection of operations which can be composed in a consistent way

An example for this idea is a group. We can view a group $G$ as a particular category, where

• There is a single object $\bullet$
• There is a morphism $\bullet \to \bullet$ for each element $g\in G$, denoted as $g:\bullet \to \bullet$
• The identity of the object $\bullet$ is the morphism given by $1\in G$
• The composition is given by the multiplication of $G$. That is, the composition $g\circ h$ of the morphisms given by $g,h\in G$ is the morphism given by the element $g\cdot h$ of $G$

Note that we didn't use the inverse $g^{-1}$ of $g\in G$, since this property is unnecessary to form a category. Therefore, if we drop the inverse requirement, we can still get a category, and this means a monoid can also be seen as a category

Sometimes, to distinguish the concept of a group/monoid and a category, we may define the category we get from a group $G$ as the delooping of $G$, denoted as $\mathbf{B}G$. Similarly, $\mathbf{B}M$ stands for the delooping of monoid $M$

Categories as spaces and maps with extra structure

By the name, we can consider a category as a collection of sets or spaces equipped with extra structure, and maps between them which are compatible with that structure

We can therefore give some examples of category in this idea.

Category Set

The category $\mathbf{Set}$ is the category whose objects are sets, and whose morphisms are maps (functions) between them.

For each set $X$, the identity function is a function $X\to X$. Functions can be composed, and the composition is associative.

Category Top

The category $\mathbf{Top}$ is the category whose objects are topological spaces, and whose morphisms are continuous maps between them

Category Mfd

The category $\mathbf{Mfd}$ is the category whose objects are smooth manifolds, and whose morphisms are smooth maps between them

Category Vect

The category $\mathbf{Vect}$ is the category whose objects are vector spaces, and whose morphisms are linear maps between them

Category Meas

The category $\mathbf{Meas}$ is the category whose objects are measurable spaces, and whose morphisms are measurable maps between them

Category Grp

The category $\mathbf{Grp}$ is the category whose objects are groups, and whose morphisms are homomorphisms between them, i.e. maps $f:G\to H$ such that for each $g,g^{\prime }\in G$, we have $f(g\cdot g^{\prime })=f(g)\cdot f(g^{\prime })$

Category Poset

The category $\mathbf{Poset}$ is the category whose objects are partially ordered sets, and whose morphisms are monotone maps between them, i.e. maps $f:X\to Y$ such that for each $x,x^{\prime }\in X$ with $x\le x^{\prime }$ in $X$, we have $f(x)\le f(x^{\prime })$ in $Y$

Other examples

Graph Theory

For graphs there are many choices of morphisms, depending on what one wants to do with graphs. For example, for undirected, unweighted graphs, a possible choice of morphisms $f:G\to H$ is function between the sets of vertices which preserve the adjacency relation: if the vertices $x$ and $y$ are connected by an edge in $G$, then $f(x)$ and $f(y)$ must be connected by an edge in $H$. With this choice of morphisms, graphs and these morphisms form a category.

Considerations with Sets

From Russell's paradox, we know that there is no such thing as the "set of all sets" since we cannot conclude whether the set contains itself. Therefore, if we want a "category of all sets", the objects of this category cannot form a set. That is why we said collections rather than sets in the definition of a category

We call a category $C$ is small if $C_0$ and $C_1$ (see the definition above) are sets. And a category $C$ is called locally small if for every to objects $X$ and $Y$ of $C$, the morphisms $X\to Y$ form a set

Based on this, if we let $X$ and $Y$ be objects of a locally small category $C$. The hom-set or hom-space of $X$ and $Y$ is the set of morphisms of $C$ from X to Y, denoted by ${\mathrm{Hom}}_C(X,Y)$. For example,

• ${\mathrm{Hom}}_{\text{Set}}(\mathbb{R},{\mathbb{R}}^2)$ is the set of all functions $\mathbb{R}\to {\mathbb{R}}^2$, i.e. the morphisms of $\mathbf{Set}$
• ${\mathrm{Hom}}_{\text{Top}}(\mathbb{R},{\mathbb{R}}^2)$ is the set of all continuous functions $\mathbb{R}\to {\mathbb{R}}^2$, i.e. the morphisms of $\text{Top}$
• ${\mathrm{Hom}}_{\text{Vect}}(\mathbb{R},{\mathbb{R}}^2)$ is the set of all linear functions $\mathbb{R}\to {\mathbb{R}}^2$