Categories are structures containing objects and arrows between them. Many mathematical structures can be viewed as objects of a category, with structure preserving morphisms as arrows. Reformulating properties of mathematical objects in the general language of category can help one see connections between seemingly different areas of mathematics.

A category $\mathcal{C}$ consists of a collection $\mathrm{Ob}(\mathcal{C})$ of objects and for any two objects $X$ and $Y$ a collection of arrows (often called morphisms) denoted $\mathrm{Hom}(X,Y)$ such that

  1. Every object $X$ comes equipped with an identity morphism $\mathbf{1}_X \in \mathrm{Hom}(X,X)$.
  2. For morphisms $f \in \mathrm{Hom}(X,Y)$ and $g \in \mathrm{Hom}(Y,Z)$, there exists a composite morphism $g\circ f \in \mathrm{Hom}(X,Z)$.

Many classes of mathematical objects can be collected and described as a category. For example:

  • $\text{Set}$ is the category with sets as objects and functions as morphisms. A subcategory of $\text{Set}$ is $\text{FinSet}$, the category of finite sets.
  • $\text{Grp}$ is the category of groups with morphisms given by group homomorphisms because these are the functions on groups that preserve their structure. $\text{Grp}$ also has a ubiquitous subcategory $\text{Ab}$, the category of abelian groups.

And the relationship between these mathematical objects, either within their own category or with another category, can be described in a coherent way using category theory. In particular many constructions within a category can be described as a limit or colimit, and many relationships between different categories can be described as a functor.

For more reading on category theory, see