Tag Info

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