# Category (category theory)

A category, $\mathcal{C}$, is a mathematical object consisting of:

• A class, $\text{Ob}(\mathcal{C})$ of objects.
• For every pair of objects $A,B\in \text{Ob}(\mathcal{C})$, a class $\text{Hom}(A,B)$ of morphisms from $A$ to $B$. (We sometimes write $f:A \to B$ to mean $f\in \text{Hom}(A,B)$.)
• For every three objects, $A,B,C \in \mathcal{C}$, a binary operation $\circ: \text{Hom}(B,C) \times \text{Hom}(A,B) \to \text{Hom}(A,C)$ called composition, which satisfies:
• (associativity) Given $f:A\to B$, $g:B\to C$ and $h:C \to D$ we have $$h\circ(g\circ f) = (h \circ g)\circ f.$$
• (identity) For and object $X$, there is an identity morphism $1_X:X\to X$ such that for any $f:A\to B$: $$1_B\circ f = f = f\circ 1_A.$$

The class of all morphisms of $\mathcal{C}$ is denoted $\text{Hom}(\mathcal{C})$.

A category $\mathcal{C}$ is called small if both $\text{Ob}(\mathcal{C})$ and $\text{Hom}(\mathcal{C})$ are sets. If $\mathcal{C}$ is not small, then it is called large. $\mathcal{C}$ is called locally small if $\text{Hom}(A,B)$ is a set for all $A,B\in \text{Ob}(\mathcal{C})$. Most important categories in math are not small, but are locally small.

Intuitively we can think of the objects of $\mathcal{C}$ as being sets (perhaps with some additional structure) and morphisms as being functions between these sets (perhaps satisfying some properties) and composition as being regular function composition, however there are examples of categories which do not satisfy this. Typically when studying category theory we deal with morphisms and composition completely abstractly (similarly to how we study multiplication abstractly in group theory), and never talk about 'plugging things in to' morphisms.

## Examples

Some common examples of categories are:

• The category Set of all sets, where morphisms are functions.
• The category Grp of all groups, where morphisms are group homomorphisms.
• The category Ab of all abelian groups, where morphisms are group homomorphisms.
• The category Ring of all rings, where morphisms are ring homomorphisms.
• The category Field of all fields, where morphisms are field homomorphisms (notice that this means all morphisms are injective, and so they can be viewed as field extensions).
• The category Vect of all vector spaces, where morphisms are linear maps.
• The category Top of all topological spaces, where morphisms are continuous functions.
• The category Cat of all small categories, where morphisms are functors.
• For any categories $\mathcal{C}$ and $\mathcal{D}$, the functor category $\mathcal{D}^\mathcal{C}$ of functors $\mathcal{C}\to \mathcal{D}$ where morphisms are natural transformations.
• For any set $S$, we can form the discrete category on $S$ whose objects are elements of $S$ and such that the only morphisms are the identity morphisms on each object.
• For any category $\mathcal{C}$ we can form the opposite category or the dual category, $\mathcal{C}^{op}$ whose objects are the objects of $\mathcal{C}$, but where all the morphisms are 'reversed' (i.e. a morphism from $X$ to $Y$ would be replaced by a morphism from $Y$ to $X$).

Examples which are more specific:

• The category in which the objects are sets and there is a morphism $U \to V$ if and only if $U \subset V$.
• The category in which the objects are positive integers and there is a morphism $s \to t$ if and only if $s$ divides $t$.
• For a fixed poset $P$, the category in which the objects are elements of $P$ and there is a morphism $s \to t$ if and only if $s \le t$.
• For a fixed ring $R$, the category in which the objects are elements of $R$ and there is a morphism $s \to t$ if and only if there exists some $u$ such that $su = t$.