Definition

A category is the data

a class of objects denoted
for each pair of objects , a set of morphisms from to denoted . that satisfies the following conditions:

There is an associative composition law

and for each , there exists and identity morphism

such that for every and

Note if a morphism has an inverse it is called an isomorphism.

Examples

Objects: Sets
Morphisms: functions between sets Composition is composition of functions, and is the identity function.

Objects: For a field , objects are vector fields over .
Morphisms: -linear maps (linear transformations). Composition is composition of linear functions (preserves linearity).

A subcategory is the data of a subclass and subsets

with make into a category using the composition rules from .

Therefore, there is a canonical inclusion functor

with is injective on objects and morphisms.