Category (mathematics) - Examples

Examples

The class of all sets together with all functions between sets, where composition is the usual function composition, forms a large category, Set. It is the most basic and the most commonly used category in mathematics. The category Rel consists of all sets, with binary relations as morphisms. Abstracting from relations instead of functions yields allegories instead of categories.

Any class can be viewed as a category whose only morphisms are the identity morphisms. Such categories are called discrete. For any given set I, the discrete category on I is the small category that has the elements of I as objects and only the identity morphisms as morphisms. Discrete categories are the simplest kind of category.

Any preordered set (P, ≤) forms a small category, where the objects are the members of P, the morphisms are arrows pointing from x to y when x ≤ y. Between any two objects there can be at most one morphism. The existence of identity morphisms and the composability of the morphisms are guaranteed by the reflexivity and the transitivity of the preorder. By the same argument, any partially ordered set and any equivalence relation can be seen as a small category. Any ordinal number can be seen as a category when viewed as an ordered set.

Any monoid (any algebraic structure with a single associative binary operation and an identity element) forms a small category with a single object x. (Here, x is any fixed set.) The morphisms from x to x are precisely the elements of the monoid, the identity morphism of x is the identity of the monoid, and the categorical composition of morphisms is given by the monoid operation. Several definitions and theorems about monoids may be generalized for categories.

Any group can be seen as a category with a single object in which every morphism is invertible (for every morphism f there is a morphism g that is both left and right inverse to f under composition) by viewing the group as acting on itself by left multiplication. A morphism which is invertible in this sense is called an isomorphism.

A groupoid is a category in which every morphism is an isomorphism. Groupoids are generalizations of groups, group actions and equivalence relations.

Any directed graph generates a small category: the objects are the vertices of the graph, and the morphisms are the paths in the graph (augmented with loops as needed) where composition of morphisms is concatenation of paths. Such a category is called the free category generated by the graph.

The class of all preordered sets with monotonic functions as morphisms forms a category, Ord. It is a concrete category, i.e. a category obtained by adding some type of structure onto Set, and requiring that morphisms are functions that respect this added structure.

The class of all groups with group homomorphisms as morphisms and function composition as the composition operation forms a large category, Grp. Like Ord, Grp is a concrete category. The category Ab, consisting of all abelian groups and their group homomorphisms, is a full subcategory of Grp, and the prototype of an abelian category. Other examples of concrete categories are given by the following table.

Fiber bundles with bundle maps between them form a concrete category.

The category Cat consists of all small categories, with functors between them as morphisms.