subcategories

A subcategory is a concept used in mathematics and information organization to describe a smaller category formed from the objects and morphisms of a larger category. In category theory, a subcategory C of a category D consists of some objects of D and some morphisms of D between those objects, such that each included object has its identity morphism in C and composition of included morphisms stays within C. Subcategories can be thought of as a subset of the structure of the parent category that still satisfies the same rules.

There are several notable types of subcategories. A wide subcategory uses the same objects as the parent category but includes only a subset of the morphisms. A full subcategory, on the other hand, takes a collection of objects and includes every morphism between those objects that exists in the parent category. A replete (or isomorphism-closed) subcategory contains, up to isomorphism, all objects that are equivalent to its objects within the parent category.

Examples illuminate the idea. The category Set has subcategories such as FinSet, whose objects are finite sets and whose morphisms are functions between them; FinSet is a full subcategory of Set. The category Grp has Ab as a full subcategory, consisting of abelian groups and group homomorphisms between them.

Beyond pure mathematics, subcategories aid information organization and navigation. In knowledge organization, a subcategory may refine a broad topic into more specific topics, support targeted searches, and help structure catalogs or ontologies in a consistent, hierarchical way.