Functors

concept in linear algebra, generalized throughout mathematics

In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in every area of mathematics where category theory is applied.

transformation between two functors studied in category theory

contravariant functor to the category of sets and functions

category containing functors with natural transformations as morphisms

homological construction in category theory

in homological algebra, the left derived functor of the tensor product of modules over a ring

functors which are surjective or injective on hom-sets

collection of objects and morphisms in a category

functor mapping hom objects to an underlying category

functor that preserves short exact sequences

derived functors of the Hom functor

construction in categorical homotopy theory; contravariant functor from the simplex category to the category of sets

drops some or all of the input's structure or properties