This tag is for questions relating to functors, which is a mapping from one category into another that is compatible with the category structure. Functors exist in both covariant and contravariant types.

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 all areas within mathematics to which category theory is applied.

Definitions: A functor $~F : \mathcal C → \mathcal D~$ relates two categories $~\mathcal C~$ and $~\mathcal D~$ in the following way:

  • To each object $~X ∈~$ ob $\mathcal C~$ it associates an object $~F X ∈~$ ob $\mathcal D~$
  • To each map $~f ∈ \mathcal C(X, Y )~$ it associates a map $~F f ∈ \mathcal D(F X, F Y )~$

such that the following properties hold:

  • For each object $~X ∈~$ ob $\mathcal C$, $~F1_X = 1_{FX}~$
  • For a map $~g ∈ \mathcal C(X, Y )~$, and a map $~f ∈ \mathcal C(Z, Y )~$, we have $~F(f ◦g) = F f ◦F g~$.