Functor

In category theory a functor or functor is a function from one category to another that takes objects to objects and morphisms to morphisms such that the composition of morphisms and identities are preserved.

Functors were first considered in algebraic topology, where algebraic objects are associated with topological spaces and algebraic homomorphisms are associated with continuous functions. Today, functors are used throughout modern mathematics to relate various categories.

Examples of typical factors are the faithful functor and the full functor.

Definition

Let C and D be categories. A functor F from C to D is a correspondence that(Jacobson, 2009, p. 19, def. 1.2)

• associates each object ${displaystyle X}$ in C object ${displaystyle F(X)}$ in D,
• associates each morphism ${displaystyle fcolon Xto Y}$ in C a morphism ${displaystyle F(f)colon F(X)to F(Y)}$ in D so that the following two conditions are maintained:
  • ${displaystyle F(mathrm {id} _{X})=mathrm {id} _{F(X)},!}$ for all objects ${displaystyle X}$ in C,
  • ${displaystyle F(gcirc f)=F(g)circ F(f)}$ for all morbidities ${displaystyle fcolon Xto Y,!}$ and ${displaystyle gcolon Yto Z}$ in C.

That is, functors must preserve identity maps and map composition.