Functor

In category theory, a functor is a map between categories satisfying certain relations. Functors, in a sense, provide for categories what group homomorphisms do for groups. To be precise, a function $F:{\mathcal {C}}\to {\mathcal {D}}$ between two categories associates to each object $X\in {\textrm {Ob}}{\mathcal {C}}$ an object $F(X)\in {\textrm {Ob}}{\mathcal {D}}$, and to each morphism $f\in {\mathcal {C}}(X,Y)$ a morphism $F(f)\in {\mathcal {D}}(FX,FY)$ such that:

$F({\textrm {id}}_{X})=id_{F(X)}$

$F(g\circ f)=F(g)\circ F(f)$.

Functors are the fundamental objects used to relate structures between different categories.

Examples

Algebraic topology was the first field in which the usefulness of the notion of a functor was recognized. A basic example is the fundamental group functor $\pi _{1}:{\textbf {Top}}*\to {\textbf {Grp}}$. The action on objects is defined by sending a topological space to its fundamental group $\pi _{1}(X)$. Recall that a map between two topological spaces $f:X\to Y$ induces a map $f_{*}:\pi _{1}(X)\to \pi (Y)$ by $f_{*}([\gamma ])=[f\circ \gamma ]$. Set $\pi _{1}(f)=f_{*}$ so defined. The functoriality of $\pi _{1}$ boils down to the fact that $F({\textrm {id}}_{X})={\textrm {id}}_{\pi _{1}(X)}$, that is, the identity map on a topological space induces the identity map on its fundamental group, together with the fact that $F(g\circ f)=F(g)\circ F(f)$, explained at fundamental group.