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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi_1(f) = f_*} so defined. The functoriality of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi_1} boils down to the fact that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(g \circ f) = F(g) \circ F(f)} , explained at fundamental group.