Functor - Imported Knowlpedia Content

Let $\mathcal C,\mathcal D$ be categories.

A (covariant) functor $F:\mathcal C\to\mathcal D$ consists of:

an assignment on objects: for each $A\in\mathcal C$ , an object $F(A)\in\mathcal D$ ;
an assignment on morphisms: for each morphism $f:A\to B$ in $\mathcal C$ , a morphism $F(f):F(A)\to F(B)$ in $\mathcal D$ ;

such that the following axioms hold:

(identity preservation) for every object $A$ , $F(\mathrm{id}_A)=\mathrm{id}_{F(A)},$ where $\mathrm{id}_A$ is the identity morphism;
(composition preservation) for every composable pair $A\xrightarrow{f}B\xrightarrow{g}C$ in $\mathcal C$ , $F(g\circ f)=F(g)\circ F(f),$ where $\circ$ denotes composition in the relevant category.

Variants

A contravariant functor $\mathcal C\to\mathcal D$ is the same thing as a covariant functor $\mathcal C^{\mathrm{op}}\to\mathcal D$ , where $\mathcal C^{\mathrm{op}}$ is the opposite category. Functors can be compared by natural transformations.

Examples

Forgetful functor $\mathbf{Grp}\to\mathbf{Set}$ : Send a group $G$ to its underlying set $U(G)$ , and a group homomorphism $\varphi:G\to H$ to the underlying function $U(\varphi):U(G)\to U(H)$ . This preserves identity maps and composition, hence is a functor.

Free abelian group functor $\mathbf{Set}\to\mathbf{Ab}$ : Send a set $X$ to the free abelian group $\mathbb Z[X]$ generated by $X$ , and a function $f:X\to Y$ to the induced homomorphism $\mathbb Z[X]\to \mathbb Z[Y]$ . This is a functor (and is left adjoint to the forgetful functor $\mathbf{Ab}\to\mathbf{Set}$ ; see adjoint functors).

Representable hom-functor: Fix an object $A\in\mathcal C$ . The assignment $X\mapsto \mathrm{Hom}_{\mathcal C}(A,X)$ defines a functor $\mathcal C\to\mathbf{Set}$ , called a representable functor (covariant in $X$ ). Dually, $X\mapsto \mathrm{Hom}_{\mathcal C}(X,A)$ is a functor $\mathcal C^{\mathrm{op}}\to\mathbf{Set}$ .