Functor

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

Definition

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

1. an assignment on objects : for each $A\in\mathcal C$, an object $F(A)\in\mathcal D$;
2. 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.

Related notions

• 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

1. 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.

2. 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 ).

3. 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 $\mathcal C^{\mathrm{op}}\to\mathbf{Set}$.