Contravariant functor - Imported Knowlpedia Content

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

A contravariant functor $F:\mathcal C\to\mathcal D$ consists of:

an assignment on objects $X\mapsto F(X)$ ,
for every morphism $f:X\to Y$ in $\mathcal C$ , a morphism $F(f):F(Y)\to F(X)$ in $\mathcal D$ ,

such that:

(Identities) $F(\mathrm{id}_X)=\mathrm{id}_{F(X)}$ for every $X$ (see identity morphism),
(Reversed composition) for composable $X\xrightarrow{f}Y\xrightarrow{g}Z$ , $F(g\circ f)=F(f)\circ F(g)$ (see composition).

Equivalent characterizations

Equivalently, a contravariant functor $F:\mathcal C\to\mathcal D$ is the same thing as an ordinary (covariant) functor

\mathcal C^{\mathrm{op}}\longrightarrow \mathcal D,

where $\mathcal C^{\mathrm{op}}$ is the opposite category of $\mathcal C$ .

Examples

Inverse-image on power sets (Set). Let $\mathbf{Set}$ be the category of sets and functions. Define $F:\mathbf{Set}^{\mathrm{op}}\to \mathbf{Set}$ by

$F(X)=\mathcal P(X)$ (the set of subsets of $X$ ),
for $f:X\to Y$ , $F(f)=f^{-1}:\mathcal P(Y)\to\mathcal P(X)$ , the preimage map. Then $f^{-1}(\mathrm{id})=\mathrm{id}$ and $(g\circ f)^{-1}=f^{-1}\circ g^{-1}$ , so this is contravariant.

Representable hom-functor $\mathrm{Hom}_{\mathcal C}(-,A)$ . Fix an object $A\in\mathcal C$ . The assignment $X\longmapsto \mathrm{Hom}_{\mathcal C}(X,A)$ extends to a contravariant functor $\mathcal C\to\mathbf{Set}$ (equivalently $\mathcal C^{\mathrm{op}}\to\mathbf{Set}$ ) by sending $f:X\to Y$ to precomposition: $\mathrm{Hom}_{\mathcal C}(Y,A)\to \mathrm{Hom}_{\mathcal C}(X,A),\quad (h:Y\to A)\mapsto h\circ f.$ This is a basic instance of a representable functor.

Linear dual (Vect $_k$ or $R$ -Mod). In the category of vector spaces over a field $k$ (or modules over a ring), define $V\mapsto V^\ast=\mathrm{Hom}_k(V,k).$ A linear map $f:V\to W$ induces $f^\ast:W^\ast\to V^\ast$ by $f^\ast(\varphi)=\varphi\circ f$ . The direction reverses, and $(g\circ f)^\ast=f^\ast\circ g^\ast$ , so $(-)^\ast$ is contravariant.