Contravariant functor

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

Definition

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:

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

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

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

Examples

1. 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.

2. 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 .

3. 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.