Contravariant functor

Definition

Definition in basic terms

Suppose ${\mathcal {C}},{\mathcal {D}}$ are categories. A contravariant functor ${\mathcal {F}}:{\mathcal {C}}\to {\mathcal {D}}$ is defined by the following data:

• A mapping ${\mathcal {F}}:\operatorname {Ob} {\mathcal {C}}\to \operatorname {Ob} {\mathcal {D}}$.
• For every $A,B\in \operatorname {Ob} {\mathcal {C}}$, a mapping ${\mathcal {F}}:{\mathcal {C}}(A,B)\to {\mathcal {D}}({\mathcal {F}}B,{\mathcal {F}}A)$.

satisfying the following conditions:

• It preserves the identity map: For any $A\in \operatorname {Ob} {\mathcal {C}}$, ${\mathcal {F}}(\operatorname {id} _{A})=\operatorname {id} _{{\mathcal {F}}A}$.
• It preserves composition, albeit reversing the order of composition: For any $A,B,C\in \operatorname {Ob} {\mathcal {C}}$, and $f\in {\mathcal {C}}(A,B),g\in {\mathcal {C}}(B,C)$, we have ${\mathcal {F}}(g\circ f)={\mathcal {F}}f\circ {\mathcal {F}}g$.

Definition in terms of the opposite category

Suppose ${\mathcal {C}},{\mathcal {D}}$ are categories. A contravariant functor from ${\mathcal {C}}$ to ${\mathcal {D}}$ can be defined as:

• A functor from ${\mathcal {C}}^{op}$ (the opposite category to ${\mathcal {C}}$) to ${\mathcal {D}}$.
• A functor from ${\mathcal {C}}$ to ${\mathcal {D}}^{op}$ (the opposite category to ${\mathcal {D}}$).

A functor in the usual sense of the word is sometimes termed a covariant functor to contrast it with a contravariant functor.