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 ${\mathcal {F}}:{\mathcal {C}}(A,B)\to {\mathcal {D}}({\mathcal {F}}B,{\mathcal {F}}A)$.

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.