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 $\mathcal{F}:\mathrm{O}\mathrm{b}\mathcal{C}\to \mathrm{O}\mathrm{b}\mathcal{D}$.
• For every $A,B\in \mathrm{O}\mathrm{b}\mathcal{C}$, 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 $A\in \mathrm{O}\mathrm{b}\mathcal{C}$, $\mathcal{F}({\mathrm{i}}_{\mathrm{d}})={\mathrm{i}}_{\mathrm{d}}$.
• It preserves composition, albeit reversing the order of composition: For any $A,B,C\in \mathrm{O}\mathrm{b}\mathcal{C}$, 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.