contravariant functor

A contravariant functor is like a functor but it reverses the directions of the morphisms. (Between groupoids, contravariant functors are essentially the same as functors.)

A **contravariant functor** $F$ from a category $C$ to a category $D$ is simply a functor from the opposite category $C^op$ to $D$.

To emphasize that one means a functor $C \to D$ as stated and *not* as a functor $C^{op} \to D$ one sometimes says **covariant functor** when referring to non-contravariant functors, for emphasis.

Equivalently, a contravariant functor from $C$ to $D$ may be thought of as a functor from $C$ to $D^op$, but the version above generalises better to functors of many variables.

Also notice that while the objects of the functor category $[C^{op}, D]$ are in canonical bijection with those in the functor category $[C, D^{op}]$ (both are contravariant functors from $C$ to $D$), the morphisms in the two functor categories are in general different, as

$[C^{op}, D] \simeq [C, D^{op}]^{op}
\,.$

This matters when discussing a natural transformation from one contravariant functor to another.

Since n-categories (and also (infinity,n)-categories) have $2^n$ different kinds of opposite category depending on which of the $k$-morphisms are reversed for $1\le k\le n$ (see for instance opposite 2-category), they also have $2^n$ different kinds of “contravariant functor”.

Categories, covariant functors, and natural transformations form a 2-category Cat. To include the contravariant functors as well, we can equip $Cat$ with a duality involution, or we can generalize to a 2-category with contravariance, or some more general structure that also includes extranatural transformations or dinatural transformations. There could also be higher-categorical versions, such as a 3-category with contravariance.

Revised on June 17, 2016 13:36:09
by Mike Shulman
(192.195.154.171)