nLab
contravariant functor

Contravariant functors

Idea

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.)

Between categories

A contravariant functor FF from a category CC to a category DD is simply a functor from the opposite category C opC^op to DD.

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

Equivalently, a contravariant functor from CC to DD may be thought of as a functor from CC to D opD^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][C^{op}, D] are in canonical bijection with those in the functor category [C,D op][C, D^{op}] (both are contravariant functors from CC to DD), the morphisms in the two functor categories are in general different, as

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

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

Between higher categories

Since n-categories (and also (infinity,n)-categories) have 2 n2^n different kinds of opposite category depending on which of the kk-morphisms are reversed for 1kn1\le k\le n (see for instance opposite 2-category), they also have 2 n2^n different kinds of “contravariant functor”.

Abstractly

Categories, covariant functors, and natural transformations form a 2-category Cat. To include the contravariant functors as well, we can equip CatCat 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)