nLab contravariant functor

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 $F$ 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.

Between higher categories

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”.

Abstractly

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.

Related concepts

• presheaf

Notions of pullback:

• pullback, fiber product (limit over a cospan)

• wide pullback

• lax pullback, comma object (lax limit over a cospan)

• (∞,1)-pullback, homotopy pullback, ((∞,1)-limit over a cospan)

• base change, context extension

• pullback bundle

• contravariant functor

  • pullback in cohomology, d-invariant

  • pullback of differential forms

  • pullback of a distribution

References

Textbook accounts:

• Saunders MacLane, §II.2 of: Categories for the Working Mathematician, Graduate Texts in Mathematics 5 Springer (1971, second ed. 1997) [doi:10.1007/978-1-4757-4721-8]

• Francis Borceux, Section 1.4 in: Handbook of Categorical Algebra Vol. 1: Basic Category Theory [doi:10.1017/CBO9780511525858]