contravariant functor (changes) in nLab

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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 C→DC \to D as stated and not as a functor C op→DC^{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 1≤k≤n1\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.

• 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

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

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](https://link.springer.com/book/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](https://doi.org/10.1017/CBO9780511525858)]