nLab copresheaf



Topos Theory

topos theory



Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory




A copresheaf, or covariant presheaf, on a category 𝒞\mathcal{C} is a presheaf on the opposite category C opC^{op}. As with presheaves, it is understood by default that they take values in Set, but one may consider copresheaves with values in any category 𝒟\mathcal{D}.

This is a concept with an attitude: a copresheaf on 𝒞\mathcal{C} is just a functor 𝒞𝒟\mathcal{C} \to \mathcal{D} (typically: 𝒞Set\mathcal{C} \to Set, but one may speak of functors as copresheaves if eventually one wants to impose a gluing condition on them and pass to cosheaves.

Accordingly, the category of copresheaves on 𝒞\mathcal{C} (and with values in 𝒟\mathcal{D}) is just the functor category Func(𝒞,𝒟)Func(\mathcal{C},\, \mathcal{D}), whose morphisms are the natural transformations.

For 𝒟=Set\mathcal{D} = Set, the opposite of the category of copresheaves on 𝒞\mathcal{C} may be understood as the free completion of 𝒞\mathcal{C}.

Last revised on November 29, 2023 at 11:55:10. See the history of this page for a list of all contributions to it.