A copresheaf, or covariant presheaf, on a category is a presheaf on the opposite category . As with presheaves, it is understood by default that they take values in Set, but one may consider copresheaves with values in any category .
This is a concept with an attitude: a copresheaf on is just a functor (typically: , 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 (and with values in ) is just the functor category , whose morphisms are the natural transformations.
For , the opposite of the category of copresheaves on may be understood as the free completion of .
Last revised on November 29, 2023 at 11:55:10. See the history of this page for a list of all contributions to it.