Introduces the category of topological spaces equipped with a presheaf (taking values in an arbitrary target category C.)
We further describe how to apply functors and natural transformations to the values of the presheaves.
A PresheafedSpace C is a topological space equipped with a presheaf of Cs.
A morphism between presheafed spaces X and Y consists of a continuous map f between the underlying topological spaces, and a (notice contravariant!) map from the presheaf on Y to the pushforward of the presheaf on X via f.
The category of PresheafedSpaces. Morphisms are pairs, a continuous map and a presheaf map from the presheaf on the target to the pushforward of the presheaf on the source.
The forgetful functor from PresheafedSpace to Top.
We can apply a functor F : C ⥤ D to the values of the presheaf in any PresheafedSpace C, giving a functor PresheafedSpace C ⥤ PresheafedSpace D