A presheaf of sets on a Category is a Contravariant Functor . It can map onto other categories, e.g. , , etc.

Presheaves which also satisfy gluing and separability axioms are called sheaves.

Presheaf of sets on a topological space

The usual example is a presheaf of sets on a Topological Space

for which we assume is a subcategory of : , and for which

where is a “restriction map” — merely a function between those sets, given by .

Example: The functor of continuous functions (over )

Given a topological space, and the set of continuous functions . Then we can see that The functor of continuous functions over a topological space is a (pre)sheaf:

can be seen as a functor. Given , we can have simply

It necessarily preserves composition of morphisms as well: Given , we’ll have

which is the same as simply , since : is already “more restrictive” than .

Therefore, the functor of continuous functions (on ) over (open sets of) is a presheaf.