Presheaf (category theory)

In category theory, a branch of mathematics, a presheaf on a category is a functor . If is the poset of open sets in a topological space, interpreted as a category, then one recovers the usual notion of presheaf on a topological space.

A morphism of presheaves is defined to be a natural transformation of functors. This makes the collection of all presheaves on into a category, and is an example of a functor category. It is often written as and it is called the category of presheaves on . A functor into is sometimes called a profunctor.

A presheaf that is naturally isomorphic to the contravariant hom-functor Hom(–, A) for some object A of C is called a representable presheaf.

Some authors refer to a functor as a -valued presheaf.

Examples

• A simplicial set is a Set-valued presheaf on the simplex category .
• A directed multigraph is a presheaf on the category with two objects and two parallel morphisms between them i.e. .
• An arrow category is a presheaf on the category with two objects and one morphism between them. i.e. .
• A right group action is a presheaf on the category created from a group , i.e. a category with one object and invertible morphisms.

Properties

• When is a small category, the functor category is cartesian closed.
• The poset of subobjects of form a Heyting algebra, whenever is an object of for small .
• For any morphism of , the pullback functor of subobjects has a right adjoint, denoted , and a left adjoint, . These are the universal and existential quantifiers.
• A locally small category embeds fully and faithfully into the category of set-valued presheaves via the Yoneda embedding which to every object of associates the hom functor .
• The category admits small limits and small colimits. See limit and colimit of presheaves for further discussion.
• The density theorem states that every presheaf is a colimit of representable presheaves; in fact, is the colimit completion of (see #Universal property below.)

Universal property

The construction is called the colimit completion of C because of the following universal property:

Proof: Given a presheaf F, by the density theorem, we can write where are objects in C. Then let which exists by assumption. Since is functorial, this determines the functor . Succinctly, is the left Kan extension of along y; hence, the name "Yoneda extension". To see commutes with small colimits, we show is a left-adjoint (to some functor). Define to be the functor given by: for each object M in D and each object U in C,

Then, for each object M in D, since by the Yoneda lemma, we have:

which is to say is a left-adjoint to .

The proposition yields several corollaries. For example, the proposition implies that the construction is functorial: i.e., each functor determines the functor .

Variants

A presheaf of spaces on an ∞-category C is a contravariant functor from C to the ∞-category of spaces (for example, the nerve of the category of CW-complexes.) It is an ∞-category version of a presheaf of sets, as a "set" is replaced by a "space". The notion is used, among other things, in the ∞-category formulation of Yoneda's lemma that says: is fully faithful (here C can be just a simplicial set.)

A copresheaf of a category C is a presheaf of C^op. In other words, it is a covariant functor from C to Set.

See also

• Topos
• Category of elements
• Simplicial presheaf (this notion is obtained by replacing "set" with "simplicial set")
• Presheaf with transfers

Notes

References

Further reading

• Daniel Dugger, Sheaves and Homotopy Theory, the pdf file provided by nlab.