# 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 . 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**.[1]

## Examples

- A simplicial set is a
**Set**-valued presheaf on the simplex category .

## Properties

- When is a small category, the functor category is cartesian closed.
- The partially ordered set 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.[2]. 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:

**Proposition[3]** — Let *C*, *D* be categories and assume *D* admits small colimits. Then each functor factorizes as

where *y* is the Yoneda embedding and is a colimit-preserving functor called the **Yoneda extension** of .

*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.)[4] 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.)[5]

## See also

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

## Notes

- co-Yoneda lemma in
*nLab* - Kashiwara–Schapira, Corollary 2.4.3.
- Kashiwara–Schapira, Proposition 2.7.1.
- Lurie, Definition 1.2.16.1.
- Lurie, Proposition 5.1.3.1.

## References

- Kashiwara, Masaki; Schapira, Pierre (2006).
*Categories and sheaves*. - Lurie, J.
*Higher Topos Theory* - Saunders Mac Lane, Ieke Moerdijk, "Sheaves in Geometry and Logic" (1992) Springer-Verlag ISBN 0-387-97710-4

## Further reading

- Presheaf in
*nLab* - Free cocompletion in
*nLab* - Daniel Dugger, Sheaves and Homotopy Theory, the pdf file provided by nlab.