Presheaf Category
Get Presheaf Category essential facts below. View Videos or join the Presheaf Category discussion. Add Presheaf Category to your topic list for future reference or share this resource on social media.
Presheaf Category

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]



  • 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 .


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


  1. ^ co-Yoneda lemma in nLab
  2. ^ Kashiwara-Schapira, Corollary 2.4.3.
  3. ^ Kashiwara-Schapira, Proposition 2.7.1.
  4. ^ Lurie, Definition
  5. ^ Lurie, Proposition


  • 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

  This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.



Music Scenes