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]



  • 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 propertyEdit

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, unique up to isomorphism, 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 alsoEdit


  1. ^ co-Yoneda lemma in nLab
  2. ^ Kashiwara & Schapira 2005, Corollary 2.4.3.
  3. ^ Kashiwara & Schapira 2005, Proposition 2.7.1.
  4. ^ Lurie, Definition
  5. ^ Lurie, Proposition


  • Kashiwara, Masaki; Schapira, Pierre (2005). Categories and sheaves. Grundlehren der mathematischen Wissenschaften. 332. Springer. ISBN 978-3-540-27950-1.
  • Lurie, J. Higher Topos Theory.
  • Mac Lane, Saunders; Moerdijk, Ieke (1992). Sheaves in Geometry and Logic. Springer. ISBN 0-387-97710-4.

Further readingEdit