Presheaf (category theory) explained

C is a functor F\colon C ^{op\toSet} . If C 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

C into a category, and is an example of a functor category. It is often written as \widehat{C} = Set ^{C ^{op}} . A functor into \widehat{C} 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

F\colon C ^{op\toV} as a V -valued presheaf.

Examples

C = \Delta .

Properties

• When

C is a small category, the functor category \widehat{C} = Set ^{C ^{op}} is cartesian closed.

• The poset of subobjects of

P form a Heyting algebra, whenever P is an object of \widehat{C} = Set ^{C ^{op}} for small C .

• For any morphism

f:X\to Y of \widehat{C} , the pullback functor of subobjects f ^{*: Sub} _{\widehat{C} }(Y)\to\mathrm_(X) has a right adjoint, denoted \forall _{f} , and a left adjoint, \exists _{f} . These are the universal and existential quantifiers.

• A locally small category

C embeds fully and faithfully into the category \widehat{C} of set-valued presheaves via the Yoneda embedding which to every object A of C associates the hom functor C( - ,A) .

• The category

\widehat{C} 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,

\widehat{C} is the colimit completion of C (see

1. Universal property

below.)

Universal property

The construction

C \mapsto \widehat{C} = Fct (C ^{op} , Set ) 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

F = \varinjlim y U _{i} where U _{i} are objects in C. Then let \widetilde{ η } F = \varinjlim η U _{i,} which exists by assumption. Since \varinjlim - is functorial, this determines the functor \widetilde{ η }: \widehat{C} \to D . Succinctly, \widetilde{ η } is the left Kan extension of η along y; hence, the name "Yoneda extension". To see \widetilde{ η } commutes with small colimits, we show \widetilde{ η } is a left-adjoint (to some functor). Define l{H}om(η, - ): D \to \widehat{C} to be the functor given by: for each object M in D and each object U in C,l{H}om(η, M)(U) = \operatorname{Hom} _{D(η} U, M). Then, for each object M in D, since l{H}om(η, M)(U _{i)} = \operatorname{Hom}(y U _{i,} l{H}om(η, M)) by the Yoneda lemma, we have:\begin{align} \operatorname{Hom} _{D(\widetilde{η}} F, M) & = \operatorname{Hom} _{D(\varinjlim} η U _{i,} M) = \varprojlim \operatorname{Hom} _{D(η} U _{i,} M) = \varprojlim l{H}om(η, M)(U _{i)} \\ & = \operatorname{Hom} _{\widehat{C} }(F, \mathcalom(\eta, M)),\endwhich is to say \widetilde{ η } is a left-adjoint to l{H}om(η, - ) . \square

The proposition yields several corollaries. For example, the proposition implies that the construction

C \mapsto \widehat{C} is functorial: i.e., each functor C \to D determines the functor \widehat{C} \to \widehat{D} .

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:

C \to PShv(C) is fully faithful (here C can be just a simplicial set.)

See also

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

References

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

Further reading

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