Presheaf (category theory)

Short description: Functor from a category's opposite category to Set

In category theory, a branch of mathematics, a presheaf on a category [math]\displaystyle{ C }[/math] is a functor [math]\displaystyle{ F\colon C^\mathrm{op}\to\mathbf{Set} }[/math]. If [math]\displaystyle{ C }[/math] 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 [math]\displaystyle{ C }[/math] into a category, and is an example of a functor category. It is often written as [math]\displaystyle{ \widehat{C} = \mathbf{Set}^{C^\mathrm{op}} }[/math]. A functor into [math]\displaystyle{ \widehat{C} }[/math] 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 [math]\displaystyle{ F\colon C^\mathrm{op}\to\mathbf{V} }[/math] as a [math]\displaystyle{ \mathbf{V} }[/math]-valued presheaf.

Examples

• A simplicial set is a Set-valued presheaf on the simplex category [math]\displaystyle{ C=\Delta }[/math].

Properties

• When [math]\displaystyle{ C }[/math] is a small category, the functor category [math]\displaystyle{ \widehat{C}=\mathbf{Set}^{C^\mathrm{op}} }[/math] is cartesian closed.
• The poset of subobjects of [math]\displaystyle{ P }[/math] form a Heyting algebra, whenever [math]\displaystyle{ P }[/math] is an object of [math]\displaystyle{ \widehat{C}=\mathbf{Set}^{C^\mathrm{op}} }[/math] for small [math]\displaystyle{ C }[/math].
• For any morphism [math]\displaystyle{ f:X\to Y }[/math] of [math]\displaystyle{ \widehat{C} }[/math], the pullback functor of subobjects [math]\displaystyle{ f^*:\mathrm{Sub}_{\widehat{C}}(Y)\to\mathrm{Sub}_{\widehat{C}}(X) }[/math] has a right adjoint, denoted [math]\displaystyle{ \forall_f }[/math], and a left adjoint, [math]\displaystyle{ \exists_f }[/math]. These are the universal and existential quantifiers.
• A locally small category [math]\displaystyle{ C }[/math] embeds fully and faithfully into the category [math]\displaystyle{ \widehat{C} }[/math] of set-valued presheaves via the Yoneda embedding which to every object [math]\displaystyle{ A }[/math] of [math]\displaystyle{ C }[/math] associates the hom functor [math]\displaystyle{ C(-,A) }[/math].
• The category [math]\displaystyle{ \widehat{C} }[/math] 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, [math]\displaystyle{ \widehat{C} }[/math] is the colimit completion of [math]\displaystyle{ C }[/math] (see #Universal property below.)

Universal property

The construction [math]\displaystyle{ C \mapsto \widehat{C} = \mathbf{Fct}(C^{\text{op}}, \mathbf{Set}) }[/math] is called the colimit completion of C because of the following universal property:

Proposition — Let C, D be categories and assume D admits small colimits. Then each functor [math]\displaystyle{ \eta: C \to D }[/math] factorizes as

[math]\displaystyle{ C \overset{y}\longrightarrow \widehat{C} \overset{\widetilde{\eta}}\longrightarrow D }[/math]

where y is the Yoneda embedding and [math]\displaystyle{ \widetilde{\eta}: \widehat{C} \to D }[/math] is a, unique up to isomorphism, colimit-preserving functor called the Yoneda extension of [math]\displaystyle{ \eta }[/math].

Proof: Given a presheaf F, by the density theorem, we can write [math]\displaystyle{ F =\varinjlim y U_i }[/math] where [math]\displaystyle{ U_i }[/math] are objects in C. Then let [math]\displaystyle{ \widetilde{\eta} F = \varinjlim \eta U_i, }[/math] which exists by assumption. Since [math]\displaystyle{ \varinjlim - }[/math] is functorial, this determines the functor [math]\displaystyle{ \widetilde{\eta}: \widehat{C} \to D }[/math]. Succinctly, [math]\displaystyle{ \widetilde{\eta} }[/math] is the left Kan extension of [math]\displaystyle{ \eta }[/math] along y; hence, the name "Yoneda extension". To see [math]\displaystyle{ \widetilde{\eta} }[/math] commutes with small colimits, we show [math]\displaystyle{ \widetilde{\eta} }[/math] is a left-adjoint (to some functor). Define [math]\displaystyle{ \mathcal{H}om(\eta, -): D \to \widehat{C} }[/math] to be the functor given by: for each object M in D and each object U in C,

[math]\displaystyle{ \mathcal{H}om(\eta, M)(U) = \operatorname{Hom}_D(\eta U, M). }[/math]

Then, for each object M in D, since [math]\displaystyle{ \mathcal{H}om(\eta, M)(U_i) = \operatorname{Hom}(y U_i, \mathcal{H}om(\eta, M)) }[/math] by the Yoneda lemma, we have:

[math]\displaystyle{ \begin{align} \operatorname{Hom}_D(\widetilde{\eta} F, M) &= \operatorname{Hom}_D(\varinjlim \eta U_i, M) = \varprojlim \operatorname{Hom}_D(\eta U_i, M) = \varprojlim \mathcal{H}om(\eta, M)(U_i) \\ &= \operatorname{Hom}_{\widehat{C}}(F, \mathcal{H}om(\eta, M)), \end{align} }[/math]

which is to say [math]\displaystyle{ \widetilde{\eta} }[/math] is a left-adjoint to [math]\displaystyle{ \mathcal{H}om(\eta, -) }[/math]. [math]\displaystyle{ \square }[/math]

The proposition yields several corollaries. For example, the proposition implies that the construction [math]\displaystyle{ C \mapsto \widehat{C} }[/math] is functorial: i.e., each functor [math]\displaystyle{ C \to D }[/math] determines the functor [math]\displaystyle{ \widehat{C} \to \widehat{D} }[/math].

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: [math]\displaystyle{ C \to PShv(C) }[/math] 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

• {{Cite book

| last1=Kashiwara | first1=Masaki | last2=Schapira | first2=Pierre | title=Categories and sheaves | year=2005 |url=https://books.google.com/books?id=mc5DAAAAQBAJ |publisher=Springer |isbn=978-3-540-27950-1 |volume=332 |series=Grundlehren der mathematischen Wissenschaften

• Lurie, J.. Higher Topos Theory. 
• Mac Lane, Saunders; Moerdijk, Ieke (1992). Sheaves in Geometry and Logic. Springer. 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.