Presheaf (category theory)

In category theory, a branch of mathematics, a presheaf on a category $$C$$ is a functor $$F\colon C^\mathrm{op}\to\mathbf{Set}$$. 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 $$\widehat{C} = \mathbf{Set}^{C^\mathrm{op}}$$. 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 $$F\colon C^\mathrm{op}\to\mathbf{V}$$ as a $$\mathbf{V}$$-valued presheaf.

Examples

 * A simplicial set is a Set-valued presheaf on the simplex category $$C=\Delta$$.

Properties

 * When $$C$$ is a small category, the functor category $$\widehat{C}=\mathbf{Set}^{C^\mathrm{op}}$$ is cartesian closed.
 * The poset of subobjects of $$P$$ form a Heyting algebra, whenever $$P$$ is an object of $$\widehat{C}=\mathbf{Set}^{C^\mathrm{op}}$$ for small $$C$$.
 * For any morphism $$f:X\to Y$$ of $$\widehat{C}$$, the pullback functor of subobjects $$f^*:\mathrm{Sub}_{\widehat{C}}(Y)\to\mathrm{Sub}_{\widehat{C}}(X)$$ 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 below.)

Universal property
The construction $$C \mapsto \widehat{C} = \mathbf{Fct}(C^{\text{op}}, \mathbf{Set})$$ 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 $$\widetilde{\eta} F = \varinjlim \eta U_i,$$ which exists by assumption. Since $$\varinjlim -$$ is functorial, this determines the functor $$\widetilde{\eta}: \widehat{C} \to D$$. Succinctly, $$\widetilde{\eta}$$ is the left Kan extension of $$\eta$$ along y; hence, the name "Yoneda extension". To see $$\widetilde{\eta}$$ commutes with small colimits, we show $$\widetilde{\eta}$$ is a left-adjoint (to some functor). Define $$\mathcal{H}om(\eta, -): D \to \widehat{C}$$ to be the functor given by: for each object M in D and each object U in C,
 * $$\mathcal{H}om(\eta, M)(U) = \operatorname{Hom}_D(\eta U, M).$$

Then, for each object M in D, since $$\mathcal{H}om(\eta, M)(U_i) = \operatorname{Hom}(y U_i, \mathcal{H}om(\eta, M))$$ by the Yoneda lemma, we have:
 * $$\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}$$ which is to say $$\widetilde{\eta}$$ is a left-adjoint to $$\mathcal{H}om(\eta, -)$$. $$\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.)