Presheaf (category theory)

Examples

• A simplicial set is a Set-valued presheaf on the simplex category C=\Delta.
• A directed multigraph is a presheaf on the category with two elements and two parallel morphisms between them i.e. C = (E \overset{s}{\underset{t}{\longrightarrow}} V).
• An arrow category is a presheaf on the category with two elements and one morphism between them. i.e. C = (E \overset{f}{\longrightarrow} V).
• A right group action is a presheaf on the category created from a group G, i.e. a category with one element and invertible morphisms.

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 #Universal property 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:

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

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

Main article: Presheaf on an ∞-category

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 \widehat{C} is fully faithful (here C can be just a simplicial set.)

A copresheaf of a category C is a presheaf of Cop. In other words, it is a covariant functor from C to Set.

Notes

References

• {{Cite book |author-link = Masaki Kashiwara|author-link2 = Pierre Schapira (mathematician)}}

References

1. "co-Yoneda lemma".
2. {{harvnb. Kashiwara. Schapira. 2005
3. {{harvnb. Kashiwara. Schapira. 2005
4. {{harvnb. Lurie
5. {{harvnb. Lurie
6. "copresheaf".