Pacific halibut

In category theory, a branch of mathematics, a $V$-valued presheaf $F$ on a category $C$ is a functor $F:C^{\mathrm{o}\mathrm{p}}\to 𝐕$. Often presheaf is defined to be a Set-valued presheaf. 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 into a category, and is an example of a functor category. It is often written as $\hat{C}={𝐒𝐞𝐭}^{C^{\mathrm{o}\mathrm{p}}}$. A functor into $\hat{C}$ 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.

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 $\hat{C}={𝐒𝐞𝐭}^{C^{\mathrm{o}\mathrm{p}}}$ is cartesian closed.
• The partially ordered set of subobjects of $P$ form a Heyting algebra, whenever $P$ is an object of $\hat{C}={𝐒𝐞𝐭}^{C^{\mathrm{o}\mathrm{p}}}$ for small $C$.
• For any morphism $f:X\to Y$ of $\hat{C}$, the pullback functor of subobjects $f^{\ast }:{\mathrm{S}\mathrm{u}\mathrm{b}}_{\hat{C}}(Y)\to {\mathrm{S}\mathrm{u}\mathrm{b}}_{\hat{C}}(X)$ has a right adjoint, denoted ${\mathrm{\forall }}_f$, and a left adjoint, ${\mathrm{\exists }}_f$. These are the universal and existential quantifiers.
• A locally small category $C$ embeds fully and faithfully into the category $\hat{C}$ of set-valued presheaves via the Yoneda embedding ${\mathrm{Y}}_c$ which to every object $A$ of $C$ associates the hom-set $C(-,A)$.
• The presheaf category $\hat{C}$ is (up to equivalence of categories) the free colimit completion of the category $C$.

See also

• Topos
• Category of elements

References

• Template:Nlab
• Saunders Mac Lane, Ieke Moerdijk, "Sheaves in Geometry and Logic" (1992) Springer-Verlag ISBN 0-387-97710-4