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 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 partially ordered set 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 $\mathrm{Y}_c$ which to every object $A$ of $C$ associates the hom functor $C(-,A)$.
• The presheaf category $\widehat{C}$ is (up to equivalence of categories) the free colimit completion of the category $C$.

See also

• Topos
• Category of elements

References

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