Presheaf (category theory)
In category theory, a branch of mathematics, a presheaf on a category is a functor . If 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 . A functor into 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 as a -valued presheaf.
Examples
- A simplicial set is a Set-valued presheaf on the simplex category .
Properties
- When is a small category, the functor category is cartesian closed.
- The partially ordered set of subobjects of form a Heyting algebra, whenever is an object of for small .
- For any morphism of , the pullback functor of subobjects has a right adjoint, denoted , and a left adjoint, . These are the universal and existential quantifiers.
- A locally small category embeds fully and faithfully into the category of set-valued presheaves via the Yoneda embedding which to every object of associates the hom functor .
- The presheaf category is (up to equivalence of categories) the free colimit completion of the category .
See also
References
- Presheaf in nLab
- Saunders Mac Lane, Ieke Moerdijk, "Sheaves in Geometry and Logic" (1992) Springer-Verlag ISBN 0-387-97710-4
This article is issued from Wikipedia - version of the 1/30/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.