Harman Patil (Editor)

Presheaf (category theory)

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In category theory, a branch of mathematics, a presheaf on a category C is a functor F : C o p → S e t . 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.

Contents

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 C ^ = S e t C o p . A functor into 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 : C o p → V as a V -valued presheaf.

Examples

  • A simplicial set is a Set-valued presheaf on the simplex category C = Δ .

    • Properties

  • When C is a small category, the functor category C ^ = S e t C o p is cartesian closed.
  • The partially ordered set of subobjects of P form a Heyting algebra, whenever P is an object of C ^ = S e t C o p for small C .
  • For any morphism f : X → Y of C ^ , the pullback functor of subobjects f ∗ : S u b C ^ ( Y ) → S u b C ^ ( X ) has a right adjoint, denoted ∀ f , and a left adjoint, ∃ f . These are the universal and existential quantifiers.
  • A locally small category C embeds fully and faithfully into the category C ^ of set-valued presheaves via the Yoneda embedding Y c which to every object A of C associates the hom functor C ( − , A ) .
  • The presheaf category C ^ is (up to equivalence of categories) the free colimit completion of the category C .

    • References

    Presheaf (category theory) Wikipedia
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