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| In [[category theory]], a branch of [[mathematics]], a <math>V</math>-valued '''presheaf''' <math>F</math> on a category <math>C</math> is a [[functor]] <math>F\colon C^\mathrm{op}\to\mathbf{V}</math>. Often ''presheaf'' is defined to be a '''Set'''-valued presheaf. If <math>C</math> is the [[poset]] of [[open set]]s in a [[topological space]], interpreted as a category, then one recovers the usual notion of [[presheaf (mathematics)|presheaf]] on a topological space.
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| 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 <math>\widehat{C} = \mathbf{Set}^{C^\mathrm{op}}</math>. A functor into <math>\widehat{C}</math> is sometimes called a [[profunctor]].
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| A presheaf that is naturally isomorphic to the contravariant [[hom-functor]] Hom(–,''A'') for some object ''A'' of '''C''' is called a [[representable presheaf]].
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| == Examples ==
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| * A [[simplicial set]] is a '''Set'''-valued presheaf on the [[simplex category]] <math>C=\Delta</math>.
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| == Properties ==
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| * When <math>C</math> is a [[small category]], the functor category <math>\widehat{C}=\mathbf{Set}^{C^\mathrm{op}}</math> is [[cartesian closed]].
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| * The partially ordered set of [[subobject]]s of <math>P</math> form a [[Heyting algebra]], whenever <math>P</math> is an object of <math>\widehat{C}=\mathbf{Set}^{C^\mathrm{op}}</math> for small <math>C</math>.
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| * For any morphism <math>f:X\to Y</math> of <math>\widehat{C}</math>, the pullback functor of subobjects <math>f^*:\mathrm{Sub}_{\widehat{C}}(Y)\to\mathrm{Sub}_{\widehat{C}}(X)</math> has a right adjoint, denoted <math>\forall_f</math>, and a left adjoint, <math>\exists_f</math>. These are the [[Universal quantifier#As adjoint|universal]] and existential quantifiers.
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| * A locally small category <math>C</math> embeds fully and faithfully into the category <math>\widehat{C}</math> of set-valued presheaves via the [[Yoneda embedding]] <math>\mathrm{Y}_c</math> which to every object <math>A</math> of <math>C</math> associates the hom-set <math>C(-,A)</math>.
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| * The presheaf category <math>\widehat{C}</math> is (up to equivalence of categories) the free [[colimit]] completion of the category <math>C</math>.
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| == See also ==
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| * [[Topos]]
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| * [[Category of elements]]
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| == References==
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| * {{nlab|id=presheaf|title=Presheaf}}
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| * Saunders Mac Lane, Ieke Moerdijk, "Sheaves in Geometry and Logic" (1992) Springer-Verlag ISBN 0-387-97710-4
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| [[Category:Functors]]
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| [[Category:Sheaf theory]]
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| [[Category:Topos theory]]
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