Definition 1.

Let T be a categoryMathworldPlanetmath, and let C be another category. Then a presheafPlanetmathPlanetmathPlanetmath on T is a contravariant functorMathworldPlanetmath F:TC. If U𝑖V is a morphismMathworldPlanetmath in T, we call F(i):F(V)F(U) the restrictionPlanetmathPlanetmathPlanetmath map obtained from i. A morphism of presheaves is a natural transformation.

To understand this definition, and what it has to do with the more familiar definition of a presheaf (, let T be a site. Then a presheaf maps the “open sets” to objects in C and provides restriction maps whenever the underlying “open sets” have an “inclusion”. Of course, on a site, there may be many non-isomorphic inclusions.

In practice, to avoid set-theoretic difficulties, one usually fixes a universePlanetmathPlanetmath 𝒰 and then requires T to be a small ( category.

Remark 1.

If T is a small category and C is a U-category, then presheaves on T form a category.

If T is not small, that is if the isomorphism classes of objects do not form a U-set, then (normally) there are as many functors as subsets of T, which leads to the resulting category failing to be a U-category. If one removes the various qualifications in terms of universes, this means that there are too many functors: Russell’s paradoxMathworldPlanetmath arises. If one uses universes, and qualifies everything in sight appropriately, a problem remains: there are too many functors from sets to sets to be allowed. While your answer is described in terms of sets, they are too big to be inside the universe of discourse. This problem underlies the set-theoretic problems with direct limitsMathworldPlanetmath.

Let T be a site, and suppose either C is an abelian categoryMathworldPlanetmathPlanetmathPlanetmath or C is the category of sets.

Definition 2.

Let F:TC be a presheaf. Then F is a sheaf if for every covering UiU in T, the productsMathworldPlanetmathPlanetmathPlanetmathPlanetmath UiF(Ui) and Ui×UUjF(Ui×UUj) exist, and the sequencePlanetmathPlanetmath


is exact.

This “exactness” really needs explaining. First of all, for every i, we have a morphism F(U)F(Ui) obtained from the covering. The map on the left (call it f) is the product of all these maps. Second, for each pair i and j, we have a morphism g1:F(Ui)F(Ui×UUj) and a morphism g2:F(Uj)F(Ui×UUj) obtained from the two projection maps of the fibred product. The two maps on the right-hand side are obtained as products of all the g1 and g2 respectively.

If C is the category of sets, then the exactness condition requires that the left-hand map is injectivePlanetmathPlanetmath and that the two right-hand maps agree exactly on its image.

If C is an abelian category, then the kernel of the left-hand map should be zero, and the kernel of the difference of the two right-hand maps should be its image. Perhaps this would be clearer if the diagram added a “0” on the left, but this usage is very standard.

This definition of a sheaf is the motivation for the definition of a site: a site captures precisely those features of a topological spaceMathworldPlanetmath that are required to form sheaves. The category of sheaves on a topological space is an example of a topos; the original definition of a topos was “a category equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to the category of sheaves on a site”, although the concept was generalized in later work. The motivation was that while a site may be badly behaved and pose set-theoretic difficulties, the topos associated to it is generally better behaved.

The category of sheaves on a site has a long list of nice properties; among them is “has enough injectives”, so that derived functorsMathworldPlanetmath can be extracted and cohomology calculated. This is how, for example, étale cohomology is obtained.


  • 1 Grothendieck et al., Séminaires en Gèometrie Algèbrique 4, tomes 1, 2, and 3, available on the web at archibal/SGA/SGA.html archibal/SGA/SGA.html
Title sheaf
Canonical name Sheaf1
Date of creation 2013-03-22 14:12:47
Last modified on 2013-03-22 14:12:47
Owner archibal (4430)
Last modified by archibal (4430)
Numerical id 9
Author archibal (4430)
Entry type Definition
Classification msc 14F20
Classification msc 18F10
Classification msc 18F20
Synonym sheaves
Related topic Sheaf
Related topic Site
Related topic Sheafification2
Related topic NaturalTransformation
Related topic FundamentalGroupoid2
Defines presheaf
Defines presheaves