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sheaf (Definition)
Definition 1   Let $ T$ be a category, and let $ C$ be another category. Then a presheaf on $ T$ is a contravariant functor $ F:T\to C$. If $ U\overset{i}{\to} V$ is a morphism in $ T$, we call $ F(i):F(V)\to F(U)$ the restriction 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 universe $ \mathcal{U}$ and then requires $ T$ to be a small category.

Remark 1   If $ T$ is a small category and $ C$ is a $ \mathcal{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 $ \mathcal{U}$-set, then (normally) there are as many functors as subsets of $ T$, which leads to the resulting category failing to be a $ \mathcal{U}$-category. If one removes the various qualifications in terms of universes, this means that there are too many functors: Russell's paradox 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 limits.

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

Definition 2   Let $ F\colon T\to C$ be a presheaf. Then $ F$ is a sheaf if for every covering $ {U_i\to U}$ in $ T$, the products $ \prod_{U_i} F(U_i)$ and $ \prod_{U_i \times_U U_j} F(U_i \times_U U_j)$ exist, and the sequence
$\displaystyle F(U)\to \prod_{U_i} F(U_i) \rightrightarrows \prod_{U_i \times_U U_j} F(U_i \times_U U_j) $
is exact.

This “exactness” really needs explaining. First of all, for every $ i$, we have a morphism $ F(U)\to F(U_i)$ 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 $ g_1\colon F(U_i)\to F(U_i\times_U U_j)$ and a morphism $ g_2\colon F(U_j)\to F(U_i\times_U U_j)$ 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 $ g_1$ and $ g_2$ respectively.

If $ C$ is the category of sets, then the exactness condition requires that the left-hand map is injective 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\to$” 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 space 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 equivalent 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 functors can be extracted and cohomology calculated. This is how, for example, étale cohomology is obtained.

Bibliography

1
Grothendieck et al., Séminaires en Gèometrie Algèbrique 4, tomes 1, 2, and 3, available on the web at http://www.math.mcgill.ca/˜archibal/SGA/SGA.html



"sheaf" is owned by archibal.
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See Also: sheaf, site, sheafification, natural transformation

Other names:  sheaves
Also defines:  presheaf, presheaves

Attachments:
sheafification (Theorem) by archibal
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Cross-references: étale, cohomology, derived functors, properties, topos, topological space, difference, kernel, image, injective, side, fibred product, projection maps, sequence, products, covering, category of sets, abelian category, direct limits, universe of discourse, Russell's paradox, terms, subsets, classes, isomorphism, small category, universe, inclusions, objects, site, natural transformation, map, restriction, morphism, contravariant functor, category
There are 47 references to this entry.

This is version 6 of sheaf, born on 2004-02-28, modified 2004-03-05.
Object id is 5648, canonical name is Sheaf2.
Accessed 8526 times total.

Classification:
AMS MSC: 14F20 (Algebraic geometry :: homology theory :: Étale and other Grothendieck topologies and cohomologies)
 18F10 (Category theory; homological algebra :: Categories and geometry :: Grothendieck topologies)
 18F20 (Category theory; homological algebra :: Categories and geometry :: Presheaves and sheaves)
Pending Errata and Addenda
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Discussion
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forum policy
Duplicate entries by archibal on 2004-02-28 05:41:11
First of all, I like the "this may be a duplicate entry" warning. Very handy.

I would like to suggest that an entry defining a term should automatically (and non-optionally) get "Related" links to every other entry defining the same term.

Such links should exist so that users who accidentally wind up at the wrong one can readily look up the others to see if they help ("So this is the kernel of a group homomorphism, but I wanted of a linear transformation! Don't give me this categorical nonsense either!" or "Is there a more general definition of this term?").

They should be automatic because on the one hand, it's a collossal pain (specifically, an O(n^2) process) to pester all the authors of a concept to link to each other. On the other hand, if we have "competing" definitions due to excessive rancor, they probably won't want to link to each other --- but readers will definitely want both points of view.

The effort required to do this should be minimal: when displaying an entry, look up each term that it defines in the index, and make sure each definition appears in the related list. (This could be done on modifications only, but this takes more care: you have to go change everyone who links here. And it doesn't require a re-render.)
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