Sheaf (mathematics)

Tool to track locally defined data attached to the open sets of a topological space

In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous functions defined on that open set. Such data is well behaved in that it can be restricted to smaller open sets, and also the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set. (Intuitively, every piece of data is the sum of its parts.)

Sheaves are understood conceptually as general and abstract objects. Their correct definition is rather technical. They are specifically defined as sheaves of sets or sheaves of rings, for example, depending on the type of data assigned to the open sets.

There are also maps (or morphisms) from one sheaf to another; sheaves (of a specific type, such as sheaves of abelian groups) with their morphisms on a fixed topological space form a category. On the other hand, to each continuous map there is associated both a direct image functor, taking sheaves and their morphisms on the domain to sheaves and morphisms on the codomain, and an inverse image functor operating in the opposite direction. These functors, and certain variants of them, are essential parts of sheaf theory.

Due to their general nature and versatility, sheaves have several applications in topology and especially in algebraic and differential geometry. First, geometric structures such as that of a differentiable manifold or a scheme can be expressed in terms of a sheaf of rings on the space. In such contexts, several geometric constructions such as vector bundles or divisors are naturally specified in terms of sheaves. Second, sheaves provide the framework for a very general cohomology theory, which encompasses also the "usual" topological cohomology theories such as singular cohomology. Especially in algebraic geometry and the theory of complex manifolds, sheaf cohomology provides a powerful link between topological and geometric properties of spaces. Sheaves also provide the basis for the theory of D-modules, which provide applications to the theory of differential equations. In addition, generalisations of sheaves to more general settings than topological spaces, such as Grothendieck topology, have provided applications to mathematical logic and number theory.

The restriction morphisms are required to satisfy two additional (functorial) properties:

Informally, the second axiom says it doesn't matter whether we restrict to W in one step or restrict first to V, then to W. A concise functorial reformulation of this definition is given further below.

Given a point x and an abelian group S, the skyscraper sheaf Sx defined as follows: If U is an open set containing x, then Sx(U) = S. If U does not contain x, then Sx(U) = 0, the trivial group. The restriction maps are either the identity on S, if both open sets contain x, or the zero map otherwise.

In addition to the constant presheaf mentioned above, which is usually not a sheaf, there are further examples of presheaves that are not sheaves:

Morphisms of sheaves are, roughly speaking, analogous to functions between them. In contrast to a function between sets, which have no additional structure, morphisms of sheaves are those functions which preserve the structure inherent in the sheaves. This idea is made precise in the following definition.

The corresponding statements do not hold for epimorphisms (of sheaves), and their failure is measured by sheaf cohomology.

the direct limit being over all open subsets of X containing the given point x. In other words, an element of the stalk is given by a section over some open neighborhood of x, and two such sections are considered equivalent if their restrictions agree on a smaller neighborhood.

The natural morphism F(U) → Fx takes a section s in F(U) to its germ at x. This generalises the usual definition of a germ.

It is frequently useful to take the data contained in a presheaf and to express it as a sheaf. It turns out that there is a best possible way to do this. It takes a presheaf F and produces a new sheaf aF called the sheafification or sheaf associated to the presheaf F. For example, the sheafification of the constant presheaf (see above) is called the constant sheaf. Despite its name, its sections are locally constant functions.

The sheaf aF can be constructed using the étalé space of F, namely as the sheaf of sections of the map

Another construction of the sheaf aF proceeds by means of a functor L from presheaves to presheaves that gradually improves the properties of a presheaf: for any presheaf F, LF is a separated presheaf, and for any separated presheaf F, LF is a sheaf. The associated sheaf aF is given by LLF.[6]

Since the data of a (pre-)sheaf depends on the open subsets of the base space, sheaves on different topological spaces are unrelated to each other in the sense that there are no morphisms between them. However, given a continuous map f : XY between two topological spaces, pushforward and pullback relate sheaves on X to those on Y and vice versa.

These functors are therefore useful in reducing sheaf-theoretic questions on X to ones on the strata of a stratification, i.e., a decomposition of X into smaller, locally closed subsets.

Presheaves with values in an arbitrary category C are defined by first considering the category of open sets on X to be the posetal category O(X) whose objects are the open sets of X and whose morphisms are inclusions. Then a C-valued presheaf on X is the same as a contravariant functor from O(X) to C. Morphisms in this category of functors, also known as natural transformations, are the same as the morphisms defined above, as can be seen by unraveling the definitions.

If C is an abelian category, this condition can also be rephrased by requiring that there is an exact sequence

The definition of sheaves by étalé spaces is older than the definition given earlier in the article. It is still common in some areas of mathematics such as mathematical analysis.

between the sheaf of holomorphic functions and non-zero holomorphic functions. This map is an epimorphism, which amounts to saying that any non-zero holomorphic function g (on some open subset in C, say), admits a complex logarithm locally, i.e., after restricting g to appropriate open subsets. However, g need not have a logarithm globally.

Sheaf cohomology captures this phenomenon. More precisely, for an exact sequence of sheaves of abelian groups

Another clean approach to the computation of some cohomology groups is the Borel–Bott–Weil theorem, which identifies the cohomology groups of some line bundles on flag manifolds with irreducible representations of Lie groups. This theorem can be used, for example, to easily compute the cohomology groups of all line bundles on projective space and grassmann manifolds.

In many cases there is a duality theory for sheaves that generalizes Poincaré duality. See Grothendieck duality and Verdier duality.

This isomorphism is an example of a base change theorem. There is another adjunction

This computation, and the compatibility of the functors with duality (see Verdier duality) can be used to obtain a high-brow explanation of Poincaré duality. In the context of quasi-coherent sheaves on schemes, there is a similar duality known as coherent duality.

A category with a Grothendieck topology is called a site. A category of sheaves on a site is called a topos or a Grothendieck topos. The notion of a topos was later abstracted by William Lawvere and Miles Tierney to define an elementary topos, which has connections to mathematical logic.

The first origins of sheaf theory are hard to pin down – they may be co-extensive with the idea of analytic continuation[clarification needed]. It took about 15 years for a recognisable, free-standing theory of sheaves to emerge from the foundational work on cohomology.

At this point sheaves had become a mainstream part of mathematics, with use by no means restricted to algebraic topology. It was later discovered that the logic in categories of sheaves is intuitionistic logic (this observation is now often referred to as Kripke–Joyal semantics, but probably should be attributed to a number of authors).