Sheaf (mathematics)


In mathematics, a sheaf is a tool for systematically tracking data 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 are well-behaved in that they can be restricted to smaller open sets, and also the data assigned to an open set are equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set.
The field of mathematics that studies sheaves is called sheaf theory.
Sheaves are understood conceptually as general and abstract objects. Their precise definition is rather technical. They are specifically defined as sheaves of sets or as sheaves of rings, for example, depending on the type of data assigned to the open sets.
There are also maps from one sheaf to another; sheaves 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 the notion of a sheaf on a category with respect to some Grothendieck topology, have provided applications to mathematical logic and to number theory.

Definitions and examples

In many mathematical branches, several structures defined on a topological space can be naturally localised or restricted to open subsets : typical examples include continuous real-valued or complex-valued functions, -times differentiable functions, bounded real-valued functions, vector fields, and sections of any vector bundle on the space. The ability to restrict data to smaller open subsets gives rise to the concept of presheaves. Roughly speaking, sheaves are then those presheaves, where local data can be glued to global data.

Presheaves

Let be a topological space. A presheaf of sets on consists of the following data:
  • For each open set, there exists a set. This set is also denoted. The elements in this set are called the sections of over. The sections of over are called the global sections of.
  • For each inclusion of open sets, a function. In view of many of the examples below, the morphisms are called restriction morphisms. If, then its restriction is often denoted by analogy with restriction of functions.
The restriction morphisms are required to satisfy two additional properties:
  • For every open set of, the restriction morphism is the identity morphism on.
  • If we have three open sets, then the composite.
Informally, the second axiom says it does not matter whether we restrict to in one step or restrict first to, then to. A concise functorial reformulation of this definition is given further below.
Many examples of presheaves come from different classes of functions: to any, one can assign the set of continuous real-valued functions on. The restriction maps are then just given by restricting a continuous function on to a smaller open subset, which again is a continuous function. The two presheaf axioms are immediately checked, thereby giving an example of a presheaf. This can be extended to a presheaf of holomorphic functions and a presheaf of smooth functions.
Another common class of examples is assigning to the set of constant real-valued functions on. This presheaf is called the constant presheaf associated to and is denoted.

Sheaves

Given a presheaf, a natural question to ask is to what extent its sections over an open set ' are specified by their restrictions to open subsets of. A sheaf is a presheaf whose sections are, in a technical sense, uniquely determined by their restrictions.
Axiomatically, a sheaf is a presheaf that satisfies both of the following axioms:
  1. Suppose is an open set, is an open cover of with for all, and are sections. If for all, then.
  2. Suppose is an open set, is an open cover of with for all, and is a family of sections. If all pairs of sections agree on the overlap of their domains, that is, if for all, then there exists a section such that for all.
In both of these axioms, the hypothesis on the open cover is equivalent to the assumption that.
The section ' whose existence is guaranteed by axiom 2 is called the gluing, concatenation, or collation of the sections. By axiom 1 it is unique. Sections ' and ' satisfying the agreement precondition of axiom 2 are often called compatible ; thus axioms 1 and 2 together state that any collection of pairwise compatible sections can be uniquely glued together. A separated presheaf, or monopresheaf, is a presheaf satisfying axiom 1.
The presheaf consisting of continuous functions mentioned above is a sheaf. This assertion reduces to checking that, given continuous functions which agree on the intersections, there is a unique continuous function whose restriction equals the. By contrast, the constant presheaf is usually not a sheaf as it fails to satisfy the locality axiom on the empty set.
Presheaves and sheaves are typically denoted by capital letters, being particularly common, presumably for the French word for sheaf, faisceau. Use of calligraphic letters such as is also common.
It can be shown that to specify a sheaf, it is enough to specify its restriction to the open sets of a basis for the topology of the underlying space. Moreover, it can also be shown that it is enough to verify the sheaf axioms above relative to the open sets of a covering. This observation is used to construct another example which is crucial in algebraic geometry, namely quasi-coherent sheaves. Here the topological space in question is the spectrum of a commutative ring, whose points are the prime ideals in. The open sets form a basis for the Zariski topology on this space. Given an -module, there is a sheaf, denoted by on the, that satisfies
There is another characterization of sheaves that is equivalent to the previously discussed.
A presheaf is a sheaf if and only if for any open and any open cover of, is the fibre product. This characterization is useful in construction of sheaves, for example, if are abelian sheaves, then the kernel of sheaves morphism is a sheaf, since projective limits commutes with projective limits. On the other hand, the cokernel is not always a sheaf because inductive limits do not necessarily commute with projective limits. One way to fix this is to consider Noetherian topological spaces; all open sets are compact so that the cokernel is a sheaf, since finite projective limits commutes with inductive limits.

Further examples

Sheaf of sections of a continuous map

Any continuous map of topological spaces determines a sheaf on by setting
Any such is commonly called a section of ', and this example is the reason why the elements in are generally called sections. This construction is especially important when is the projection of a fiber bundle onto its base space. For example, the sheaves of smooth functions are the sheaves of sections of the trivial bundle.
Another example: the sheaf of sections of
is the sheaf which assigns to any ' the set of branches of the complex logarithm on .
Given a point and an abelian group, the skyscraper sheaf is defined as follows: if is an open set containing, then. If does not contain, then, the trivial group. The restriction maps are either the identity on, if both open sets contain, or the zero map otherwise.

Sheaves on manifolds

On an -dimensional -manifold, there are a number of important sheaves, such as the sheaf of -times continuously differentiable functions . Its sections on some open are the -functions. For, this sheaf is called the structure sheaf and is denoted. The nonzero functions also form a sheaf, denoted. Differential forms also form a sheaf. In all these examples, the restriction morphisms are given by restricting functions or forms.
The assignment sending to the compactly supported functions on is not a sheaf, since there is, in general, no way to preserve this property by passing to a smaller open subset. Instead, this forms a cosheaf, a dual concept where the restriction maps go in the opposite direction than with sheaves. However, taking the dual of these vector spaces does give a sheaf, the sheaf of distributions.

Presheaves that are not sheaves

In addition to the constant presheaf mentioned above, which is usually not a sheaf, there are further examples of presheaves that are not sheaves:
  • Let be the two-point topological space with the discrete topology. Define a presheaf as follows: The restriction map is the projection of onto its first coordinate, and the restriction map is the projection of onto its second coordinate. is a presheaf that is not separated: a global section is determined by three numbers, but the values of that section over and determine only two of those numbers. So while we can glue any two sections over and, we cannot glue them uniquely.
  • Let be the real line, and let be the set of bounded continuous functions on. This is not a sheaf because it is not always possible to glue. For example, let be the set of all such that. The identity function is bounded on each. Consequently, we get a section on. However, these sections do not glue, because the function is not bounded on the real line. Consequently is a presheaf, but not a sheaf. In fact, is separated because it is a sub-presheaf of the sheaf of continuous functions.