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In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and 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 one. For example, such data can consist of the rings of continuous or smooth real-valued functions defined on each open set. Sheaves are by design quite general and abstract objects, and their correct definition is rather technical. They are variously defined, for example, as sheaves of sets or sheaves of rings, depending on the type of data assigned to 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.
In topology, differential geometry, and algebraic geometry, several structures defined on a topological space (e.g., a differentiable manifold) can be naturally localised or restricted to open subsets of the space: typical examples include continuous real-valued or complex-valued functions, n times differentiable (real-valued or complex-valued) functions, bounded real-valued functions, vector fields, and sections of any vector bundle on the space.
Presheaves formalise the situation common to the examples above: a presheaf (of sets) on a topological space is a structure that associates to each open set U of the space a set F(U) of sections on U, and to each open set V included in U a map F(U) -> F(V) giving restrictions of sections over U to V. Each of the examples above defines a presheaf by taking the restriction maps to be the usual restriction of functions, vector fields and sections of a vector bundle. Moreover, in each of these examples the sets of sections have additional algebraic structure: pointwise operations make them abelian groups, and in the examples of real and complex-valued functions the sets of sections even have a ring structure. In addition, in each example the restriction maps are homomorphisms of the corresponding algebraic structure. This observation leads to the natural definition of presheaves with additional algebraic structure such as presheaves of groups, of abelian groups, of rings: sets of sections are required to have the specified algebraic structure, and the restrictions are required to be homomorphisms. Thus for example continuous real-valued functions on a topological space form a presheaf of rings on the space.
Given a presheaf, a natural question to ask is to what extent its sections over an open set U are specified by their restrictions to smaller open sets V_{i} of an open cover of U. A presheaf is separated if its sections are "locally determined": whenever two sections over U coincide when restricted to each of V_{i}, the two sections are identical. All examples of presheaves discussed above are separated, since in each case the sections are specified by their values at the points of the underlying space. Finally, a separated presheaf is a sheaf if compatible sections can be glued together, i.e., whenever there is a section of the presheaf over each of the covering sets V_{i}, chosen so that they match on the overlaps of the covering sets, these sections correspond to a (unique) section on U, of which they are restrictions. It is easy to verify that all examples above except the presheaf of bounded functions are in fact sheaves: in all cases the criterion of being a section of the presheaf is local in a sense that it is enough to verify it in an arbitrary neighbourhood of each point.
On the other hand, a function can be bounded on each set of an (infinite) open cover of a space without being bounded on all of the space; thus bounded functions provide an example of a (separated) presheaf that in general fails to be a sheaf. Another example of a presheaf that fails to be a sheaf is a constant presheaf that associates the same fixed set (or abelian group, or ring,...) to each open set: it follows from the gluing property of sheaves that the set of sections on a disjoint union of two open sets is the Cartesian product of the sets of sections over the two open sets. The correct way to define the constant sheaf F_{A} (associated to for instance a set A) on a topological space is to require sections on an open set U to be continuous maps from U to A equipped with the discrete topology; then in particular F_{A}(U) ≅ A for connected U.
Maps between sheaves or presheaves (called morphisms), consist of maps between the sets of sections over each open set of the underlying space, compatible with restrictions of sections. If the presheaves or sheaves considered are provided with additional algebraic structure, these maps are assumed to be homomorphisms. Sheaves endowed with nontrivial endomorphisms, such as the action of an algebraic torus or a Galois group, are of particular interest.
Presheaves and sheaves are typically denoted by capital letters, F being particularly common, presumably for the French word for sheaf, faisceau. Use of calligraphic letters such as is also common.
The first step in defining a sheaf is to define a presheaf, which captures the idea of associating data and restriction maps to the open sets of a topological space. The second step is to require the normalisation and gluing axioms. A presheaf that satisfies these axioms is a sheaf.
Let X be a topological space, and let C be a category. Usually C is the category of sets, the category of groups, the category of abelian groups, or the category of commutative rings. A presheaf F on X is a functor with values in C given by the following data:
The morphisms res_{V,U} are called restriction morphisms. If , then its restriction is often denoted s|_{V} by analogy with restriction of functions. The restriction morphisms are required to satisfy two 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.
There is a compact way to express the notion of a presheaf in terms of category theory. First we define 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. This definition can be generalized to the case when the source category is not of the form O(X) for any X; see presheaf (category theory).
If F is a C-valued presheaf on X, and U is an open subset of X, then F(U) is called the sections of F over U. If C is a concrete category, then each element of F(U) is called a section. A section over X is called a global section. A common notation (used also below) for the restriction res_{V,U}(s) of a section is s|_{V}. This terminology and notation is by analogy with sections of fiber bundles or sections of the étalé space of a sheaf; see below. F(U) is also often denoted (U, F), especially in contexts such as sheaf cohomology where U tends to be fixed and F tends to be variable.
For simplicity, consider first the case where the sheaf takes values in the category of sets. In fact, this definition applies more generally to the situation where the category is a concrete category whose underlying set functor is conservative, meaning that if the underlying map of sets is a bijection, then the original morphism is an isomorphism.
A sheaf is a presheaf with values in the category of sets that satisfies the following two axioms:
The section s whose existence is guaranteed by axiom 2 is called the gluing, concatenation, or collation of the sections s_{i}. By axiom 1 it is unique. Sections s_{i} satisfying the condition of axiom 2 are often called compatible; thus axioms 1 and 2 together state that compatible sections can be uniquely glued together. A separated presheaf, or monopresheaf, is a presheaf satisfying axiom 1.^{[1]}
If C (the category in which the sheaf takes values) has products, the sheaf axioms are equivalent to the requirement that, for any open covering U_{i}, the first arrow in the following diagram is an equalizer:
Here the first map is the product of the restriction maps
and the pair of arrows the products of the two sets of restrictions
and
For a separated presheaf, the first arrow need only be injective.
In general, for an open set U and open covering (U_{i}), construct a category J whose objects are the sets U_{i} and the intersections and whose morphisms are the inclusions of in U_{i} and U_{j}. The sheaf axioms for U and (U_{i}) are that the limit of the functor F restricted to the category J must be isomorphic to F(U).
Notice that the empty subset of a topological space is covered by the empty family of sets. The product of an empty family or the limit of an empty family is a terminal object, and consequently the value of a sheaf on the empty set must be a terminal object. If sheaf values are in the category of sets, applying the local identity axiom to the empty family shows that over the empty set, there is at most one section, and applying the gluing axiom to the empty family shows that there is at least one section. This property is called the normalisation axiom.
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. Thus a sheaf can often be defined by giving its values on the open sets of a basis, and verifying the sheaf axioms relative to the basis. (see Gluing axiom § Sheaves on a basis of open sets.)
Heuristically speaking, a morphism of sheaves is analogous to a function between them. However, because sheaves contain data relative to every open set of a topological space, a morphism of sheaves is defined as a collection of functions, one for each open set, that satisfy a compatibility condition.
Let F and G be two sheaves on X with values in the category C. A morphism consists of a morphism for each open set U of X, subject to the condition that this morphism is compatible with restrictions. In other words, for every open subset V of an open set U, the following diagram is commutative.
Recall that we could also express a sheaf as a special kind of functor. In this language, a morphism of sheaves is a natural transformation of the corresponding functors. With this notion of morphism, there is a category of C-valued sheaves on X for any C. The objects are the C-valued sheaves on X, and the morphisms are morphisms of sheaves on X. An isomorphism of sheaves is an isomorphism in this category.
It can be proved that an isomorphism of sheaves is an isomorphism on each open set U. In other words, is an isomorphism if and only if for each U, is an isomorphism. It is also true that a morphism of sheaves is an isomorphism if and only if there exists an open cover such that are isomorphisms of sheaves for all . The same facts are true for monomorphisms. However, they are false for epimorphisms, and their failure is measured by sheaf cohomology.
Notice that we used neither the locality nor the gluing axiom in defining a morphism of sheaves. Consequently, the above definition makes sense for presheaves as well. The category of C-valued presheaves on X is then a functor category, the category of contravariant functors from O(X) to C.
Because sheaves encode exactly the data needed to pass between local and global situations, there are many examples of sheaves occurring throughout mathematics. Here are some additional examples of sheaves:
In the following examples M is an n-dimensional C^{k}-manifold. The table lists the values of certain sheaves over open subsets U of M and their restriction maps.
Sheaf | Sections over an open set U | Restriction maps | Remarks |
---|---|---|---|
Sheaf of j-times continuously differentiable functions , j k | C^{j}-functions U -> R | Restriction of functions. | This is a sheaf of rings with addition and multiplication given by pointwise addition and multiplication. When j = k, this sheaf is called the structure sheaf and is denoted . |
Sheaf of nonzero k-times continuously differentiable functions | Nowhere zero C^{k}-functions U -> R | Restriction of functions. | A sheaf of groups under pointwise multiplication. |
Cotangent sheaves ?^{p}_{M} | Differential forms of degree p on U | Restriction of differential forms. | ?^{1}_{M} and ?^{n}_{M} are commonly denoted ?_{M} and ?_{M}, respectively. |
Sheaf of distributions DB | Distributions on U | The dual map to extension of smooth compactly supported functions by zero. | Here M is assumed to be smooth. |
Sheaf of differential operators | Finite-order differential operators on U | Restriction of differential operators. | Here M is assumed to be smooth. |
Here are two examples of presheaves that are not sheaves:
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 sheaving, sheafification or sheaf associated to the presheaf F. The functor a is called the sheaving functor, sheafification functor, or associated sheaf functor. There is a natural morphism of presheaves that has the universal property that for any sheaf G and any morphism of presheaves , there is a unique morphism of sheaves such that . In fact a is the left adjoint functor to the inclusion functor (or forgetful functor) from the category of sheaves to the category of presheaves, and i is the unit of the adjunction. In this way, the category of sheaves turns into a Giraud subcategory of presheaves.
One concrete way of constructing the sheaf aF is to identify it with the sheaf of sections of an appropriate topological space, called the étalé space of the presheaf. Briefly, the underlying set of the topological space is the disjoint union of the stalks of F, denoted . There is a natural map that sends each germ to the point of X over which it lies. For each open set U and each section s of F over U, we define a section of φ that sends x to the germ s_{x}. Then is given the finest topology for which all sections are continuous, and aF is the sheaf of continuous sections of φ for this topology.
There are other constructions of the sheaf aF. In particular, Alexander Grothendieck and Jean-Louis Verdier (SGA 4 II 3.0.5) define a functor L from presheaves to presheaves which, when applied to a presheaf, yields a separated presheaf and, when applied to a separated presheaf, yields a sheaf. Applying the functor L twice therefore turns a presheaf into a sheaf, and in fact LLF is the associated sheaf aF.
If K is a subsheaf of a sheaf F of abelian groups, then the quotient sheaf Q is the sheaf associated to the presheaf ; in other words, the quotient sheaf fits into an exact sequence of sheaves of abelian groups;
(this is also called a sheaf extension.)
Let F, G be sheaves of abelian groups. The set of morphisms of sheaves from F to G forms an abelian group (by the abelian group structure of G). The sheaf hom of F and G, denoted by,
is the sheaf of abelian groups where is the sheaf on U given by (Note sheafification is not needed here). The direct sum of F and G is the sheaf given by , and the tensor product of F and G is the sheaf associated to the presheaf .
All of these operations extend to sheaves of modules over a sheaf of rings A; the above is the special case when A is the constant sheaf .
The definition of a morphism on sheaves makes sense only for sheaves on the same space X. This is because the data contained in a sheaf is indexed by the open sets of the space. If we have two sheaves on different spaces, then their data is indexed differently. There is no way to go directly from one set of data to the other.
However, it is possible to move a sheaf from one space to another using a continuous function. Let f : X -> Y be a continuous function from a topological space X to a topological space Y. If we have a sheaf on X, we can move it to Y, and vice versa. There are four ways in which sheaves can be moved.
The twisted inverse image functor is, in general, only defined as a functor between derived categories. These functors come in adjoint pairs: and are left and right adjoints of each other, and and are left and right adjoints of each other. ( is the exceptional inverse image functor.) The functors are intertwined with each other by Grothendieck duality and Verdier duality.
There is a different inverse image functor for sheaves of modules over sheaves of rings. This functor is usually denoted and it is distinct from . See inverse image functor.
The stalk of a sheaf captures the properties of a sheaf "around" a point x ? X, generalizing the germs of functions. Here, "around" means that, conceptually speaking, one looks at smaller and smaller neighborhoods of the point. Of course, no single neighborhood will be small enough, so we will have to take a limit of some sort.
The stalk is defined by
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) -> F_{x} takes a section s in F(U) to its germ at x. This generalises the usual definition of a germ.
A different way of defining the stalk is
where i is the inclusion of the one-point space {x} into X. The equivalence follows from the definition of the inverse image.
In many situations, knowing the stalks of a sheaf is enough to control the sheaf itself. For example, whether or not a morphism of sheaves is a monomorphism, epimorphism, or isomorphism can be tested on the stalks. They also find use in constructions such as Godement resolutions.
A pair consisting of a topological space X and a sheaf of rings on X is called a ringed space. Many types of spaces can be defined as certain types of ringed spaces. The sheaf is called the structure sheaf of the space. A very common situation is when all the stalks of the structure sheaf are local rings, in which case the pair is called a locally ringed space. Here are examples of definitions made in this way:
Let be a ringed space. A sheaf of modules is a sheaf such that on every open set U of X, is an -module and for every inclusion of open sets V ? U, the restriction map is compatible with the restriction map O(U) -> O(V): the restriction of fs is the restriction of f times that of s for any f in O(U) and s in F(U).
Most important geometric objects are sheaves of modules. For example, there is a one-to-one correspondence between vector bundles and locally free sheaves of -modules. Sheaves of solutions to differential equations are D-modules, that is, modules over the sheaf of differential operators.
A particularly important case is abelian sheaves, which are modules over the constant sheaf . Every sheaf of modules is an abelian sheaf.
The condition that a module is finitely generated or finitely presented can also be formulated for a sheaf of modules. is finitely generated if, for every point x of X, there exists an open neighborhood U of x, a natural number n (possibly depending on U), and a surjective morphism of sheaves . Similarly, is finitely presented if in addition there exists a natural number m (again possibly depending on U) and a morphism of sheaves such that the sequence of morphisms is exact. Equivalently, the kernel of the morphism is itself a finitely generated sheaf.
These, however, are not the only possible finiteness conditions on a sheaf. The most important finiteness condition for a sheaf is coherence. is coherent if it is of finite type and if, for every open set U and every morphism of sheaves (not necessarily surjective), the kernel of ? is of finite type. is coherent if it is coherent as a module over itself. Note that coherence is a strictly stronger condition than finite presentation: is always finitely presented as a module over itself, but it is not always coherent. For example, let X be a point, let be the ring of complex polynomials in countably many indeterminates. Choose , and for the morphism ?, take the map that sends every variable to zero. The kernel of this map is not finitely generated, so is not coherent.
In the examples above it was noted that some sheaves occur naturally as sheaves of sections. In fact, all sheaves of sets can be represented as sheaves of sections of a topological space called the étalé space, from the French word étalé [etale], meaning roughly "spread out". If F is a sheaf over X, then the étalé space of F is a topological space E together with a local homeomorphism ? : E -> X such that the sheaf of sections of ? is F. The space E is usually very strange, and even if the sheaf F arises from a natural topological situation, E may not have any clear topological interpretation. For example, if F is the sheaf of sections of a continuous function f : Y -> X, then E = Y if and only if f is a local homeomorphism.
The étalé space E is constructed from the stalks of F over X. As a set, it is their disjoint union and ? is the obvious map that takes the value x on the stalk of F over x ? X. The topology of E is defined as follows. For each element s of F(U) and each x in U, we get a germ of s at x. These germs determine points of E. For any U and s ? F(U), the union of these points (for all x ? U) is declared to be open in E. Notice that each stalk has the discrete topology as subspace topology. Two morphisms between sheaves determine a continuous map of the corresponding étalé spaces that is compatible with the projection maps (in the sense that every germ is mapped to a germ over the same point). This makes the construction into a functor.
The construction above determines an equivalence of categories between the category of sheaves of sets on X and the category of étalé spaces over X. The construction of an étalé space can also be applied to a presheaf, in which case the sheaf of sections of the étalé space recovers the sheaf associated to the given presheaf.
This construction makes all sheaves into representable functors on certain categories of topological spaces. As above, let F be a sheaf on X, let E be its étalé space, and let ? : E -> X be the natural projection. Consider the category Top/X of topological spaces over X, that is, the category of topological spaces together with fixed continuous maps to X. Every object of this category is a continuous map f : Y -> X, and a morphism from Y -> X to Z -> X is a continuous map Y -> Z that commutes with the two maps to X. There is a functor ? from Top/X to the category of sets that takes an object f : Y -> X to (f^{−1}F)(Y). For example, if i : U -> X is the inclusion of an open subset, then ?(i) = (i^{−1}F)(U) agrees with the usual F(U), and if i : {x} -> X is the inclusion of a point, then ?({x}) = (i^{−1}F)({x}) is the stalk of F at x. There is a natural isomorphism
which shows that ? represents the functor ?.
E is constructed so that the projection map ? is a covering map. In algebraic geometry, the natural analog of a covering map is called an étale morphism. Despite its similarity to "étalé", the word étale [etal] has a different meaning in French. It is possible to turn E into a scheme and ? into a morphism of schemes in such a way that ? retains the same universal property, but ? is not in general an étale morphism because it is not quasi-finite. It is, however, formally étale.
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.
It was noted above that the functor preserves isomorphisms and monomorphisms, but not epimorphisms. If F is a sheaf of abelian groups, or more generally a sheaf with values in an abelian category, then is actually a left exact functor. This means that it is possible to construct derived functors of . These derived functors are called the cohomology groups (or modules) of F and are written . Grothendieck proved in his "Tohoku paper" (Grothendieck (1957)) that every category of sheaves of abelian groups contains enough injective objects, so these derived functors always exist.
However, computing sheaf cohomology using injective resolutions is nearly impossible. In practice, it is much more common to find a different and more tractable resolution of F. A general construction is provided by Godement resolutions, and particular resolutions may be constructed using soft sheaves, fine sheaves, and flabby sheaves (also known as flasque sheaves from the French flasque meaning flabby). As a consequence, it can become possible to compare sheaf cohomology with other cohomology theories. For example, the de Rham complex is a resolution of the constant sheaf on any smooth manifold, so the sheaf cohomology of is equal to its de Rham cohomology. In fact, comparing sheaf cohomology to de Rham cohomology and singular cohomology provides a proof of de Rham's theorem that the two cohomology theories are isomorphic.
A different approach is by ?ech cohomology. ?ech cohomology was the first cohomology theory developed for sheaves and it is well-suited to concrete calculations. It relates sections on open subsets of the space to cohomology classes on the space. In most cases, ?ech cohomology computes the same cohomology groups as the derived functor cohomology. However, for some pathological spaces, ?ech cohomology will give the correct but incorrect higher cohomology groups. To get around this, Jean-Louis Verdier developed hypercoverings. Hypercoverings not only give the correct higher cohomology groups but also allow the open subsets mentioned above to be replaced by certain morphisms from another space. This flexibility is necessary in some applications, such as the construction of Pierre Deligne's mixed Hodge structures.
A much cleaner 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.
In many cases there is a duality theory for sheaves that generalizes Poincaré duality. See Grothendieck duality and Verdier duality.
André Weil's Weil conjectures stated that there was a cohomology theory for algebraic varieties over finite fields that would give an analogue of the Riemann hypothesis. The cohomology of a complex manifold can be defined as the sheaf cohomology of the locally constant sheaf in the Euclidean topology, which suggests defining a Weil cohomology theory in positive characteristic as the sheaf cohomology of a constant sheaf. But the only classical topology on such a variety is the Zariski topology, and the Zariski topology has very few open sets, so few that the cohomology of any Zariski-constant sheaf on an irreducible variety vanishes (except in degree zero). Alexandre Grothendieck solved this problem by introducing Grothendieck topologies, which axiomatize the notion of covering. Grothendieck's insight was that the definition of a sheaf depends only on the open sets of a topological space, not on the individual points. Once he had axiomatized the notion of covering, open sets could be replaced by other objects. A presheaf takes each one of these objects to data, just as before, and a sheaf is a presheaf that satisfies the gluing axiom with respect to our new notion of covering. This allowed Grothendieck to define étale cohomology and l-adic cohomology, which eventually were used to prove the Weil conjectures.
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.
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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). This shows that some of the facets of sheaf theory can also be traced back as far as Leibniz.