SHEAF (MATHEMATICS)
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In mathematics, a 'sheaf' is the basic tool for expressing relationships between small regions of a space and large regions. Beginning with a topological space ''X'', a sheaf assigns to every region (technically, open set) ''U'' of ''X'' some data ''F''(''U''), such as a set, a group, or a ring. Often these data are a collection of geometric objects defined on that region, such as functions, vector fields, or differential forms. The data can be restricted to smaller regions, and compatible collections of data can be glued to give data over larger regions.
It is common to write a sheaf using the variable ''F''. This comes from the French word for sheaf, ''faisceau''.
Sheaves are used to keep track of the relationship between local and global data. For this reason they are prominent in topology, differential geometry, and algebraic geometry, but they have also found uses in number theory, analysis, and category theory. Roughly speaking, a sheaf ''F'' on a topological space ''X'' consists of two types of data and two properties. The first piece of data is a function which takes every open set ''U'' of ''X'' to a set ''F''(''U''). (We can require that ''F''(''U'') have additional structure, but for now we will require only that it is a set.) The second piece of data takes two open sets ''U'' and ''V'', with ''V'' contained in ''U'', and gives a map
:''res''''V'',''U'' : ''F''(''U'') → ''F''(''V'')
called the ''restriction map''. Conceptually, the restriction map is analogous to restricting the domain of a function. These data satisfy two properties. The first is a normalization axiom and states that ''F''(∅) is a one-element set. The second is usually called the gluing axiom. Roughly speaking, it says that if an open set ''U'' is covered by smaller open sets {''U''''i''}''i'' ∈ ''I'', then an element of ''F''(''U'') corresponds to compatible choices of elements from each ''F''(''U''i). That is, given one element from each ''F''(''U''''i''), and assuming that, for all ''i'' and ''j'', the chosen elements of ''F''(''U''''i'') and ''F''(''U''''j'') become equal when restricted to the overlaps ''U''''i'' ∩ ''U''''j'', there exists one and only one element of ''F''(''U'') which restricts to the original element of each ''F''(''U''''i'').
Before giving the formal definition, we list several examples.
The most basic example is the sheaf of continuous real-valued functions on a topological space ''X''. A continuous function can be restricted to give a continuous function on an open subset, and continuous functions on open subsets can be used to construct a continuous function on the union of the open sets.
To be precise, on each open set ''U'' of ''X'', we let ''F''(''U'') be the set of continuous real-valued functions ''f'' : ''U'' → 'R'. Given an open set ''V'' contained in ''U'' and a function ''f'' in ''F''(''U''), we can restrict the domain of ''f'' to ''V'' to get ''f''|''V''. The restriction ''f''|''V'' is a continuous real-valued function ''V'' → 'R', so it is member of ''F''(''V''). This defines the restriction map ''res''''V'',''U''.
The normalization axiom is clear, because there is a unique function from the empty set to 'R', namely the empty function. To show that the gluing axiom holds, suppose that we have a collection of open sets {''U''''i''}''i'' ∈ ''I'', and let ''U'' be the union of the {''U''''i''}. For each ''i'', choose an ''f''''i'' in ''F''(''U''''i''), that is, a continuous real-valued function ''U''''i'' → 'R'. The hypothesis of the gluing axiom is that the {''f''''i''} agree on overlaps. This means that when we restrict ''f''''i'' and ''f''''j'' to ''U''''i'' ∩ ''U''''j'', they must be equal. In symbols, ''f''''i''|''U''''i''∩''U''''j'' = ''f''''j''|''U''''i''∩''U''''j''. Assuming this, we define a function ''f'' : ''U'' → 'R' as follows: Every point ''x'' of ''U'' lies in some ''U''''i''. Choose such a ''U''''i'', and define ''f''(''x'') to be ''f''''i''(''x''). Because of our assumption that the functions {''f''''i''} agreed on overlaps, this is unambiguous, so ''f'' is well-defined. ''f'' is continuous because each ''f''''i'' is continuous and continuity is a local property of functions. Furthermore, ''f'' is the only possible function that could restrict to ''f''''i'' on ''U''''i'', because functions are determined by their values on points. Consequently there is one and only one function gluing the {''f''''i''}, namely ''f''.
In fact, this sheaf is not just a sheaf of sets. Because functions can be added pointwise, it is also a sheaf of groups. Because they can be multiplied pointwise, it is a sheaf of rings. Since they form a vector space, it is a sheaf of algebras.
For simplicity, we will work on 'R'. Suppose that we have a differential equation ''F''(''x'', ''y'', ''y''′, ''y''″, … ) = 0. and that we are looking for smooth solutions, that is, smooth functions ''y'' : 'R' → 'R' that satisfy ''F''. In the previous example, we found that there was a sheaf of continuous real-valued functions on 'R'. A similar construction gives a sheaf of smooth real-valued functions on 'R'. We will call this sheaf ''G''. ''G''(''U'') is the set of smooth functions ''U'' → 'R'. Some of the members of ''G''(''U'') are solutions to the differential equation ''F'' = 0. It turns out that these solutions themselves form a sheaf.
For each open set ''U'', let ''H''(''U'') be the set of smooth functions ''y'' : ''U'' → 'R' such that ''F''(''x'', ''y'', ''y''′, ''y''″, … ) = 0. The restriction maps are still restriction of functions, just like for ''G''. ''H''(∅) is still the empty function. To check the gluing axiom, let {''U''''i''}''i'' ∈ ''I'' be a collection of open sets, and let ''U'' be the union of the {''U''''i''}. For each ''i'', choose ''f''''i'' in ''H''(''U''''i''), and assume that the {''f''''i''} agree on overlaps, that is, ''f''''i''|''U''''i''∩''U''''j'' = ''f''''j''|''U''''i''∩''U''''j''. Construct ''f'' in the same way as before: ''f''(''x'') = ''f''''i''(''x'') whenever ''f''''i'' is defined. To see that ''f'' is still a solution to the differential equation, notice that ''f'' satisfies the differential equation near a point ''x'' if and only if ''f'' satisfies the differential equation after restricting. We can always restrict to some ''f''''i'', and we know that ''f''''i'' satisfies the differential equation. Therefore ''f'' is a solution to ''F'' = 0. To see that ''f'' is unique, notice that just as before, ''f'' is determined by its values on points, and those values must restrict to give the values of the ''f''''i''. Consequently ''f'' is the unique gluing of the {''f''''i''}, so ''H'' is a sheaf.
Notice that ''H''(''U'') is contained in ''G''(''U'') for each ''U''. Also, if ''f'' is in both ''H''(''U'') and ''G''(''U''), and if ''V'' is contained in ''U'', then applying the restriction function of ''H'' to ''f'' is the same as applying the restriction function of ''G'' to ''f''. This tells us that ''H'' is a ''subsheaf'' of ''G''.
Depending on the differential equation ''F'', it may be possible to add two solutions to get a third—for example, if ''F'' is linear. If this is the case, then ''H'' is a sheaf of groups, with the group law given by pointwise addition of functions. In general, however, ''H'' is only a sheaf of sets, not a sheaf of groups or a sheaf of rings.
Let ''M'' be a smooth manifold. A vector field ''V'' on ''M'' associates to every point ''x'' of ''M'' a vector ''V''(''x'') in ''T''''x''''M'', the tangent space to ''M'' at ''x''. ''V''(''x'') is required to vary smoothly with ''x''. We will define a sheaf which gives information about the vector fields on ''M''. For each open set ''U'', we consider ''U'' as a smooth manifold and let be the set of all vector fields on ''U''. In other words, is a set of functions ''V'' which take a point ''x'' of ''U'' to a vector ''V''(''x'') in ''T''''x''''U'' in a smooth varying manner. Note that because ''U'' is open, ''T''''x''''U'' = ''T''''x''''M''. We define the restriction maps to be restriction of vector fields.
To show that is a sheaf, first notice that is the empty function because there are no points in the empty set. To check the gluing axiom, let {''U''''i''}''i'' ∈ ''I'' be a collection of open sets, and let ''U'' be the union of the {''U''''i''}. On each open set ''U''''i'', we choose a vector field ''V''''i'', and we assume that these vector fields agree on overlaps, that is, ''V''''i''|''U''''i''∩''U''''j'' = ''V''''j''|''U''''i''∩''U''''j''. Now we define a new vector field ''V'' on ''U'' as follows: For each ''x'' in ''U'', choose a ''U''''i'' containing ''x''. Define ''V''(''x'') to be ''V''''i''(''x''). Because of our assumption that the ''V''''i'' agreed on overlaps, ''V'' is well-defined. Furthermore, ''V''(''x'') is a vector in ''T''''x''''M'', and that vector varies smoothly with ''x'' because ''V''''i''(''x'') varies smoothly with ''x'' and "varying smoothly" is a local property. Lastly, ''V'' is the only possible gluing of the set of ''V''''i'', because ''V'' is determined by its values on each ''x'', and those values must restrict to the values of ''V''''i'' on ''U''''i''.
There is another way of expressing which involves the tangent bundle ''TM'' of ''M''. There is a natural projection map ''p'' : ''TM'' → ''M'' which takes a pair (''x'', ''v''), where ''x'' is a point in ''M'' and ''v'' is a vector in ''T''''x''''M'', to the point ''x''. A vector field on an open set ''U'' is the same as a section of ''p'', that is, it is a smooth map ''s'' : ''U'' → ''TM'' such that ''ps'' = ''id''''U'', where ''id''''U'' is the identity function on ''U''. In other words, ''s'' takes points ''x'' to a pair (''x'', ''v'') in a smooth fashion. ''s'' cannot take a point ''x'' to a pair (''y'', ''v'') with ''y'' ≠''x'' because of the restriction ''ps'' = ''id''''U''. This lets us express the tangent sheaf as a ''sheaf of sections''. In other words, over each ''U'', is the collection of all sections of the projection map ''p'', and the restriction maps are restriction of functions. There is an analogous sheaf of sections for any continuous map of topological spaces.
Notice that is always a sheaf of groups, with addition given by pointwise addition of vectors. However, is not naturally a sheaf of rings because there is no natural multiplication of vectors.
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 normalization and gluing axioms. A presheaf which 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'' with values in 'C' is given by the following data:
★ For each open set ''U'' of ''X'', an object ''F''(''U'') in 'C'
★ For each inclusion of open sets , a morphism ''res''''V'',''U'' : ''F''(''U'') → ''F''(''V'') in the category 'C'.
The morphisms ''res''''V'',''U'' are called 'restriction morphisms'. The restriction morphisms are required to satisfy two properties.
★ For every open set ''U'' of ''X'', the restriction morphism ''res''''U'',''U'' : ''F''(''U'') → ''F''(''U'') is the identity morphism on ''F''(''U'').
★ If we have three open sets , then ''res''''W'',''V'' o ''res''''V'',''U'' = ''res''''W'',''U''.
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 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'. This 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 Γ(''U'',''F''), especially in contexts such as sheaf cohomology where ''U'' tends to be fixed and ''F'' tends to be variable.
Sheaves are presheaves subject to two axioms. The first is the normalization axiom:
★ ''F''(∅) is the terminal object of 'C'.
For this definition to make sense, 'C' must have a terminal object, but in practice this is usually the case.
More important is the gluing axiom. Recall that in our examples above, the gluing axiom required that we could paste together sections which agreed on overlaps. For simplicity, we will state the gluing axiom only when 'C' is a concrete category. For a more abstract and general formulation, see the article gluing axiom.
Let be a collection of open subsets of , and let . For each , choose a section . We say that are 'compatible' if, for all and , . The gluing axiom states:
★ For every set of compatible sections on , there exists a unique section such that .
The section ''s'' is called the 'gluing', 'concatenation', or 'collation' of the sections {''si''}.
In the examples we gave above, the sections of the sheaf corresponded to functions. When this is the case, the hypothesis of the gluing axiom is that the two functions are equal where they overlap, and the conclusion is that there is one and only one function on ''U'' which pastes together all of functions on the ''U''''i''. This is what we showed above to demonstrate that our examples were sheaves.
Sometimes the gluing axiom is split into two axioms, one for existence and one for uniqueness. A presheaf that satisfies only uniqueness but not existence is called a 'separated presheaf'.
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:
★ Any continuous map of topological spaces determines a sheaf of sets. Let ''f'' : ''Y'' → ''X'' be a continuous map. We define a sheaf by setting equal to the sections ''U'' → ''Y'', that is, is the set of all functions ''s'' : ''U'' → ''Y'' such that ''fs'' = ''id''''U''. Restriction is given by restriction of functions. This sheaf is called the 'sheaf of sections' of ''f'', and it is especially important when ''f'' is the projection of a fiber bundle onto its base space. Notice that if the image of ''f'' does not contain ''U'', then is empty. For a concrete example, take , , and . is the set of branches of the logarithm on .
★ Let ''M'' be a ''C''''k''-manifold. For each open subset ''U'' of ''M'' set equal to the set of all ''C''''k''-functions ''U'' → 'R'. Restriction is given by restriction of functions. Then is a sheaf of rings with addition and multiplication given by pointwise addition and multiplication. This is called the 'structure sheaf' of ''M''.
★ For every ''j'' ≤ ''k'', ''M'' also has a sheaf called the 'sheaf of ''j''-times continuously differentiable functions' on ''M''. is the subsheaf of which, on the open set ''U'', gives the set of all ''C''''j'' functions on ''M''.
★ ''M'' has a sheaf of nonzero functions. That is, for each ''U'', equals the set of all non-zero real-valued functions on ''U''. Restriction is given by restriction of functions. This is a sheaf of groups where the group operation is given by pointwise multiplication.
★ ''M'' also has a 'cotangent sheaf' Ω''M''. On each open set ''U'', Ω''M''(''U'') is the set of degree one differential forms on ''U''. Restriction is given by restriction of differential forms. Similarly, for every ''p'' > 0, there is a sheaf Ω''p'' of differential p-forms.
★ If ''M'' is smooth, then for each open set ''U'', we have a set of real-valued distributions on ''U''. Restriction is given by restriction of functions. Then is a sheaf called the 'sheaf of distributions'.
★ If ''X'' is a complex manifold and ''U'' is an open set of ''X'', let be the set of finite-order holomorphic differential operators on ''U''. Letting restriction be given by restriction of functions, we get a sheaf called the 'sheaf of holomorphic differential operators'.
★ For any set ''S'' and any topological space ''X'', there is a constant presheaf ''F'' which has ''F''(''U'') = ''S'' for all ''U'' and restriction maps equal to the identity. ''F'' is not a sheaf: Let ''U'' and ''V'' be disjoint open sets and ''s'' and ''t'' be distinct elements of ''S''. ''s'' determines a section in ''F''(''U''), because ''s'' is in ''S'' and ''S'' = ''F''(''U''). Similarly, ''t'' determines a section in ''F''(''V''). Since ''U'' and ''V'' are disjoint, the hypothesis of the gluing axiom is vacuously true. Consequently there must be a section in ''F''(''U'' ∪ ''V'') which restricts to ''s'' on ''U'' and to ''t'' on ''V'', but that's impossible. So ''F'' is a presheaf, and even a separated presheaf, but not a sheaf.
★ However, there is a sheaf, called the 'constant sheaf on S', which is very similar to the constant presheaf. We let be the set of all functions from ''U'' to ''S'' which are constant on each connected component. In other words, if ''U'' has a single connected component, then is ''S''. If ''U'' has two connected components, then is ''S'' × ''S''; one factor of ''S'' is the section over one component, and the other factor is the section over the other component. Restriction corresponds to restriction of functions. It can be checked that this makes a sheaf.
★ More generally, if ''S'' is an object in a concrete category 'C' which has all set-indexed products, then we define the constant sheaf to be the sheaf which takes an open set ''U'' to the set of all functions ''U'' → ''S'' which are constant on the connected components of ''U''. For example, this can be done when ''S'' is a ring such as 'Z' to get the constant sheaf . If 'C' is a category such as the category of groups or the category of commutative rings, this will give a sheaf of groups or a sheaf of commutative rings, respectively.
★ Fix a point ''x'' in ''X'' and an object ''S'' in a category 'C'. The 'skyscraper sheaf over ''x'' with stalk ''S''' is the sheaf ''S''''x'' defined as follows: If ''U'' is an open set containing ''x'', then ''S''''x''(''U'') = ''S''. If ''U'' does not contain ''x'', then ''S''''x''(''U'') is the terminal object of 'C'. The restriction maps are either the identity on ''S'', if both open sets contain ''x'', or the unique map from ''S'' to the terminal object of 'C'.
Some types of structure are defined by a space and a fixed sheaf on it. For example, a space together with a sheaf of rings is called a ringed space. If the stalks (see below) are all local rings, then it is a locally ringed space. If the sheaf of rings is locally the same as the elements of a commutative ring, we get a scheme.
Here are two examples of presheaves which are not sheaves:
★ Let ''X'' be the two-point topological space {''x'', ''y''} with the discrete topology. Define a presheaf ''F'' as follows: ''F''(∅) = ∅, ''F''({''x''}) = 'R', ''F''({''y''}) = 'R', ''F''({''x'', ''y''}) = 'R' × 'R' × 'R'. The restriction map ''F''({''x'', ''y''}) → ''F''({''x''}) is the projection of 'R' × 'R' × 'R' onto its first coordinate, and the restriction map ''F''({''x'', ''y''}) → ''F''({''y''}) is the projection of 'R' × 'R' × 'R' onto its second coordinate. ''F'' is a presheaf which is not separated: A global section is determined by three numbers, but the values of that section over {''x''} and {''y''} determine only two of those numbers. So while we can glue any two sections over {''x''} and {''y''}, we cannot glue them uniquely.
★ Let ''X'' be the complex plane, and let ''F''(''U'') be the set of bounded holomorphic functions on ''U''. This is not a sheaf because it is not always possible to glue. For example, let ''U''''i'' be the set of all ''z'' such that |''z''| < ''i''. The function ''f''(''z'') = ''z'' is bounded on each ''U''''i''. Consequently we get a section ''s''''i'' on ''U''''i'' which is the restriction of the constant function to ''U''''i''. However, these sections do not glue, because the function ''f'' is not bounded on the complex plane. Consequently ''F'' is a presheaf, but not a sheaf. In fact, ''F'' is separated because it is a sub-presheaf of the sheaf of holomorphic functions.
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, which satisfy a compatibility condition.
Let and be two sheaves on ''X'' with values in the category 'C'. A ''morphism'' φ : → takes each open set ''U'' of ''X'' to a morphism φ(''U'') : → , subject to the condition that this morphism is compatible with restriction. In other words, for every open subset ''U'' of an open set ''V'', we must have a commutative diagram:
This compatibility condition says that if we have a section ''s'' in , then mapping ''s'' to its image φ(''U'')(''s'') in and then restricting to ''U'' gives the same result as first restricting to ''U'' and then mapping the restriction to its image in .
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, and the morphisms are morphisms of sheaves. 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, φ is an isomorphism if and only if for each ''U'', φ(''U'') is an isomorphism. The same is true of monomorphisms, but not of epimorphisms. See sheaf cohomology.
Notice that we did not use 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 is then a functor category, the category of contravariant functors from ''O''(''X'') to 'C'.
Main articles: Gluing axiom
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''. ''a'' is called the 'sheaving functor', 'sheafification functor', or 'associated sheaf functor'. There is a natural morphism of presheaves ''i'' : ''F'' → ''aF'' which has the universal property that for any sheaf ''G'' and any morphism of presheaves ''f'' : ''F'' → ''G'', there is a unique morphism of sheaves such that . In fact ''a'' is the adjoint functor to the inclusion functor from the category of sheaves to the category of presheaves, and ''i'' is the unit of the adjunction.
Main articles: Direct image functor
Main articles: Inverse image functor
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.
Concretely, let be a sheaf on ''X''. We define the 'direct image' or 'pushforward' of to be the sheaf on ''Y'' that takes open sets ''U'' of ''Y'' to the object . If ''V'' is an open subset of ''U'', then the restriction map ''res''''V'',''U'' is defined to be the restriction map . It can be checked that this is still a sheaf.
Suppose instead that we have a sheaf on ''Y'' and that we want to transport to ''X'' using ''f''. We will call the result the 'inverse image' or 'pullback' sheaf . If we try to imitate the direct image by setting for each open set ''U'' of ''X'', we immediately run into a problem: ''f''(''U'') is not necessarily open. The best we can do is to approximate it by open sets, and even then we will get a presheaf, not a sheaf. Consequently we define to be the sheaf associated to the presheaf:
:
To define the restriction maps, we use the universal property of direct limits.
It is possible to define the direct image and the inverse image of a morphism of sheaves as well, and using this definition, ''f''
★ and ''f'' -1 become functors. In fact, ''f'' -1 is the left adjoint of ''f''
★ . This implies that there are natural unit and counit morphisms and . However, these are ''almost never'' isomorphisms.
There is a different inverse image functor ''f''
★ which appears when working with sheaves of modules on ringed spaces. It is related to, but not the same as, the inverse image functor ''f'' -1. See the main article on the inverse image functor.
Sheaves are defined on open sets, but the underlying topological space ''X'' consists of points. It is reasonable to attempt to isolate the behavior of a sheaf at a single fixed point ''x'' of ''X''. Conceptually speaking, we do this by looking at small neighborhoods of the point. If we look at a sufficiently small neighborhood of ''x'', the behavior of the sheaf on that small neighborhood should be the same as the behavior of at that point. Of course, no single neighborhood will be small enough, so we will have to take a limit of some sort.
To make this precise, remember that if we have an inclusion of an open set ''V'' into an open set ''U'', we get a restriction map . Every restriction map gets us closer to a small neighborhood of ''x'', so to get the local behavior of at ''x'', we want to take a limit over all the open sets and all the restriction maps. In other words, we want to take a direct limit indexed over all the open sets containing ''x''. We define the 'stalk' of at ''x'' to be:
:.
For some categories 'C' this may not exist. However, it exists for most categories which occur in practice, such as the category of sets or most categories of algebraic objects such as abelian groups or rings.
Because we defined the stalk as a direct limit over open sets, there is a natural morphism ''F''(''U'') → ''F''''x'' for any open set ''U'' containing ''x''. This takes a section ''s'' in ''F''(''U'') to its ''germ''. This is a generalization of the usual concept of a germ, which can be recovered by looking at the stalks of the sheaf of continuous functions on ''X''.
Germs are more useful for some sheaves than for others. For example, in the sheaf of analytic functions on an analytic manifold, a germ of a function at a point determines the function in a small neighborhood of a point. This is because the germ records the function's power series expansion, and all analytic functions are by definition equal to their power series. Using analytic continuation, we find that the germ at a point determines the function on any connected open set where the function can be everywhere defined. (This does not imply that all the restriction maps of this sheaf are injective!)
In contrast, for the sheaf of smooth functions on a smooth manifold, germs contain some local information, but are not enough to reconstruct the function on any open neighborhood. For example, let ''f'' : 'R' → 'R' be a bump function which is identically one in a neighborhood of the origin and identically zero far away from the origin. On any sufficiently small neighborhood containing the origin, ''f'' is identically one, so at the origin it has the same germ as the constant function with value 1. Suppose that we want to reconstruct ''f'' from its germ. Even if we know in advance that ''f'' is a bump function, the germ does not tell us how large its bump is. From what the germ tells us, the bump could be infinitely wide, that is, ''f'' could equal the constant function with value 1. We cannot even reconstruct ''f'' on a small open neighborhood ''U'' containing the origin, because we cannot tell whether the bump of ''f'' fits entirely in ''U'' or whether it is so large that ''f'' is identically one in ''U''.
On the other hand, germs of smooth functions can distinguish between the constant function with value one and the function , because the latter function is not identically one on any neighborhood of the origin. This example shows that germs contain more information than the power series expansion of a function, because the power series of is identically one. (This extra information is related to the fact that the stalk of the sheaf of smooth functions at the origin is a non-Noetherian ring. The Krull intersection theorem says that this cannot happen for a Noetherian ring.)
There is another approach to defining a germ which is useful in some contexts. Choose a point ''x'' of ''X'', and let ''i'' be the inclusion of the one point space {''x''} into ''X''. Then the stalk is the same as the inverse image sheaf . Notice that the only open sets of the one point space {''x''} are {''x''} and ∅, and there is no data over the empty set. Over {''x''}, however, we get:
:.
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 ''étale space''. If ''F'' is a sheaf over ''X'', then the 'étale space' of ''F'' is a topological space ''E'' together with a local homeomorphism ''π'': ''E'' → ''X''; the sheaf of sections of ''π'' is ''F''. ''E'' is usually a very strange space, 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 covering map.
The étale space ''E'' is constructed from the stalks of ''F'' over ''X''. As a set, it is their disjoint union and ''Ï€'' is the obvious map which 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''. The union of these points (for all such ''s'' and ''x'') is declared to be open in ''E''. Notice that each stalk has the discrete topology. Two morphisms between sheaves determine a continuous map of the corresponding étale spaces which 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.
This gives an example of an étale space over ''X''. An ''étale space'' is a topological space ''E'' together with a continuous map ''π'': ''E'' → ''X'' which is a local homeomorphism such that each fiber of ''π'' has the discrete topology. 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 étale space can also be applied to a presheaf, in which case the sheaf of sections of the étale space recovers the sheaf associated to the given presheaf.
The map ''π'' is an example of what is sometimes called an ''étale map''. "Étale" here means the same thing as "local homeomorphism". However, the terminology "étale map" is more common in contexts where the right analogue of a local homeomorphism of manifolds is not characterized by the property of being a local homeomorphism. This is the case in algebraic geometry. For more information see the article étale morphism.
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 étale space, and let ''Ï€'': ''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 space is a continuous map ''f'' : ''Y'' → ''X'', and a morphism from ''Y'' → ''X'' to ''Z'' → ''X'' is a continuous map ''Y'' → ''Z'' which commutes with the two maps to ''X''. There is a functor Γ from 'Top'/''X'' to the category of sets which 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 Γ(''i'') = (''i''-1''F'')(''U'') agrees with the usual ''F''(''U''), and if ''i'' : {''x''} → ''X'' is the inclusion of a point, then Γ({''x''}) = (''i''-1''F'')({''x''}) is the stalk of ''F'' at ''x''. There is a natural isomorphism
:
which shows that ''E'' represents the functor Γ.
The definition of sheaves by étale spaces is older than the definition given earlier in the article. It is still common in some areas of mathematics such as mathematical analysis.
Main articles: Sheaf cohomology
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 .
Unfortunately, applying this definition to a computation is nearly impossible. To make computations, we can apply ÄŒech cohomology. ÄŒech 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, ÄŒech cohomology computes the same cohomology groups as the derived functor cohomology. However, for some pathological spaces, ÄŒech 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.
Main articles: Grothendieck topology
Main articles: Topos
André Weil's Weil conjectures stated that there was a cohomology theory for algebraic varieties over finite fields which would give an analogue of the Riemann hypothesis. The only natural topology on such a variety, however, is the Zariski topology, but sheaf cohomology in the Zariski topology is badly behaved because there are very few open sets. 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.
The first origins of 'sheaf theory' are hard to pin down — they may be co-extensive with the idea of analytic continuation. It took about 15 years for a recognisable, free-standing theory of sheaves to emerge from the foundational work on cohomology.
★ 1936 Eduard ÄŒech introduces the ''nerve'' construction, for associating a simplicial complex to an open covering.
★ 1938 Hassler Whitney gives a 'modern' definition of cohomology, summarizing the work since J. W. Alexander and Kolmogorov first defined ''cochains''.
★ 1943 Norman Steenrod publishes on homology ''with local coefficients''.
★ 1945 Jean Leray publishes work carried out as a POW, motivated by proving fixed point theorems for application to PDE theory; it is the start of sheaf theory and spectral sequences.
★ 1947 Henri Cartan reproves the de Rham theorem by sheaf methods, in correspondence with André Weil (see De Rham-Weil theorem). Leray gives a sheaf definition in his courses via closed sets (the later ''carapaces'').
★ 1948 The Cartan seminar writes up sheaf theory for the first time.
★ 1950 The 'second edition' sheaf theory from the Cartan seminar: the sheaf space (''espace étalé'') definition is used, with stalkwise structure. Supports are introduced, and cohomology with supports. Continuous mappings give rise to spectral sequences. At the same time Kiyoshi Oka introduces an idea (adjacent to that) of a sheaf of ideals, in several complex variables.
★ 1951 The Cartan seminar proves the Theorems A and B based on Oka's work.
★ 1953 The finiteness theorem for coherent sheaves in the analytic theory is proved by Cartan and Jean-Pierre Serre, as is Serre duality.
★ 1954 Serre's paper ''Faisceaux algébriques cohérents'' (published 1955) introduces sheaves into algebraic geometry. These ideas are immediately exploited by Hirzebruch, who writes a major 1956 book on topological methods.
★ 1955 Alexander Grothendieck in lectures in Kansas defines abelian category and ''presheaf'', and by using injective resolutions allows direct use of sheaf cohomology on all topological spaces, as derived functors.
★ 1956 Oscar Zariski's report ''Algebraic sheaf theory, Scientific report on the Second summer Institute : Several complex variables [1954, Boulder (Col.)]'', Part III., Bull. Amer. math. Soc., t. 62, 1956, p. 117-141.
★ 1957 Grothendieck's ''Tohoku'' paper rewrites homological algebra; he proves Grothendieck duality (i.e., Serre duality for possibly singular algebraic varieties).
★ 1958 Godement's book on sheaf theory is published. At around this time Mikio Sato proposes his hyperfunctions, which will turn out to have sheaf-theoretic nature.
★ 1957 onwards: Grothendieck extends sheaf theory in line with the needs of algebraic geometry, introducing: schemes and general sheaves on them, local cohomology, derived categories (with Verdier), and Grothendieck topologies. There emerges also his influential schematic idea of 'six operations' in homological algebra.
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.
★ Gerbe
★ Stack (category theory)
★ (oriented towards conventional topological applications)
★
★ (updated edition of a classic using enough sheaf theory to show its power)
★ (advanced techniques such as the derived category and vanishing cycles on the most reasonable spaces)
★ (category theory and toposes emphasised)
★ (concise lecture notes)
★ (pedagogic treatment)
★
It is common to write a sheaf using the variable ''F''. This comes from the French word for sheaf, ''faisceau''.
Introduction
Sheaves are used to keep track of the relationship between local and global data. For this reason they are prominent in topology, differential geometry, and algebraic geometry, but they have also found uses in number theory, analysis, and category theory. Roughly speaking, a sheaf ''F'' on a topological space ''X'' consists of two types of data and two properties. The first piece of data is a function which takes every open set ''U'' of ''X'' to a set ''F''(''U''). (We can require that ''F''(''U'') have additional structure, but for now we will require only that it is a set.) The second piece of data takes two open sets ''U'' and ''V'', with ''V'' contained in ''U'', and gives a map
:''res''''V'',''U'' : ''F''(''U'') → ''F''(''V'')
called the ''restriction map''. Conceptually, the restriction map is analogous to restricting the domain of a function. These data satisfy two properties. The first is a normalization axiom and states that ''F''(∅) is a one-element set. The second is usually called the gluing axiom. Roughly speaking, it says that if an open set ''U'' is covered by smaller open sets {''U''''i''}''i'' ∈ ''I'', then an element of ''F''(''U'') corresponds to compatible choices of elements from each ''F''(''U''i). That is, given one element from each ''F''(''U''''i''), and assuming that, for all ''i'' and ''j'', the chosen elements of ''F''(''U''''i'') and ''F''(''U''''j'') become equal when restricted to the overlaps ''U''''i'' ∩ ''U''''j'', there exists one and only one element of ''F''(''U'') which restricts to the original element of each ''F''(''U''''i'').
Before giving the formal definition, we list several examples.
Sheaves of functions
The most basic example is the sheaf of continuous real-valued functions on a topological space ''X''. A continuous function can be restricted to give a continuous function on an open subset, and continuous functions on open subsets can be used to construct a continuous function on the union of the open sets.
To be precise, on each open set ''U'' of ''X'', we let ''F''(''U'') be the set of continuous real-valued functions ''f'' : ''U'' → 'R'. Given an open set ''V'' contained in ''U'' and a function ''f'' in ''F''(''U''), we can restrict the domain of ''f'' to ''V'' to get ''f''|''V''. The restriction ''f''|''V'' is a continuous real-valued function ''V'' → 'R', so it is member of ''F''(''V''). This defines the restriction map ''res''''V'',''U''.
The normalization axiom is clear, because there is a unique function from the empty set to 'R', namely the empty function. To show that the gluing axiom holds, suppose that we have a collection of open sets {''U''''i''}''i'' ∈ ''I'', and let ''U'' be the union of the {''U''''i''}. For each ''i'', choose an ''f''''i'' in ''F''(''U''''i''), that is, a continuous real-valued function ''U''''i'' → 'R'. The hypothesis of the gluing axiom is that the {''f''''i''} agree on overlaps. This means that when we restrict ''f''''i'' and ''f''''j'' to ''U''''i'' ∩ ''U''''j'', they must be equal. In symbols, ''f''''i''|''U''''i''∩''U''''j'' = ''f''''j''|''U''''i''∩''U''''j''. Assuming this, we define a function ''f'' : ''U'' → 'R' as follows: Every point ''x'' of ''U'' lies in some ''U''''i''. Choose such a ''U''''i'', and define ''f''(''x'') to be ''f''''i''(''x''). Because of our assumption that the functions {''f''''i''} agreed on overlaps, this is unambiguous, so ''f'' is well-defined. ''f'' is continuous because each ''f''''i'' is continuous and continuity is a local property of functions. Furthermore, ''f'' is the only possible function that could restrict to ''f''''i'' on ''U''''i'', because functions are determined by their values on points. Consequently there is one and only one function gluing the {''f''''i''}, namely ''f''.
In fact, this sheaf is not just a sheaf of sets. Because functions can be added pointwise, it is also a sheaf of groups. Because they can be multiplied pointwise, it is a sheaf of rings. Since they form a vector space, it is a sheaf of algebras.
Sheaves of solutions to differential equations
For simplicity, we will work on 'R'. Suppose that we have a differential equation ''F''(''x'', ''y'', ''y''′, ''y''″, … ) = 0. and that we are looking for smooth solutions, that is, smooth functions ''y'' : 'R' → 'R' that satisfy ''F''. In the previous example, we found that there was a sheaf of continuous real-valued functions on 'R'. A similar construction gives a sheaf of smooth real-valued functions on 'R'. We will call this sheaf ''G''. ''G''(''U'') is the set of smooth functions ''U'' → 'R'. Some of the members of ''G''(''U'') are solutions to the differential equation ''F'' = 0. It turns out that these solutions themselves form a sheaf.
For each open set ''U'', let ''H''(''U'') be the set of smooth functions ''y'' : ''U'' → 'R' such that ''F''(''x'', ''y'', ''y''′, ''y''″, … ) = 0. The restriction maps are still restriction of functions, just like for ''G''. ''H''(∅) is still the empty function. To check the gluing axiom, let {''U''''i''}''i'' ∈ ''I'' be a collection of open sets, and let ''U'' be the union of the {''U''''i''}. For each ''i'', choose ''f''''i'' in ''H''(''U''''i''), and assume that the {''f''''i''} agree on overlaps, that is, ''f''''i''|''U''''i''∩''U''''j'' = ''f''''j''|''U''''i''∩''U''''j''. Construct ''f'' in the same way as before: ''f''(''x'') = ''f''''i''(''x'') whenever ''f''''i'' is defined. To see that ''f'' is still a solution to the differential equation, notice that ''f'' satisfies the differential equation near a point ''x'' if and only if ''f'' satisfies the differential equation after restricting. We can always restrict to some ''f''''i'', and we know that ''f''''i'' satisfies the differential equation. Therefore ''f'' is a solution to ''F'' = 0. To see that ''f'' is unique, notice that just as before, ''f'' is determined by its values on points, and those values must restrict to give the values of the ''f''''i''. Consequently ''f'' is the unique gluing of the {''f''''i''}, so ''H'' is a sheaf.
Notice that ''H''(''U'') is contained in ''G''(''U'') for each ''U''. Also, if ''f'' is in both ''H''(''U'') and ''G''(''U''), and if ''V'' is contained in ''U'', then applying the restriction function of ''H'' to ''f'' is the same as applying the restriction function of ''G'' to ''f''. This tells us that ''H'' is a ''subsheaf'' of ''G''.
Depending on the differential equation ''F'', it may be possible to add two solutions to get a third—for example, if ''F'' is linear. If this is the case, then ''H'' is a sheaf of groups, with the group law given by pointwise addition of functions. In general, however, ''H'' is only a sheaf of sets, not a sheaf of groups or a sheaf of rings.
Sheaves of vector fields
Let ''M'' be a smooth manifold. A vector field ''V'' on ''M'' associates to every point ''x'' of ''M'' a vector ''V''(''x'') in ''T''''x''''M'', the tangent space to ''M'' at ''x''. ''V''(''x'') is required to vary smoothly with ''x''. We will define a sheaf which gives information about the vector fields on ''M''. For each open set ''U'', we consider ''U'' as a smooth manifold and let be the set of all vector fields on ''U''. In other words, is a set of functions ''V'' which take a point ''x'' of ''U'' to a vector ''V''(''x'') in ''T''''x''''U'' in a smooth varying manner. Note that because ''U'' is open, ''T''''x''''U'' = ''T''''x''''M''. We define the restriction maps to be restriction of vector fields.
To show that is a sheaf, first notice that is the empty function because there are no points in the empty set. To check the gluing axiom, let {''U''''i''}''i'' ∈ ''I'' be a collection of open sets, and let ''U'' be the union of the {''U''''i''}. On each open set ''U''''i'', we choose a vector field ''V''''i'', and we assume that these vector fields agree on overlaps, that is, ''V''''i''|''U''''i''∩''U''''j'' = ''V''''j''|''U''''i''∩''U''''j''. Now we define a new vector field ''V'' on ''U'' as follows: For each ''x'' in ''U'', choose a ''U''''i'' containing ''x''. Define ''V''(''x'') to be ''V''''i''(''x''). Because of our assumption that the ''V''''i'' agreed on overlaps, ''V'' is well-defined. Furthermore, ''V''(''x'') is a vector in ''T''''x''''M'', and that vector varies smoothly with ''x'' because ''V''''i''(''x'') varies smoothly with ''x'' and "varying smoothly" is a local property. Lastly, ''V'' is the only possible gluing of the set of ''V''''i'', because ''V'' is determined by its values on each ''x'', and those values must restrict to the values of ''V''''i'' on ''U''''i''.
There is another way of expressing which involves the tangent bundle ''TM'' of ''M''. There is a natural projection map ''p'' : ''TM'' → ''M'' which takes a pair (''x'', ''v''), where ''x'' is a point in ''M'' and ''v'' is a vector in ''T''''x''''M'', to the point ''x''. A vector field on an open set ''U'' is the same as a section of ''p'', that is, it is a smooth map ''s'' : ''U'' → ''TM'' such that ''ps'' = ''id''''U'', where ''id''''U'' is the identity function on ''U''. In other words, ''s'' takes points ''x'' to a pair (''x'', ''v'') in a smooth fashion. ''s'' cannot take a point ''x'' to a pair (''y'', ''v'') with ''y'' ≠''x'' because of the restriction ''ps'' = ''id''''U''. This lets us express the tangent sheaf as a ''sheaf of sections''. In other words, over each ''U'', is the collection of all sections of the projection map ''p'', and the restriction maps are restriction of functions. There is an analogous sheaf of sections for any continuous map of topological spaces.
Notice that is always a sheaf of groups, with addition given by pointwise addition of vectors. However, is not naturally a sheaf of rings because there is no natural multiplication of vectors.
The formal definition
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 normalization and gluing axioms. A presheaf which satisfies these axioms is a sheaf.
Definition of a presheaf
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'' with values in 'C' is given by the following data:
★ For each open set ''U'' of ''X'', an object ''F''(''U'') in 'C'
★ For each inclusion of open sets , a morphism ''res''''V'',''U'' : ''F''(''U'') → ''F''(''V'') in the category 'C'.
The morphisms ''res''''V'',''U'' are called 'restriction morphisms'. The restriction morphisms are required to satisfy two properties.
★ For every open set ''U'' of ''X'', the restriction morphism ''res''''U'',''U'' : ''F''(''U'') → ''F''(''U'') is the identity morphism on ''F''(''U'').
★ If we have three open sets , then ''res''''W'',''V'' o ''res''''V'',''U'' = ''res''''W'',''U''.
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 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'. This 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 Γ(''U'',''F''), especially in contexts such as sheaf cohomology where ''U'' tends to be fixed and ''F'' tends to be variable.
Definition of a sheaf
Sheaves are presheaves subject to two axioms. The first is the normalization axiom:
★ ''F''(∅) is the terminal object of 'C'.
For this definition to make sense, 'C' must have a terminal object, but in practice this is usually the case.
More important is the gluing axiom. Recall that in our examples above, the gluing axiom required that we could paste together sections which agreed on overlaps. For simplicity, we will state the gluing axiom only when 'C' is a concrete category. For a more abstract and general formulation, see the article gluing axiom.
Let be a collection of open subsets of , and let . For each , choose a section . We say that are 'compatible' if, for all and , . The gluing axiom states:
★ For every set of compatible sections on , there exists a unique section such that .
The section ''s'' is called the 'gluing', 'concatenation', or 'collation' of the sections {''si''}.
In the examples we gave above, the sections of the sheaf corresponded to functions. When this is the case, the hypothesis of the gluing axiom is that the two functions are equal where they overlap, and the conclusion is that there is one and only one function on ''U'' which pastes together all of functions on the ''U''''i''. This is what we showed above to demonstrate that our examples were sheaves.
Sometimes the gluing axiom is split into two axioms, one for existence and one for uniqueness. A presheaf that satisfies only uniqueness but not existence is called a 'separated presheaf'.
Examples
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:
★ Any continuous map of topological spaces determines a sheaf of sets. Let ''f'' : ''Y'' → ''X'' be a continuous map. We define a sheaf by setting equal to the sections ''U'' → ''Y'', that is, is the set of all functions ''s'' : ''U'' → ''Y'' such that ''fs'' = ''id''''U''. Restriction is given by restriction of functions. This sheaf is called the 'sheaf of sections' of ''f'', and it is especially important when ''f'' is the projection of a fiber bundle onto its base space. Notice that if the image of ''f'' does not contain ''U'', then is empty. For a concrete example, take , , and . is the set of branches of the logarithm on .
★ Let ''M'' be a ''C''''k''-manifold. For each open subset ''U'' of ''M'' set equal to the set of all ''C''''k''-functions ''U'' → 'R'. Restriction is given by restriction of functions. Then is a sheaf of rings with addition and multiplication given by pointwise addition and multiplication. This is called the 'structure sheaf' of ''M''.
★ For every ''j'' ≤ ''k'', ''M'' also has a sheaf called the 'sheaf of ''j''-times continuously differentiable functions' on ''M''. is the subsheaf of which, on the open set ''U'', gives the set of all ''C''''j'' functions on ''M''.
★ ''M'' has a sheaf of nonzero functions. That is, for each ''U'', equals the set of all non-zero real-valued functions on ''U''. Restriction is given by restriction of functions. This is a sheaf of groups where the group operation is given by pointwise multiplication.
★ ''M'' also has a 'cotangent sheaf' Ω''M''. On each open set ''U'', Ω''M''(''U'') is the set of degree one differential forms on ''U''. Restriction is given by restriction of differential forms. Similarly, for every ''p'' > 0, there is a sheaf Ω''p'' of differential p-forms.
★ If ''M'' is smooth, then for each open set ''U'', we have a set of real-valued distributions on ''U''. Restriction is given by restriction of functions. Then is a sheaf called the 'sheaf of distributions'.
★ If ''X'' is a complex manifold and ''U'' is an open set of ''X'', let be the set of finite-order holomorphic differential operators on ''U''. Letting restriction be given by restriction of functions, we get a sheaf called the 'sheaf of holomorphic differential operators'.
★ For any set ''S'' and any topological space ''X'', there is a constant presheaf ''F'' which has ''F''(''U'') = ''S'' for all ''U'' and restriction maps equal to the identity. ''F'' is not a sheaf: Let ''U'' and ''V'' be disjoint open sets and ''s'' and ''t'' be distinct elements of ''S''. ''s'' determines a section in ''F''(''U''), because ''s'' is in ''S'' and ''S'' = ''F''(''U''). Similarly, ''t'' determines a section in ''F''(''V''). Since ''U'' and ''V'' are disjoint, the hypothesis of the gluing axiom is vacuously true. Consequently there must be a section in ''F''(''U'' ∪ ''V'') which restricts to ''s'' on ''U'' and to ''t'' on ''V'', but that's impossible. So ''F'' is a presheaf, and even a separated presheaf, but not a sheaf.
★ However, there is a sheaf, called the 'constant sheaf on S', which is very similar to the constant presheaf. We let be the set of all functions from ''U'' to ''S'' which are constant on each connected component. In other words, if ''U'' has a single connected component, then is ''S''. If ''U'' has two connected components, then is ''S'' × ''S''; one factor of ''S'' is the section over one component, and the other factor is the section over the other component. Restriction corresponds to restriction of functions. It can be checked that this makes a sheaf.
★ More generally, if ''S'' is an object in a concrete category 'C' which has all set-indexed products, then we define the constant sheaf to be the sheaf which takes an open set ''U'' to the set of all functions ''U'' → ''S'' which are constant on the connected components of ''U''. For example, this can be done when ''S'' is a ring such as 'Z' to get the constant sheaf . If 'C' is a category such as the category of groups or the category of commutative rings, this will give a sheaf of groups or a sheaf of commutative rings, respectively.
★ Fix a point ''x'' in ''X'' and an object ''S'' in a category 'C'. The 'skyscraper sheaf over ''x'' with stalk ''S''' is the sheaf ''S''''x'' defined as follows: If ''U'' is an open set containing ''x'', then ''S''''x''(''U'') = ''S''. If ''U'' does not contain ''x'', then ''S''''x''(''U'') is the terminal object of 'C'. The restriction maps are either the identity on ''S'', if both open sets contain ''x'', or the unique map from ''S'' to the terminal object of 'C'.
Some types of structure are defined by a space and a fixed sheaf on it. For example, a space together with a sheaf of rings is called a ringed space. If the stalks (see below) are all local rings, then it is a locally ringed space. If the sheaf of rings is locally the same as the elements of a commutative ring, we get a scheme.
Here are two examples of presheaves which are not sheaves:
★ Let ''X'' be the two-point topological space {''x'', ''y''} with the discrete topology. Define a presheaf ''F'' as follows: ''F''(∅) = ∅, ''F''({''x''}) = 'R', ''F''({''y''}) = 'R', ''F''({''x'', ''y''}) = 'R' × 'R' × 'R'. The restriction map ''F''({''x'', ''y''}) → ''F''({''x''}) is the projection of 'R' × 'R' × 'R' onto its first coordinate, and the restriction map ''F''({''x'', ''y''}) → ''F''({''y''}) is the projection of 'R' × 'R' × 'R' onto its second coordinate. ''F'' is a presheaf which is not separated: A global section is determined by three numbers, but the values of that section over {''x''} and {''y''} determine only two of those numbers. So while we can glue any two sections over {''x''} and {''y''}, we cannot glue them uniquely.
★ Let ''X'' be the complex plane, and let ''F''(''U'') be the set of bounded holomorphic functions on ''U''. This is not a sheaf because it is not always possible to glue. For example, let ''U''''i'' be the set of all ''z'' such that |''z''| < ''i''. The function ''f''(''z'') = ''z'' is bounded on each ''U''''i''. Consequently we get a section ''s''''i'' on ''U''''i'' which is the restriction of the constant function to ''U''''i''. However, these sections do not glue, because the function ''f'' is not bounded on the complex plane. Consequently ''F'' is a presheaf, but not a sheaf. In fact, ''F'' is separated because it is a sub-presheaf of the sheaf of holomorphic functions.
Morphisms of sheaves
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, which satisfy a compatibility condition.
Let and be two sheaves on ''X'' with values in the category 'C'. A ''morphism'' φ : → takes each open set ''U'' of ''X'' to a morphism φ(''U'') : → , subject to the condition that this morphism is compatible with restriction. In other words, for every open subset ''U'' of an open set ''V'', we must have a commutative diagram:
SheafMorphism-01.png
This compatibility condition says that if we have a section ''s'' in , then mapping ''s'' to its image φ(''U'')(''s'') in and then restricting to ''U'' gives the same result as first restricting to ''U'' and then mapping the restriction to its image in .
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, and the morphisms are morphisms of sheaves. 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, φ is an isomorphism if and only if for each ''U'', φ(''U'') is an isomorphism. The same is true of monomorphisms, but not of epimorphisms. See sheaf cohomology.
Notice that we did not use 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 is then a functor category, the category of contravariant functors from ''O''(''X'') to 'C'.
Turning a presheaf into a sheaf
Main articles: Gluing axiom
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''. ''a'' is called the 'sheaving functor', 'sheafification functor', or 'associated sheaf functor'. There is a natural morphism of presheaves ''i'' : ''F'' → ''aF'' which has the universal property that for any sheaf ''G'' and any morphism of presheaves ''f'' : ''F'' → ''G'', there is a unique morphism of sheaves such that . In fact ''a'' is the adjoint functor to the inclusion functor from the category of sheaves to the category of presheaves, and ''i'' is the unit of the adjunction.
Direct and inverse images
Main articles: Direct image functor
Main articles: Inverse image functor
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.
Concretely, let be a sheaf on ''X''. We define the 'direct image' or 'pushforward' of to be the sheaf on ''Y'' that takes open sets ''U'' of ''Y'' to the object . If ''V'' is an open subset of ''U'', then the restriction map ''res''''V'',''U'' is defined to be the restriction map . It can be checked that this is still a sheaf.
Suppose instead that we have a sheaf on ''Y'' and that we want to transport to ''X'' using ''f''. We will call the result the 'inverse image' or 'pullback' sheaf . If we try to imitate the direct image by setting for each open set ''U'' of ''X'', we immediately run into a problem: ''f''(''U'') is not necessarily open. The best we can do is to approximate it by open sets, and even then we will get a presheaf, not a sheaf. Consequently we define to be the sheaf associated to the presheaf:
:
To define the restriction maps, we use the universal property of direct limits.
It is possible to define the direct image and the inverse image of a morphism of sheaves as well, and using this definition, ''f''
★ and ''f'' -1 become functors. In fact, ''f'' -1 is the left adjoint of ''f''
★ . This implies that there are natural unit and counit morphisms and . However, these are ''almost never'' isomorphisms.
There is a different inverse image functor ''f''
★ which appears when working with sheaves of modules on ringed spaces. It is related to, but not the same as, the inverse image functor ''f'' -1. See the main article on the inverse image functor.
Stalks of a sheaf
Sheaves are defined on open sets, but the underlying topological space ''X'' consists of points. It is reasonable to attempt to isolate the behavior of a sheaf at a single fixed point ''x'' of ''X''. Conceptually speaking, we do this by looking at small neighborhoods of the point. If we look at a sufficiently small neighborhood of ''x'', the behavior of the sheaf on that small neighborhood should be the same as the behavior of at that point. Of course, no single neighborhood will be small enough, so we will have to take a limit of some sort.
To make this precise, remember that if we have an inclusion of an open set ''V'' into an open set ''U'', we get a restriction map . Every restriction map gets us closer to a small neighborhood of ''x'', so to get the local behavior of at ''x'', we want to take a limit over all the open sets and all the restriction maps. In other words, we want to take a direct limit indexed over all the open sets containing ''x''. We define the 'stalk' of at ''x'' to be:
:.
For some categories 'C' this may not exist. However, it exists for most categories which occur in practice, such as the category of sets or most categories of algebraic objects such as abelian groups or rings.
Because we defined the stalk as a direct limit over open sets, there is a natural morphism ''F''(''U'') → ''F''''x'' for any open set ''U'' containing ''x''. This takes a section ''s'' in ''F''(''U'') to its ''germ''. This is a generalization of the usual concept of a germ, which can be recovered by looking at the stalks of the sheaf of continuous functions on ''X''.
Germs are more useful for some sheaves than for others. For example, in the sheaf of analytic functions on an analytic manifold, a germ of a function at a point determines the function in a small neighborhood of a point. This is because the germ records the function's power series expansion, and all analytic functions are by definition equal to their power series. Using analytic continuation, we find that the germ at a point determines the function on any connected open set where the function can be everywhere defined. (This does not imply that all the restriction maps of this sheaf are injective!)
In contrast, for the sheaf of smooth functions on a smooth manifold, germs contain some local information, but are not enough to reconstruct the function on any open neighborhood. For example, let ''f'' : 'R' → 'R' be a bump function which is identically one in a neighborhood of the origin and identically zero far away from the origin. On any sufficiently small neighborhood containing the origin, ''f'' is identically one, so at the origin it has the same germ as the constant function with value 1. Suppose that we want to reconstruct ''f'' from its germ. Even if we know in advance that ''f'' is a bump function, the germ does not tell us how large its bump is. From what the germ tells us, the bump could be infinitely wide, that is, ''f'' could equal the constant function with value 1. We cannot even reconstruct ''f'' on a small open neighborhood ''U'' containing the origin, because we cannot tell whether the bump of ''f'' fits entirely in ''U'' or whether it is so large that ''f'' is identically one in ''U''.
On the other hand, germs of smooth functions can distinguish between the constant function with value one and the function , because the latter function is not identically one on any neighborhood of the origin. This example shows that germs contain more information than the power series expansion of a function, because the power series of is identically one. (This extra information is related to the fact that the stalk of the sheaf of smooth functions at the origin is a non-Noetherian ring. The Krull intersection theorem says that this cannot happen for a Noetherian ring.)
There is another approach to defining a germ which is useful in some contexts. Choose a point ''x'' of ''X'', and let ''i'' be the inclusion of the one point space {''x''} into ''X''. Then the stalk is the same as the inverse image sheaf . Notice that the only open sets of the one point space {''x''} are {''x''} and ∅, and there is no data over the empty set. Over {''x''}, however, we get:
:.
The étale space of a sheaf
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 ''étale space''. If ''F'' is a sheaf over ''X'', then the 'étale space' of ''F'' is a topological space ''E'' together with a local homeomorphism ''π'': ''E'' → ''X''; the sheaf of sections of ''π'' is ''F''. ''E'' is usually a very strange space, 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 covering map.
The étale space ''E'' is constructed from the stalks of ''F'' over ''X''. As a set, it is their disjoint union and ''Ï€'' is the obvious map which 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''. The union of these points (for all such ''s'' and ''x'') is declared to be open in ''E''. Notice that each stalk has the discrete topology. Two morphisms between sheaves determine a continuous map of the corresponding étale spaces which 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.
This gives an example of an étale space over ''X''. An ''étale space'' is a topological space ''E'' together with a continuous map ''π'': ''E'' → ''X'' which is a local homeomorphism such that each fiber of ''π'' has the discrete topology. 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 étale space can also be applied to a presheaf, in which case the sheaf of sections of the étale space recovers the sheaf associated to the given presheaf.
The map ''π'' is an example of what is sometimes called an ''étale map''. "Étale" here means the same thing as "local homeomorphism". However, the terminology "étale map" is more common in contexts where the right analogue of a local homeomorphism of manifolds is not characterized by the property of being a local homeomorphism. This is the case in algebraic geometry. For more information see the article étale morphism.
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 étale space, and let ''Ï€'': ''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 space is a continuous map ''f'' : ''Y'' → ''X'', and a morphism from ''Y'' → ''X'' to ''Z'' → ''X'' is a continuous map ''Y'' → ''Z'' which commutes with the two maps to ''X''. There is a functor Γ from 'Top'/''X'' to the category of sets which 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 Γ(''i'') = (''i''-1''F'')(''U'') agrees with the usual ''F''(''U''), and if ''i'' : {''x''} → ''X'' is the inclusion of a point, then Γ({''x''}) = (''i''-1''F'')({''x''}) is the stalk of ''F'' at ''x''. There is a natural isomorphism
:
which shows that ''E'' represents the functor Γ.
The definition of sheaves by étale spaces is older than the definition given earlier in the article. It is still common in some areas of mathematics such as mathematical analysis.
Sheaf cohomology
Main articles: Sheaf cohomology
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 .
Unfortunately, applying this definition to a computation is nearly impossible. To make computations, we can apply ÄŒech cohomology. ÄŒech 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, ÄŒech cohomology computes the same cohomology groups as the derived functor cohomology. However, for some pathological spaces, ÄŒech 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.
Sites and topoi
Main articles: Grothendieck topology
Main articles: Topos
André Weil's Weil conjectures stated that there was a cohomology theory for algebraic varieties over finite fields which would give an analogue of the Riemann hypothesis. The only natural topology on such a variety, however, is the Zariski topology, but sheaf cohomology in the Zariski topology is badly behaved because there are very few open sets. 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.
History
The first origins of 'sheaf theory' are hard to pin down — they may be co-extensive with the idea of analytic continuation. It took about 15 years for a recognisable, free-standing theory of sheaves to emerge from the foundational work on cohomology.
★ 1936 Eduard ÄŒech introduces the ''nerve'' construction, for associating a simplicial complex to an open covering.
★ 1938 Hassler Whitney gives a 'modern' definition of cohomology, summarizing the work since J. W. Alexander and Kolmogorov first defined ''cochains''.
★ 1943 Norman Steenrod publishes on homology ''with local coefficients''.
★ 1945 Jean Leray publishes work carried out as a POW, motivated by proving fixed point theorems for application to PDE theory; it is the start of sheaf theory and spectral sequences.
★ 1947 Henri Cartan reproves the de Rham theorem by sheaf methods, in correspondence with André Weil (see De Rham-Weil theorem). Leray gives a sheaf definition in his courses via closed sets (the later ''carapaces'').
★ 1948 The Cartan seminar writes up sheaf theory for the first time.
★ 1950 The 'second edition' sheaf theory from the Cartan seminar: the sheaf space (''espace étalé'') definition is used, with stalkwise structure. Supports are introduced, and cohomology with supports. Continuous mappings give rise to spectral sequences. At the same time Kiyoshi Oka introduces an idea (adjacent to that) of a sheaf of ideals, in several complex variables.
★ 1951 The Cartan seminar proves the Theorems A and B based on Oka's work.
★ 1953 The finiteness theorem for coherent sheaves in the analytic theory is proved by Cartan and Jean-Pierre Serre, as is Serre duality.
★ 1954 Serre's paper ''Faisceaux algébriques cohérents'' (published 1955) introduces sheaves into algebraic geometry. These ideas are immediately exploited by Hirzebruch, who writes a major 1956 book on topological methods.
★ 1955 Alexander Grothendieck in lectures in Kansas defines abelian category and ''presheaf'', and by using injective resolutions allows direct use of sheaf cohomology on all topological spaces, as derived functors.
★ 1956 Oscar Zariski's report ''Algebraic sheaf theory, Scientific report on the Second summer Institute : Several complex variables [1954, Boulder (Col.)]'', Part III., Bull. Amer. math. Soc., t. 62, 1956, p. 117-141.
★ 1957 Grothendieck's ''Tohoku'' paper rewrites homological algebra; he proves Grothendieck duality (i.e., Serre duality for possibly singular algebraic varieties).
★ 1958 Godement's book on sheaf theory is published. At around this time Mikio Sato proposes his hyperfunctions, which will turn out to have sheaf-theoretic nature.
★ 1957 onwards: Grothendieck extends sheaf theory in line with the needs of algebraic geometry, introducing: schemes and general sheaves on them, local cohomology, derived categories (with Verdier), and Grothendieck topologies. There emerges also his influential schematic idea of 'six operations' in homological algebra.
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.
See also
★ Gerbe
★ Stack (category theory)
References
★ (oriented towards conventional topological applications)
★
★ (updated edition of a classic using enough sheaf theory to show its power)
★ (advanced techniques such as the derived category and vanishing cycles on the most reasonable spaces)
★ (category theory and toposes emphasised)
★ (concise lecture notes)
★ (pedagogic treatment)
External link
★
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