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INTRODUCTION Sheaves are used in Topology , Algebraic Geometry and Differential Geometry whenever one wants to keep track of algebraic data that vary with every open set of the given geometrical space. They are a ''global tool'' to study objects which ''vary locally'' (that is, depend on the open sets). As such, they are a natural instrument to study the global behaviour of entities which are of local nature, such as Open Set s, Analytic Function s, Manifold s, and so on. For a typical example, consider a topological space ''X'', and for every open set ''U'' in ''X'', let ''F''(''U'') be the set of all and the restriction maps are Ring Homomorphism s, making ''F'' a sheaf of rings on ''X''. For a very similar example, consider a differentiable Manifold ''X'', and for every open set ''U'' of ''X'', let ''F''(''U'') be the set of differentiable functions ''U'' R. Here too, gluing works and we obtain a sheaf of rings on ''X''. Another sheaf on ''X'' assigns to every open set ''U'' of ''X'' the Vector Space of all differentiable Vector Field s defined on ''U''. Restriction and gluing of vector fields works like that of functions, and we obtain a sheaf of vector spaces on the manifold ''X''. THE FORMAL DEFINITION To define sheaves we will proceed in two steps. The first step is to introduce the concept of a ''presheaf'', which captures the idea of associating local information to a topological space. The second step is to introduce an additional axiom, called the ''gluing axiom'' or the ''sheaf axiom'', which captures the idea of gluing local information to get global information. Definition of a presheaf Suppose ''X'' is a topological space, and C is a Category (often, this is the category of Set s, the category of Abelian Group s, the category of Commutative Ring s, or the category of Modules over a fixed Ring ). A '''presheaf''' ''F'' of objects in C on the space ''X'' is given by the following data:
This definition can be expressed naturally in terms of Category Theory , see Presheaf (category Theory) . First we define the category of Open Set s on ''X'' to be the category Top''X'' whose objects are the open sets of ''X'' and whose morphisms are inclusions. Top''X'' is then the category corresponding to the partial order ⊂ on the open sets of ''X''. A C-presheaf on ''X'' is then a Contravariant Functor from Top''X'' to C. 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''. (This is by analogy with sections of Fiber Bundle s; see below) If C is a Concrete Category , then each element of ''F(U)'' is called a ''section''. ''F(U)'' is also often denoted Γ(''U'',''F''). The gluing axiom ''See main article Gluing Axiom for a higher-level discussion'' Sheaves are presheaves on which sections over small open sets can be glued to give sections over larger open sets. Here the gluing axiom will be given in a form that requires C to be a concrete category. Let ''U'' be the union of the collection of open sets {''Ui''}. For each ''Ui'', choose a section ''fi'' on ''Ui''. We say that the ''fi'' are ''compatible'' if for any ''i'' and ''j'', :res''Ui'',''Ui''∩''Uj''(''fi'') = res''Uj'',''Ui''∩''Uj''(''fj''). Intuitively speaking, if the ''fi'' represent functions, this says that any two compatible functions agree where they overlap. The sheaf axiom says that we can produce from the ''fi'' a unique section ''f'' over ''U'' whose restriction to each ''Ui'' is ''fi'', i.e., res''U'',''Ui''(''f'')=''fi''. Sometimes this is split into two axioms, one guaranteeing existence, and the other guaranteeing uniqueness. A presheaf satisfying only the uniqueness part of the sheaf axiom is sometimes called a monopresheaf. EXAMPLES In addition to the sheaves of continuous functions, differentiable functions and vector fields given in the introduction, sheaves of sections are very important examples. Suppose ''E'' and ''X'' are topological spaces and π : ''E'' → ''X'' is a continuous map. For every open set ''U'' in ''X'', let ''F''(''U'') be the set of all continuous maps ''f'' : ''U'' ''E'' such that π(''f''(''x'')) = ''x'' for all ''x'' in ''U''. Such a function ''f'' is called a section of π. It is not difficult to check that ''F'' is a sheaf of sets on ''X''. In fact, every sheaf of sets on ''X'' is essentially of this type, for very special maps π; see Below . Given a sheaf ''F'' on ''X'', the elements of ''F''(''X'') are also called the global sections, a terminology motivated by the previous example. Further examples:
MORPHISMS OF SHEAVES Let ''F'' and ''G'' be two sheaves on ''X'' both with values in the category ''C''. We define a for each pair of open sets ''U'' ⊆ ''V'' in ''X''. If ''F'' and ''G'' are considered as Contravariant Functor s from Top''X'' to ''C'' then a morphism between them is nothing more than a Natural Transformation . With this definition the set of all ''C''-valued sheaves on ''X'' forms a category (a Functor Category ). An ''isomorphism'' of sheaves on ''X'' is just an isomorphism in this category. One can generalize this notion to morphisms between sheaves on different spaces. Let ''f'' : ''X'' ''Y'' be a Continuous Function between two topological spaces, and let ''F'' be a sheaf on ''X'' and ''G'' a sheaf on ''Y'' both with values in ''C''. Then a morphism from ''G'' to ''F'' relative to ''f'' is given by a family of morphisms φ''U'' : ''G''(''U'') ''F''(''f''−1(''U'')) for each open set ''U'' in ''Y'' such that the diagram commutes for each pair of open sets ''U'' ⊆ ''V'' in ''Y''. The previous definition is the special case resulting when ''f'' is the identity map on ''X''. The category theoretical description is slightly more complicated in the general case. Let Top be the contravariant functor from the Category Of Topological Spaces Top to the Category Of Small Categories '''Cat''' which sends each space ''X'' to the poset category of its open sets Top''X''. Here Top(''f'') is a covariant functor from Top''Y'' to Top''X'' sending each open set to its Preimage . Composing ''F'' with Top(''f'') we obtain a contravariant functor from Top''Y'' to ''C''. A morphism from ''G'' to ''F'' relative to ''f'' is then a natural transformation from ''G'' to ''F'' ○ Top(''f''). Note that all of the above makes sense if we are working only with presheaves instead of sheaves. STALKS OF A SHEAF AT A POINT AND GERMS OF FUNCTIONS Fix a point ''x'' of ''X''. We would like to study the behavior of ''F'' near the point ''x''. In Analytical terms, we would like to somehow take the limit as we get nearer and nearer to the point ''x''. The corresponding concept is to take the Direct Limit of ''F''(''N'') as ''N'' runs over the open neighbourhoods of ''x'' ordered by inclusion (in categorical terminology, this is an example of a Colimit ). We denote this limit by ''Fx'' and call it the stalk of ''F'' at ''x''. If ''F'' is a '''C'''-valued sheaf on ''X'', then the stalk ''Fx'' is an object of '''C''', for '''C''' a category such as the Category Of Abelian Groups or the Category Of Commutative Rings . For any open set ''U'' containing ''x'' there is a morphism from ''F''(''U'') to ''Fx''. If C is a concrete category, then applying this morphism to an element ''f'' in ''F''(''U'') gives an element of ''Fx'' called the '''germ''' of ''f'' at ''x''. This corresponds to the notion of '' Germ Of A Function '' used elsewhere in mathematics. Intuitively, the germ of the function ''f'' at ''x'' describes the local behavior of ''f'' at the point ''x''; it is a kind of 'ghost' of ''f'', looked at only very near ''x''. See also the detailed example given at Local Ring . For some sheaves, germs behave well, and can give good local information; the germ of an Analytic Function around a point determines the function in a small neighbourhood of the point, using its Power Series expansion. However, some sheaves do not behave well; the germ of a Smooth Function at any point does not determine the function in any small neighbourhood of the point. As an example, take any Bump Function . Its local behavior on the interval where it is one is that of a constant function, but knowing that a bump function is the constant one near a given point does not tell you where the function begins to decay; from its local behavior, you cannot even conclude that it is a bump function! THE SHEAF SPACE (''ESPACE éTALé'') OF A SHEAF In early developments of sheaf theory, it was shown that giving a sheaf ''F'' on ''X'' is as good as giving a certain topological space ''E'' together with a continuous map from ''E'' to ''X''. More precisely: to every sheaf ''F'' of sets on ''X'' there exists a Local Homeomorphism :π: ''E'' ''X'' such that ''F'' is isomorphic to the sheaf of sections of π that was described in the example section above. Furthermore, the space ''E'' is determined , and we take the disjoint union of all the stalks, with π mapping all of the stalks ''F''''x'' to ''x''. The topology on this space of stalks can be chosen so that the sheaf ''F'' can be recovered as the sheaf of sections of π. At a higher level of abstraction, we can say that the category of sheaves of sets on ''X'' is Equivalent to the category of local homeomorphisms to ''X''. (One can also consider such a space in the light of the theory of Representable Functor s; the history shows that this theory developed also in the mid-1950s.) The space ''E'' was called espace étalé in Godement's influential book about Homological Algebra and sheaf theory (''Topologie Algébrique et Théorie des Faisceaux'', R. Godement ); in that book, sheaves are in fact ''defined'' as coming from sections of local homeomorphisms; the functorial approach we gave above came later and is now more common. The space should not be referred to as an "etale space", as the word "etale" has other mathematical meanings. The above considerations remain true for sheaves of C on ''X'': we can still form the space of stalks, each stalk is an object in C, and the sections naturally become objects in ''C'' as well. Given an arbitrary continuous map ''g'' : ''Z'' ''X'', the corresponding sheaf of sections gives rise in the above manner to a space of stalks ''E'' and a local homeomorphism π : ''E'' ''X''. In a sense this deals with all the ' Ramification ' in the map ''g'', in the 'best possible way'. This may be expressed by Adjoint Functor s; but is also important as an intuition about sheaves of sets. This collection of ideas is related to Topos theory, but in a sense that more general notion of sheaf moves away from geometric intuition. GENERALIZATIONS It is possible to define a Cohomology theory for sheaves of abelian groups ( Sheaf Cohomology ) that can give much useful, more concrete information. The main issue is the existence of the Long Exact Sequence coming from an Exact Sequence of sheaves. In applications emphasis was placed on sheaves on spaces that were less Well-behaved than Finite Complex es. For example, in algebraic geometry spaces carrying the Zariski Topology are rarely Hausdorff Space s. The algebraic geometry case was first tackled by Jean-Pierre Serre by developing an analogue of Čech Cohomology ; this worked, though in general the construction doesn't have such good properties. Then Alexander Grothendieck used Derived Functor s of the global section functor, providing a more definitive solution. Grothendieck was motivated to develop a cohomology theory for sheaves that would give stronger results, and that would, in particular, allow a proof of the Weil Conjectures . By precisely analyzing the properties of ''X'' needed to define sheaves, he defined the notion of a Grothendieck Topology on a category (this came in a somewhat roundabout fashion — see Background And Genesis Of Topos Theory ). A category together with a Grothendieck topology is called a ''site''. It is possible to define the notion of a sheaf on any site. The notion of sites later led Lawvere to develop the notion of an Elementary Topos . 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 .
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 REFERENCES
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