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displaystyle (560), the (476), mathcal (184), sheaf (180), sheaves (136), and (125), for (98), open (78), this (73), are (68), that (67), with (65), #cohomology (61), space (60), functions (44), presheaf (42), theory (39), sections (38), edit (37), mathbb (37), topological (34), complex (34), spaces (33), set (33), from (32), there (32), sets (32), not (32), category (31), such (31), can (30), which (29), groups (27), example (26), functor (26), also (25), map (25), called (24), over (24), image (24), restriction (24), res (24), all (23), grothendieck (23), any (23), morphism (23), algebraic (22), continuous (22), geometry (21), abelian (21), then (21), its (20), given (20), morphisms (20), defined (19), constant (19), mathematics (18), topology (18), these (18), locally (18), modules (18), some (18), derived (17), holomorphic (17), coherent (17), general (17), denoted (17), above (17), presheaves (17), categories (16), étalé (16), data (16), between (16), natural (16), two (16), 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functoriality 2 5 1 direct image 2 5 2 inverse image 2 5 3 extension by zero 3 complements toggle complements subsection 3 1 sheaves in more general categories 3 2 ringed spaces and sheaves of modules 3 2 1 finiteness conditions for sheaves of modules 3 3 the étalé space of a sheaf 4 sheaf cohomology toggle sheaf cohomology subsection 4 1 computing sheaf cohomology 4 2 derived categories of sheaves 4 2 1 derived categories of coherent sheaves and the grothendieck group 5 sites and topoi 6 history 7 see also 8 notes 9 references toggle the table of contents sheaf mathematics 29 languages العربية català čeština deutsch ελληνικά esperanto español euskara فارسی suomi français עברית italiano 日本語 한국어 македонски မြန်မာဘာသာ nederlands norsk bokmål polski português русский српски srpski svenska türkçe українська tiếng việt 粵語 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia tool to track locally defined data attached to the open sets of a topological space this article is about sheaves on topological spaces for sheaves on a site see grothendieck topology and topos look up sheaf in wiktionary the free dictionary in mathematics a sheaf pl sheaves is a tool for systematically tracking data such as sets abelian groups rings attached to the open sets of a topological space and defined locally with regard to them for example for each open set the data could be the ring of continuous functions defined on that open set such data are well behaved in that they can be restricted to smaller open sets and also the data assigned to an open set are equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set intuitively every datum is the sum of its constituent data the field of mathematics that studies sheaves is called sheaf theory sheaves are understood conceptually as general and abstract objects their precise definition is rather technical they are specifically defined as sheaves of sets or as sheaves of rings for example depending on the type of data assigned to the open sets there are also maps or morphisms from one sheaf to another sheaves of a specific type such as sheaves of abelian groups with their morphisms on a fixed topological space form a category on the other hand to each continuous map there is associated both a direct image functor taking sheaves and their morphisms on the domain to sheaves and morphisms on the codomain and an inverse image functor operating in the opposite direction these functors and certain variants of them are essential parts of sheaf theory due to their general nature and versatility sheaves have several applications in topology and especially in algebraic and differential geometry first geometric structures such as that of a differentiable manifold or a scheme can be expressed in terms of a sheaf of rings on the space in such contexts several geometric constructions such as vector bundles or divisors are naturally specified in terms of sheaves second sheaves provide the framework for a very general cohomology theory which encompasses also the usual topological cohomology theories such as singular cohomology especially in algebraic geometry and the theory of complex manifolds sheaf cohomology provides a powerful link between topological and geometric properties of spaces sheaves also provide the basis for the theory of d modules which provide applications to the theory of differential equations in addition generalisations of sheaves to more general settings than topological spaces such as the notion of a sheaf on a category with respect to some grothendieck topology have provided applications to mathematical logic and to number theory definitions and examples edit in many mathematical branches several structures defined on a topological space x displaystyle x e g a differentiable manifold can be naturally localised or restricted to open subsets u x displaystyle u subseteq x typical examples include continuous real valued or complex valued functions n displaystyle n times differentiable real valued or complex valued functions bounded real valued functions vector fields and sections of any vector bundle on the space the ability to restrict data to smaller open subsets gives rise to the concept of presheaves roughly speaking sheaves are then those presheaves where local data can be glued to global data presheaves edit see also presheaf category theory let x displaystyle x be a topological space a presheaf f displaystyle mathcal f of sets on x displaystyle x consists of the following data for each open set u x displaystyle u subseteq x there exists a set f u displaystyle mathcal f u this set is also denoted γ u f displaystyle gamma u mathcal f the elements in this set are called the sections of f displaystyle mathcal f over u displaystyle u the sections of f displaystyle mathcal f over x displaystyle x are called the global sections of f displaystyle mathcal f for each inclusion of open sets v u displaystyle v subseteq u a function res v u f u f v displaystyle operatorname res _ v u colon mathcal f u rightarrow mathcal f v in view of many of the examples below the morphisms res v u displaystyle text res _ v u are called restriction morphisms if s f u displaystyle s in mathcal f u then its restriction res v u s displaystyle text res _ v u s is often denoted s v displaystyle s _ v by analogy with restriction of functions the restriction morphisms are required to satisfy two additional functorial properties for every open set u displaystyle u of x displaystyle x the restriction morphism res u u f u f u displaystyle operatorname res _ u u colon mathcal f u rightarrow mathcal f u is the identity morphism on f u displaystyle mathcal f u if we have three open sets w v u displaystyle w subseteq v subseteq u then the composite res w v res v u res w u displaystyle text res _ w v circ text res _ v u text res _ w u informally the second axiom says it does not matter whether we restrict to w displaystyle w in one step or restrict first to v displaystyle v then to w displaystyle w a concise functorial reformulation of this definition is given further below many examples of presheaves come from different classes of functions to any u displaystyle u one can assign the set c 0 u displaystyle c 0 u of continuous real valued functions on u displaystyle u the restriction maps are then just given by restricting a continuous function on u displaystyle u to a smaller open subset v u displaystyle v subseteq u which again is a continuous function the two presheaf axioms are immediately checked thereby giving an example of a presheaf this can be extended to a presheaf of holomorphic functions h displaystyle mathcal h and a presheaf of smooth functions c displaystyle c infty another common class of examples is assigning to u displaystyle u the set of constant real valued functions on u displaystyle u this presheaf is called the constant presheaf associated to r displaystyle mathbb r and is denoted r _ psh displaystyle underline mathbb r text psh sheaves edit given a presheaf a natural question to ask is to what extent its sections over an open set u displaystyle u are specified by their restrictions to open subsets of u displaystyle u a sheaf is a presheaf whose sections are in a technical sense uniquely determined by their restrictions axiomatically a sheaf is a presheaf that satisfies both of the following axioms locality suppose u displaystyle u is an open set u i i i displaystyle u_ i _ i in i is an open cover of u displaystyle u with u i u displaystyle u_ i subseteq u for all i i displaystyle i in i and s t f u displaystyle s t in mathcal f u are sections if s u i t u i displaystyle s _ u_ i t _ u_ i for all i i displaystyle i in i then s t displaystyle s t gluing suppose u displaystyle u is an open set u i i i displaystyle u_ i _ i in i is an open cover of u displaystyle u with u i u displaystyle u_ i subseteq u for all i i displaystyle i in i and s i f u i i i displaystyle s_ i in mathcal f u_ i _ i in i is a family of sections if all pairs of sections agree on the overlap of their domains that is if s i u i u j s j u i u j displaystyle s_ i _ u_ i cap u_ j s_ j _ u_ i cap u_ j for all i j i displaystyle i j in i then there exists a section s f u displaystyle s in mathcal f u such that s u i s i displaystyle s _ u_ i s_ i for all i i displaystyle i in i 1 sections over two opens of the two point space gluing compatible local sections to a section over the union in both of these axioms the hypothesis on the open cover is equivalent to the assumption that i i u i u textstyle bigcup _ i in i u_ i u the section s displaystyle s whose existence is guaranteed by axiom 2 is called the gluing concatenation or collation of the sections s i displaystyle s_ i by axiom 1 it is unique sections s i displaystyle s_ i and s j displaystyle s_ j satisfying the agreement precondition of axiom 2 are often called compatible thus axioms 1 and 2 together state that any collection of pairwise compatible sections can be uniquely glued together a separated presheaf or monopresheaf is a presheaf satisfying axiom 1 2 the presheaf consisting of continuous functions mentioned above is a sheaf this assertion reduces to checking that given continuous functions f i u i r displaystyle f_ i u_ i to mathbb r which agree on the intersections u i u j displaystyle u_ i cap u_ j there is a unique continuous function f u r displaystyle f u to mathbb r whose restriction equals the f i displaystyle f_ i by contrast the constant presheaf is usually not a sheaf as it fails to satisfy the locality axiom on the empty set this is explained in more detail at constant sheaf presheaves and sheaves are typically denoted by capital letters f displaystyle f being particularly common presumably for the french word for sheaf faisceau use of calligraphic letters such as f displaystyle mathcal f is also common it can be shown that to specify a sheaf it is enough to specify its restriction to the open sets of a basis for the topology of the underlying space moreover it can also be shown that it is enough to verify the sheaf axioms above relative to the open sets of a covering this observation is used to construct another example which is crucial in algebraic geometry namely quasi coherent sheaves here the topological space in question is the spectrum of a commutative ring r displaystyle r whose points are the prime ideals p displaystyle mathfrak p in r displaystyle r the open sets d f p r f p displaystyle d_ f mathfrak p subseteq r f notin mathfrak p form a basis for the zariski topology on this space given an r displaystyle r module m displaystyle m there is a sheaf denoted by m displaystyle tilde m on the spec r displaystyle operatorname spec r that satisfies m d f m 1 f displaystyle tilde m d_ f m 1 f where m 1 f displaystyle m 1 f is the localization of m displaystyle m at f displaystyle f there is another characterization of sheaves that is equivalent to the previously discussed a presheaf f displaystyle mathcal f is a sheaf if and only if for any open u displaystyle u and any open cover u a displaystyle u_ a of u displaystyle u f u displaystyle mathcal f u is the fibre product f u f u a f u a u b f u b displaystyle mathcal f u cong mathcal f u_ a times _ mathcal f u_ a cap u_ b mathcal f u_ b this characterization is useful in construction of sheaves for example if f g displaystyle mathcal f mathcal g are abelian sheaves then the kernel of sheaves morphism f g displaystyle mathcal f to mathcal g is a sheaf since projective limits commute with projective limits on the other hand the cokernel is not always a sheaf because inductive limits do not necessarily commute with projective limits one way to fix this is to consider noetherian topological spaces all open sets are compact so that the cokernel is a sheaf since finite projective limits commute with inductive limits further examples edit sheaf of sections of a continuous map edit any continuous map f y x displaystyle f y to x of topological spaces determines a sheaf γ y x displaystyle gamma y x on x displaystyle x by setting γ y x u s u y f s id u displaystyle gamma y x u s u to y f circ s operatorname id _ u any such s displaystyle s is commonly called a section of f displaystyle f and this example is the reason why the elements in f u displaystyle mathcal f u are generally called sections this construction is especially important when f displaystyle f is the projection of a fiber bundle onto its base space for example the sheaves of smooth functions are the sheaves of sections of the trivial bundle another example the sheaf of sections of c exp c 0 displaystyle mathbb c stackrel exp longrightarrow mathbb c setminus 0 is the sheaf which assigns to any u c 0 displaystyle u subseteq mathbb c setminus 0 the set of branches of the complex logarithm on u displaystyle u given a point x displaystyle x and an abelian group s displaystyle s the skyscraper sheaf s x displaystyle s_ x is defined as follows if u displaystyle u is an open set containing x displaystyle x then s x u s displaystyle s_ x u s if u displaystyle u does not contain x displaystyle x then s x u 0 displaystyle s_ x u 0 the trivial group the restriction maps are either the identity on s displaystyle s if both open sets contain x displaystyle x or the zero map otherwise sheaves on manifolds edit on an n displaystyle n dimensional c k displaystyle c k manifold m displaystyle m there are a number of important sheaves such as the sheaf of j displaystyle j times continuously differentiable functions o m j displaystyle mathcal o _ m j with j k displaystyle j leq k its sections on some open u displaystyle u are the c j displaystyle c j functions u r displaystyle u to mathbb r for j k displaystyle j k this sheaf is called the structure sheaf and is denoted o m displaystyle mathcal o _ m the nonzero c k displaystyle c k functions also form a sheaf denoted o x displaystyle mathcal o _ x times differential forms of degree p displaystyle p also form a sheaf ω m p displaystyle omega _ m p in all these examples the restriction morphisms are given by restricting functions or forms the assignment sending u displaystyle u to the compactly supported functions on u displaystyle u is not a sheaf since there is in general no way to preserve this property by passing to a smaller open subset instead this forms a cosheaf a dual concept where the restriction maps go in the opposite direction than with sheaves 3 however taking the dual of these vector spaces does give a sheaf the sheaf of distributions presheaves that are not sheaves edit in addition to the constant ...
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