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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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s 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 presheaf mentioned above which is usually not a sheaf there are further examples of presheaves that are not sheaves let x displaystyle x be the two point topological space x y displaystyle x y with the discrete topology define a presheaf f displaystyle f as follows f f x r f y r f x y r r r displaystyle f varnothing varnothing f x mathbb r f y mathbb r f x y mathbb r times mathbb r times mathbb r the restriction map f x y f x displaystyle f x y to f x is the projection of r r r displaystyle mathbb r times mathbb r times mathbb r onto its first coordinate and the restriction map f x y f y displaystyle f x y to f y is the projection of r r r displaystyle mathbb r times mathbb r times mathbb r onto its second coordinate f displaystyle f is a presheaf that is not separated a global section is determined by three numbers but the values of that section over x displaystyle x and y displaystyle y determine only two of those numbers so while we can glue any two sections over x displaystyle x and y displaystyle y we cannot glue them uniquely let x r displaystyle x mathbb r be the real line and let f u displaystyle f u be the set of bounded continuous functions on u displaystyle u this is not a sheaf because it is not always possible to glue for example let u i displaystyle u_ i be the set of all x displaystyle x such that x i displaystyle x i the identity function f x x displaystyle f x x is bounded on each u i displaystyle u_ i consequently we get a section s i displaystyle s_ i on u i displaystyle u_ i however these sections do not glue because the function f displaystyle f is not bounded on the real line consequently f displaystyle f is a presheaf but not a sheaf in fact f displaystyle f is separated because it is a sub presheaf of the sheaf of continuous functions motivating sheaves from complex analytic spaces and algebraic geometry edit one of the historical motivations for sheaves have come from studying complex manifolds 4 complex analytic geometry 5 and scheme theory from algebraic geometry this is because in all of the previous cases we consider a topological space x displaystyle x together with a structure sheaf o displaystyle mathcal o giving it the structure of a complex manifold complex analytic space or scheme this perspective of equipping a topological space with a sheaf is essential to the theory of locally ringed spaces see below technical challenges with complex manifolds edit one of the main historical motivations for introducing sheaves was constructing a device which keeps track of holomorphic functions on complex manifolds for example on a compact complex manifold x displaystyle x like complex projective space or the vanishing locus in projective space of a homogeneous polynomial the only holomorphic functions f x c displaystyle f x to mathbb c are the constant functions 6 7 this means there exist two compact complex manifolds x x displaystyle x x which are not isomorphic but nevertheless their rings of global holomorphic functions denoted h x h x displaystyle mathcal h x mathcal h x are isomorphic contrast this with smooth manifolds where every manifold m displaystyle m can be embedded inside some r n displaystyle mathbb r n hence its ring of smooth functions c m displaystyle c infty m comes from restricting the smooth functions from c r n displaystyle c infty mathbb r n of which there exist plenty another complexity when considering the ring of holomorphic functions on a complex manifold x displaystyle x is given a small enough open set u x displaystyle u subseteq x the holomorphic functions will be isomorphic to h u h c n displaystyle mathcal h u cong mathcal h mathbb c n sheaves are a direct tool for dealing with this complexity since they make it possible to keep track of the holomorphic structure on the underlying topological space of x displaystyle x on arbitrary open subsets u x displaystyle u ...