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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a sheaf is a tool for systematically tracking data (such as sets,
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
s, rings) attached to the
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suf ...
s of a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
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 is well behaved in that it can be restricted to smaller open sets, and also the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set (intuitively, every piece of data is the sum of its parts). The field of mathematics that studies sheaves is called sheaf theory. Sheaves are understood conceptually as general and abstract objects. Their correct 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 A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes. Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
(or
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
s) from one sheaf to another; sheaves (of a specific type, such as sheaves of
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
s) with their
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
s on a fixed topological space form a
category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...
. On the other hand, to each
continuous map In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
there is associated both a
direct image functor In mathematics, the direct image functor is a construction in sheaf theory that generalizes the global sections functor to the relative case. It is of fundamental importance in topology and algebraic geometry. Given a sheaf ''F'' defined on a topolo ...
, taking sheaves and their morphisms on the
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...
to sheaves and morphisms on the
codomain In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either the ...
, and an
inverse image functor In mathematics, specifically in algebraic topology and algebraic geometry, an inverse image functor is a contravariant construction of sheaves; here “contravariant” in the sense given a map f : X \to Y, the inverse image functor is a functor ...
operating in the opposite direction. These
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
s, 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 Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
. First, geometric structures such as that of a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
or a
scheme A scheme is a systematic plan for the implementation of a certain idea. Scheme or schemer may refer to: Arts and entertainment * ''The Scheme'' (TV series), a BBC Scotland documentary series * The Scheme (band), an English pop band * ''The Schem ...
can be expressed in terms of a sheaf of rings on the space. In such contexts, several geometric constructions such as
vector bundles In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
or
divisors In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
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 In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
. Especially in algebraic geometry and the theory of
complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a com ...
s, 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 equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s. In addition, generalisations of sheaves to more general settings than topological spaces, such as
Grothendieck topology In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category ''C'' that makes the objects of ''C'' act like the open sets of a topological space. A category together with a choice of Grothendieck topology is ca ...
, have provided applications to
mathematical logic Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ...
and to
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
.


Definitions and examples

In many mathematical branches, several structures defined on a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
X (e.g., a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
) can be naturally ''localised'' or ''restricted'' to
open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' (YF ...
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
s U \subset X: typical examples include
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
-valued or
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
-valued functions, n-times
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
(real-valued or complex-valued) functions, bounded real-valued functions, vector fields, and
sections Section, Sectioning or Sectioned may refer to: Arts, entertainment and media * Section (music), a complete, but not independent, musical idea * Section (typography), a subdivision, especially of a chapter, in books and documents ** Section sig ...
of any
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
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

Let X be a topological space. A ''presheaf of sets'' F on X consists of the following data: *For each open set U of X, a set F(U). This set is also denoted \Gamma(U, F). The elements in this set are called the ''sections'' of F over U. The sections of F over X are called the ''global sections'' of F. *For each inclusion of open sets V \subseteq U, a function \operatorname_ \colon F(U) \rightarrow F(V). In view of many of the examples below, the morphisms \text_ are called ''restriction morphisms''. If s \in F(U), then its restriction \text_(s) is often denoted s, _V by analogy with restriction of functions. The restriction morphisms are required to satisfy two additional (
functorial In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ma ...
) properties: *For every open set U of X, the restriction morphism \operatorname_ \colon F(U) \rightarrow F(U) is the identity morphism on F(U). *If we have three open sets W \subseteq V \subseteq U, then the
composite Composite or compositing may refer to: Materials * Composite material, a material that is made from several different substances ** Metal matrix composite, composed of metal and other parts ** Cermet, a composite of ceramic and metallic materials ...
\text_\circ\text_=\text_ 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''. A concise functorial reformulation of this definition is given further below. Many examples of presheaves come from different classes of functions: to any ''U'', one can assign the set C^0(U) of continuous real-valued functions on ''U''. The restriction maps are then just given by restricting a continuous function on ''U'' to a smaller open subset ''V'', 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 sheaf of holomorphic functions \mathcal(-) and a sheaf of smooth functions C^\infty(-). Another common class of examples is assigning to U the set of constant real-valued functions on U. This presheaf is called the ''constant presheaf'' associated to \mathbb and is denoted \underline^.


Sheaves

Given a presheaf, a natural question to ask is to what extent its sections over an open set ''U'' are specified by their restrictions to smaller open sets U_i of an
open cover In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\s ...
\mathcal = \_ of ''U''. A ''sheaf'' is a presheaf that satisfies both of the following two additional axioms: # (''Locality'') Suppose U is an open set, \_ is an open cover of U, and s, t \in F(U) are sections. If s, _ = t, _ for all i \in I, then s = t. # ( ''Gluing'') Suppose U is an open set, \_ is an open cover of U, and \_ is a family of sections. If all pairs of sections agree on the overlap of their domains, that is, if s_i, _ = s_j, _ for all i, j \in I, then there exists a section s \in F(U) such that s, _ = s_i for all i \in I. The section ''s'' whose existence is guaranteed by axiom 2 is called the ''gluing'', ''concatenation'', or ''collation'' of the sections ''s''''i''. By axiom 1 it is unique. Sections ''s_i'' and ''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. The presheaf consisting of continuous functions mentioned above is a sheaf. This assertion reduces to checking that, given continuous functions f_i : U_i \to \R which agree on the intersections U_i \cap U_j, there is a unique continuous function f: U \to \R whose restriction equals the 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 Constant or The Constant may refer to: Mathematics * Constant (mathematics), a non-varying value * Mathematical constant, a special number that arises naturally in mathematics, such as or Other concepts * Control variable or scientific cons ...
). Presheaves and sheaves are typically denoted by capital letters, F being particularly common, presumably for the French word for sheaf, ''faisceau''. Use of calligraphic letters such as \mathcal 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 Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
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, whose points are the
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...
s p in R. The open sets D_f := \ form a basis for the
Zariski topology In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
on this space. Given an R-module M, there is a sheaf, denoted by \tilde M on the Spec R, that satisfies :\tilde M(D_f) := M /f the
localization Localization or localisation may refer to: Biology * Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence * Localization of sensation, ability to tell what part of the body is a ...
of M at f.


Further examples


Sheaf of sections of a continuous map

Any continuous map f:Y\to X of topological spaces determines a sheaf \Gamma(Y/X) on X by setting :\Gamma(Y/X)(U) = \. Any such s is commonly called a
section Section, Sectioning or Sectioned may refer to: Arts, entertainment and media * Section (music), a complete, but not independent, musical idea * Section (typography), a subdivision, especially of a chapter, in books and documents ** Section sign ...
of ''f'', and this example is the reason why the elements in F(U) are generally called sections. This construction is especially important when f is the projection of a
fiber bundle In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
onto its base space. For example, the sheaves of smooth functions are the sheaves of sections of the
trivial bundle In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
. Another example: the sheaf of sections of :\C \stackrel \to \C\setminus \ is the sheaf which assigns to any ''U'' the set of branches of the
complex logarithm In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
on ''U''. Given a point x and an abelian group S, the skyscraper sheaf S_x is 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)=0, the
trivial group In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usually ...
. The restriction maps are either the identity on S, if both open sets contain x, or the zero map otherwise.


Sheaves on manifolds

On an n-dimensional C^k-manifold M, there are a number of important sheaves, such as the sheaf of j-times continuously differentiable functions \mathcal^j_M (with j \leq k). Its sections on some open U are the C^j-functions U \to \R. For j = k, this sheaf is called the ''structure sheaf'' and is denoted \mathcal_M. The nonzero C^k functions also form a sheaf, denoted \mathcal_X^\times.
Differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s (of degree p) also form a sheaf \Omega^p_M. In all these examples, the restriction morphisms are given by restricting functions or forms. The assignment sending U to the compactly supported functions on 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 In topology, a branch of mathematics, a cosheaf with values in an ∞-category ''C'' that admits colimits is a functor ''F'' from the category of open subsets of a topological space ''X'' (more precisely its nerve) to ''C'' such that *(1) The ''F ...
, a dual concept where the restriction maps go in the opposite direction than with sheaves. However, taking the dual of these vector spaces does give a sheaf, the sheaf of distributions.


Presheaves that are not sheaves

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 be the two-point topological space \ with the discrete topology. Define a presheaf F as follows: F(\varnothing) = \,\ F(\) = \R,\ F(\) = \R,\ F(\) = \R\times\R\times\RThe restriction map F(\) \to F(\) is the projection of \R \times\R\times\R onto its first coordinate, and the restriction map F(\) \to F(\) is the projection of \R \times\R\times\R onto its second coordinate. F is a presheaf that is not separated: a global section is determined by three numbers, but the values of that section over \ and \ determine only two of those numbers. So while we can glue any two sections over \ and \, we cannot glue them uniquely. * Let X = \R be the
real line In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
, and let F(U) be the set of bounded continuous 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 x such that , x, . The identity function f(x)=x is bounded on each U_i. Consequently we get a section s_i on U_i. However, these sections do not glue, because the function f is not bounded on the real line. Consequently F is a presheaf, but not a sheaf. In fact, F is separated because it is a sub-presheaf of the sheaf of continuous functions.


Motivating sheaves from complex analytic spaces and algebraic geometry

One of the historical motivations for sheaves have come from studying
complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a com ...
s,
complex analytic geometry In mathematics, and in particular differential geometry and complex geometry, a complex analytic variety Complex analytic variety (or just variety) is sometimes required to be irreducible and (or) Reduced ring, reduced or complex analytic space i ...
, and
scheme theory In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations ''x'' = 0 and ''x''2 = 0 define the same algebraic variety but different sc ...
from
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
. This is because in all of the previous cases, we consider a topological space X together with a structure sheaf \mathcal 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

One of the main historical motivations for introducing sheaves was constructing a device which keeps track of
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
s on
complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a com ...
s. For example, on a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
complex manifold X (like
complex projective space In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a ...
or the
vanishing locus In geometry, a locus (plural: ''loci'') (Latin word for "place", "location") is a set of all points (commonly, a line, a line segment, a curve or a surface), whose location satisfies or is determined by one or more specified conditions.. In ...
of a
homogeneous polynomial In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; t ...
), the ''only'' holomorphic functions
f:X \to \C
are the constant functions. This means there could exist two compact complex manifolds X,X' which are not isomorphic, but nevertheless their ring of global holomorphic functions, denoted \mathcal(X), \mathcal(X'), are isomorphic. Contrast this with
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
s where every manifold M can be embedded inside some \R^n, hence its ring of smooth functions C^\infty(M) comes from restricting the smooth functions from C^\infty(\R^n). Another complexity when considering the ring of holomorphic functions on a complex manifold X is given a small enough open set U \subset X, the holomorphic functions will be isomorphic to \mathcal(U) \cong \mathcal(\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 on arbitrary open subsets U \subset X. This means as U becomes more complex topologically, the ring \mathcal(U) can be expressed from gluing the \mathcal(U_i). Note that sometimes this sheaf is denoted \mathcal(-) or just \mathcal, or even \mathcal_X when we want to emphasize the space the structure sheaf is associated to.


Tracking submanifolds with sheaves

Another common example of sheaves can be constructed by considering a complex submanifold Y \hookrightarrow X. There is an associated sheaf \mathcal_Y which takes an open subset U \subset X and gives the ring of holomorphic functions on U \cap Y. This kind of formalism was found to be extremely powerful and motivates a lot of
homological algebra Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
such as
sheaf cohomology In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when i ...
since an
intersection theory In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theore ...
can be built using these kinds of sheaves from the Serre intersection formula.


Operations with sheaves


Morphisms

Morphisms of sheaves are, roughly speaking, analogous to functions between them. In contrast to a function between sets, which have no additional structure, morphisms of sheaves are those functions which preserve the structure inherent in the sheaves. This idea is made precise in the following definition. Let F and G be two sheaves on X. A ''
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
'' \varphi:G\to F consists of a morphism \varphi_U:G(U)\to F(U) for each open set U of X, subject to the condition that this morphism is compatible with restrictions. In other words, for every open subset V of an open set U, the following diagram is
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
. :\begin G(U) & \xrightarrow & F(U)\\ r_\Biggl\downarrow & & \Biggl\downarrow r_\\ G(V) & \xrightarrow[] & F(V) \end For example, taking the derivative gives a morphism of sheaves on \R: \mathcal O^n_ \to \mathcal O^_. Indeed, given an (n-times continuously differentiable) function f : U \to \R (with U in \R open), the restriction (to a smaller open subset V) of its derivative equals the derivative of f, _V. With this notion of morphism, sheaves on a fixed topological space X form a
category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...
. The general categorical notions of
mono- Numeral or number prefixes are prefixes derived from numerals or occasionally other numbers. In English and many other languages, they are used to coin numerous series of words. For example: * unicycle, bicycle, tricycle (1-cycle, 2-cycle, 3-cyc ...
, epi- and
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
s can therefore be applied to sheaves. A sheaf morphism \varphi is an isomorphism (resp. monomorphism) if and only if each \varphi_U is a bijection (resp. injective map). Moreover, a morphism of sheaves \varphi is an isomorphism if and only if there exists an open cover \ such that \varphi, _ are isomorphisms of sheaves for all \alpha. This statement, which also holds for monomorphisms, but does not hold for presheaves, is another instance of the idea that sheaves are of a local nature. The corresponding statements do not hold for
epimorphism In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms , : g_1 \circ f = g_2 \circ f \ ...
s (of sheaves), and their failure is measured by
sheaf cohomology In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when i ...
.


Stalks of a sheaf

The ''stalk'' \mathcal_x of a sheaf \mathcal captures the properties of a sheaf "around" a point x\in X, generalizing the germs of functions. Here, "around" means that, conceptually speaking, one looks at smaller and smaller
neighborhoods A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographically localised community ...
of the point. Of course, no single neighborhood will be small enough, which requires considering a limit of some sort. More precisely, the stalk is defined by :\mathcal_x = \varinjlim_ \mathcal(U), the
direct limit In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any categor ...
being over all open subsets of X containing the given point x. In other words, an element of the stalk is given by a section over some open neighborhood of x, and two such sections are considered equivalent if their restrictions agree on a smaller neighborhood. The natural morphism F(U)\to F_x takes a section x in F(U) to its ''germ'' at x. This generalises the usual definition of a
germ Germ or germs may refer to: Science * Germ (microorganism), an informal word for a pathogen * Germ cell, cell that gives rise to the gametes of an organism that reproduces sexually * Germ layer, a primary layer of cells that forms during embry ...
. In many situations, knowing the stalks of a sheaf is enough to control the sheaf itself. For example, whether or not a morphism of sheaves is a monomorphism, epimorphism, or isomorphism can be tested on the stalks. In this sense, a sheaf is determined by its stalks, which are a local data. By contrast, the ''global'' information present in a sheaf, i.e., the ''global sections'', i.e., the sections \mathcal F(X) on the whole space X, typically carry less information. For example, for a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
complex manifold X, the global sections of the sheaf of holomorphic functions are just \C, since any holomorphic function :X \to \C is constant by Liouville's theorem.


Turning a presheaf into a sheaf

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 ''sheafification'' or ''sheaf associated to the presheaf'' F. For example, the sheafification of the constant presheaf (see above) is called the ''
constant sheaf Constant or The Constant may refer to: Mathematics * Constant (mathematics), a non-varying value * Mathematical constant, a special number that arises naturally in mathematics, such as or Other concepts * Control variable or scientific cons ...
''. Despite its name, its sections are ''locally'' constant functions. The sheaf aF can be constructed using the
étalé space In mathematics, a sheaf 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 ...
of F, namely as the sheaf of sections of the map :\mathrm(F) \to X. Another construction of the sheaf aF proceeds by means of a functor L from presheaves to presheaves that gradually improves the properties of a presheaf: for any presheaf F, LF is a separated presheaf, and for any separated presheaf F, LF is a sheaf. The associated sheaf aF is given by LLF. The idea that the sheaf aF is the best possible approximation to F by a sheaf is made precise using the following
universal property In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fro ...
: there is a natural morphism of presheaves i\colon F\to aF so that for any sheaf G and any morphism of presheaves f\colon F\to G, there is a unique morphism of sheaves \tilde f \colon aF \rightarrow G such that f = \tilde f i. In fact a is the left
adjoint functor In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
to the inclusion functor (or
forgetful functor In mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given signa ...
) from the category of sheaves to the category of presheaves, and i is the
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (alb ...
of the adjunction. In this way, the category of sheaves turns into a
Giraud subcategory In mathematics, Giraud subcategories form an important class of subcategories of Grothendieck categories. They are named after Jean Giraud. Definition Let \mathcal be a Grothendieck category. A full subcategory \mathcal is called ''reflective' ...
of presheaves. This categorical situation is the reason why the sheafification functor appears in constructing cokernels of sheaf morphisms or tensor products of sheaves, but not for kernels, say.


Subsheaves, quotient sheaves

If K is a subsheaf of a sheaf F of abelian groups, then the quotient sheaf Q is the sheaf associated to the presheaf U \mapsto F(U)/K(U); in other words, the quotient sheaf fits into an exact sequence of sheaves of abelian groups; :0 \to K \to F \to Q \to 0. (this is also called a
sheaf extension In mathematics, a sheaf of ''O''-modules or simply an ''O''-module over a ringed space (''X'', ''O'') is a sheaf ''F'' such that, for any open subset ''U'' of ''X'', ''F''(''U'') is an ''O''(''U'')-module and the restriction maps ''F''(''U'')  ...
.) Let F,G be sheaves of abelian groups. The set \operatorname(F, G) of morphisms of sheaves from F to G forms an abelian group (by the abelian group structure of G). The sheaf hom of F and G, denoted by, :\mathcal(F, G) is the sheaf of abelian groups U \mapsto \operatorname(F, _U, G, _U) where F, _U is the sheaf on U given by (F, _U)(V) = F(V) (note sheafification is not needed here). The direct sum of F and G is the sheaf given by U \mapsto F(U) \oplus G(U) , and the tensor product of F and G is the sheaf associated to the presheaf U \mapsto F(U) \otimes G(U). All of these operations extend to
sheaves of modules ''Sheaves'' is the plural of either of two nouns: * Sheaf (disambiguation) * Sheave A sheave () or pulley wheel is a grooved wheel often used for holding a belt, wire rope, or rope and incorporated into a pulley A pulley is a wheel ...
over a
sheaf of rings In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf of ...
A; the above is the special case when A is the
constant sheaf Constant or The Constant may refer to: Mathematics * Constant (mathematics), a non-varying value * Mathematical constant, a special number that arises naturally in mathematics, such as or Other concepts * Control variable or scientific cons ...
\underline.


Basic functoriality

Since the data of a (pre-)sheaf depends on the open subsets of the base space, sheaves on different topological spaces are unrelated to each other in the sense that there are no morphisms between them. However, given a continuous map f:X\to Y between two topological spaces, pushforward and pullback relate sheaves on X to those on Y and vice versa.


Direct image

The pushforward (also known as
direct image In mathematics, the direct image functor is a construction in sheaf theory that generalizes the global sections functor to the relative case. It is of fundamental importance in topology and algebraic geometry. Given a sheaf ''F'' defined on a topolo ...
) of a sheaf \mathcal on X is the sheaf defined by :(f_* \mathcal F)(V) = \mathcal F(f^(V)). Here V is an open subset of Y, so that its preimage is open in X by the continuity of f. This construction recovers the skyscraper sheaf S_x mentioned above: :S_x = i_* (S) where i: \ \to X is the inclusion, and S is regarded as a sheaf on the
singleton Singleton may refer to: Sciences, technology Mathematics * Singleton (mathematics), a set with exactly one element * Singleton field, used in conformal field theory Computing * Singleton pattern, a design pattern that allows only one instance ...
(by S(\)=S, S(\emptyset) = \emptyset. For a map between
locally compact space In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
s, the
direct image with compact support In mathematics, the direct image with compact (or proper) support is an image functor for sheaves that extends the compactly supported global sections functor to the relative setting. It is one of Grothendieck's six operations. Definition Let ...
is a subsheaf of the direct image. By definition, (f_! \mathcal F)(V) consists of those f \in \mathcal F(f^(V)) whose
support Support may refer to: Arts, entertainment, and media * Supporting character Business and finance * Support (technical analysis) * Child support * Customer support * Income Support Construction * Support (structure), or lateral support, a ...
is
proper map In mathematics, a function between topological spaces is called proper if inverse images of compact subsets are compact. In algebraic geometry, the analogous concept is called a proper morphism. Definition There are several competing definitio ...
over V. If f is proper itself, then f_! \mathcal F = f_* \mathcal F, but in general they disagree.


Inverse image

The pullback or
inverse image In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) ...
goes the other way: it produces a sheaf on X, denoted f^ \mathcal G out of a sheaf \mathcal G on Y. If f is the inclusion of an open subset, then the inverse image is just a restriction, i.e., it is given by (f^ \mathcal G)(U) = \mathcal G(U) for an open U in X. A sheaf F (on some space X) is called
locally constant In mathematics, a locally constant function is a function from a topological space into a set with the property that around every point of its domain, there exists some neighborhood of that point on which it restricts to a constant function. ...
if X= \bigcup_ U_i by some open subsets U_i such that the restriction of F to all these open subsets is constant. One a wide range of topological spaces X, such sheaves are
equivalent Equivalence or Equivalent may refer to: Arts and entertainment *Album-equivalent unit, a measurement unit in the music industry * Equivalence class (music) *'' Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre *''Equiva ...
to
representations ''Representations'' is an interdisciplinary journal in the humanities published quarterly by the University of California Press. The journal was established in 1983 and is the founding publication of the New Historicism movement of the 1980s. It ...
of the
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
\pi_1(X). For general maps f, the definition of f^ \mathcal G is more involved; it is detailed at
inverse image functor In mathematics, specifically in algebraic topology and algebraic geometry, an inverse image functor is a contravariant construction of sheaves; here “contravariant” in the sense given a map f : X \to Y, the inverse image functor is a functor ...
. The stalk is an essential special case of the pullback in view of a natural identification, where i is as above: :\mathcal G_x = i^\mathcal(\). More generally, stalks satisfy (f^ \mathcal G)_x = \mathcal G_.


Extension by zero

For the inclusion j : U \to X of an open subset, the
extension by zero Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (predicate logic), the set of tuples of values that satisfy the predicate * E ...
of a sheaf of abelian groups on U is defined as :(j_! \mathcal F)(V) = \mathcal F(V) if V \subset U and (j_! \mathcal F)(V) = 0 otherwise. For a sheaf \mathcal G on X, this construction is in a sense complementary to i_*, where i is the inclusion of the complement of U: :(j_! j^* \mathcal G)_x = \mathcal G_x for x in U, and the stalk is zero otherwise, while :(i_* i^* \mathcal G)_x = 0 for x in U, and equals \mathcal G_x otherwise. These functors are therefore useful in reducing sheaf-theoretic questions on X to ones on the strata of a
stratification Stratification may refer to: Mathematics * Stratification (mathematics), any consistent assignment of numbers to predicate symbols * Data stratification in statistics Earth sciences * Stable and unstable stratification * Stratification, or st ...
, i.e., a decomposition of X into smaller, locally closed subsets.


Complements


Sheaves in more general categories

In addition to (pre-)sheaves as introduced above, where \mathcal F(U) is merely a set, it is in many cases important to keep track of additional structure on these sections. For example, the sections of the sheaf of continuous functions naturally form a real
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
, and restriction is a
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
between these vector spaces. Presheaves with values in an arbitrary category C are defined by first considering the category of open sets on X to be the
posetal category In mathematics, specifically category theory, a posetal category, or thin category, is a category whose homsets each contain at most one morphism. As such, a posetal category amounts to a preordered class (or a preordered set, if its objects for ...
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 In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ma ...
from O(X) to C. Morphisms in this category of functors, also known as
natural transformations In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natural ...
, are the same as the morphisms defined above, as can be seen by unraveling the definitions. If the target category C admits all
limits Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
, a C-valued presheaf is a sheaf if the following diagram is an equalizer for every open cover \mathcal = \_ of any open set ''U'': :F(U) \rightarrow \prod_ F(U_i) \prod_ F(U_i \cap U_j). Here the first map is the product of the restriction maps :\operatorname_ \colon F(U) \rightarrow F(U_i) and the pair of arrows the products of the two sets of restrictions :\operatorname_ \colon F(U_i) \rightarrow F(U_i \cap U_j) and :\operatorname_ \colon F(U_j) \rightarrow F(U_i \cap U_j). If C is an
abelian category In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ab ...
, this condition can also be rephrased by requiring that there is an
exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context o ...
:0 \to F(U) \to \prod_i F(U_i) \xrightarrow \prod_ F(U_i \cap U_j). A particular case of this sheaf condition occurs for U being the empty set, and the index set I also being empty. In this case, the sheaf condition requires \mathcal F(\emptyset) to be the
terminal object In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): ...
in C.


Ringed spaces and sheaves of modules

In several geometrical disciplines, including
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
and
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, the spaces come along with a natural sheaf of rings, often called the structure sheaf and denoted by \mathcal_X. Such a pair (X, \mathcal O_X) is called a ''
ringed space In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf of ...
''. Many types of spaces can be defined as certain types of ringed spaces. Commonly, all the stalks \mathcal O_ of the structure sheaf are
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic num ...
s, in which case the pair is called a ''locally ringed space''. For example, an n-dimensional C^k manifold M is a locally ringed space whose structure sheaf consists of C^k-functions on the open subsets of M. The property of being a ''locally'' ringed space translates into the fact that such a function, which is nonzero at a point x, is also non-zero on a sufficiently small open neighborhood of x. Some authors actually ''define'' real (or complex) manifolds to be locally ringed spaces that are locally isomorphic to the pair consisting of an open subset of \R^n (resp. \C^n) together with the sheaf of C^k (resp. holomorphic) functions. Similarly,
scheme A scheme is a systematic plan for the implementation of a certain idea. Scheme or schemer may refer to: Arts and entertainment * ''The Scheme'' (TV series), a BBC Scotland documentary series * The Scheme (band), an English pop band * ''The Schem ...
s, the foundational notion of spaces in algebraic geometry, are locally ringed spaces that are locally isomorphic to the
spectrum of a ring In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with the ...
. Given a ringed space, a ''sheaf of modules'' is a sheaf \mathcal such that on every open set U of X, \mathcal(U) is an \mathcal_X(U)-module and for every inclusion of open sets V\subseteq U, the restriction map \mathcal(U) \to \mathcal(V) is compatible with the restriction map \mathcal(U) \to \mathcal(V): the restriction of ''fs'' is the restriction of f times that of s for any f in \mathcal(U) and s in \mathcal(U). Most important geometric objects are sheaves of modules. For example, there is a one-to-one correspondence between
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
s and
locally free sheaves In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with ref ...
of \mathcal_X-modules. This paradigm applies to real vector bundles, complex vector bundles, or vector bundles in algebraic geometry (where \mathcal O consists of smooth functions, holomorphic functions, or regular functions, respectively). Sheaves of solutions to differential equations are D-modules, that is, modules over the sheaf of
differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...
s. On any topological space, modules over the constant sheaf \underline are the same as sheaves of abelian groups in the sense above. There is a different inverse image functor for sheaves of modules over sheaves of rings. This functor is usually denoted f^* and it is distinct from f^. See
inverse image functor In mathematics, specifically in algebraic topology and algebraic geometry, an inverse image functor is a contravariant construction of sheaves; here “contravariant” in the sense given a map f : X \to Y, the inverse image functor is a functor ...
.


Finiteness conditions for sheaves of modules

Finiteness conditions for module over
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
s give rise to similar finiteness conditions for sheaves of modules: \mathcal is called ''finitely generated'' (resp. ''finitely presented'') if, for every point x of X, there exists an open neighborhood U of x, a natural number n (possibly depending on U), and a surjective morphism of sheaves \mathcal_X^n, _U \to \mathcal, _U (respectively, in addition a natural number m, and an exact sequence \mathcal_X^m, _U \to \mathcal_X^n, _U \to \mathcal, _U \to 0.) Paralleling the notion of a
coherent module In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring ''R'' may also be called a finite ''R''-module, finite over ''R'', or a module of finite type. Related concepts inclu ...
, \mathcal is called a ''
coherent sheaf In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refe ...
'' if it is of finite type and if, for every open set U and every morphism of sheaves \phi : \mathcal_X^n \to \mathcal (not necessarily surjective), the kernel of \phi is of finite type. \mathcal_X is ''coherent'' if it is coherent as a module over itself. Like for modules, coherence is in general a strictly stronger condition than finite presentation. The
Oka coherence theorem In mathematics, the Oka coherence theorem, proved by , states that the sheaf \mathcal := \mathcal_ of germs of holomorphic functions on \mathbb^n over a complex manifold is coherent.In paper it was called the idéal de domaines indéterminés. ...
states that the sheaf of holomorphic functions on a
complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a com ...
is coherent.


The étalé 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 ''étalé space'', from the French word étalé , meaning roughly "spread out". If F \in \text(X) is a sheaf over X, then the étalé space of F is a topological space E together with a
local homeomorphism In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure. If f : X \to Y is a local homeomorphism, X is said to be an ...
\pi: E \to X such that the sheaf of sections \Gamma(\pi, -) of \pi is F. The space ''E'' is usually very strange, and even if the sheaf ''F'' arises from a natural topological situation, ''E'' may not have any clear topological interpretation. For example, if ''F'' is the sheaf of sections of a continuous function f: Y \to X, then E=Y if and only if f is a
local homeomorphism In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure. If f : X \to Y is a local homeomorphism, X is said to be an ...
. The étalé space ''E'' is constructed from the stalks of ''F'' over ''X''. As a set, it is their
disjoint union In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A (th ...
and ''\pi'' is the obvious map that takes the value x on the stalk of F over x \in X. The topology of ''E'' is defined as follows. For each element s \in F(U) and each x \in U, we get a germ of s at x, denoted x or s_x. These germs determine points of ''E''. For any U and s \in F(U), the union of these points (for all x \in U) is declared to be open in ''E''. Notice that each stalk has the
discrete topology In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
as subspace topology. Two morphisms between sheaves determine a continuous map of the corresponding étalé spaces that is compatible with the projection maps (in the sense that every germ is mapped to a germ over the same point). This makes the construction into a functor. The construction above determines an
equivalence of categories In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences fro ...
between the category of sheaves of sets on ''X'' and the category of étalé spaces over ''X''. The construction of an étalé space can also be applied to a presheaf, in which case the sheaf of sections of the étalé space recovers the sheaf associated to the given presheaf. This construction makes all sheaves into
representable functor In mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures (i.e. sets and ...
s on certain categories of topological spaces. As above, let ''F'' be a sheaf on ''X'', let ''E'' be its étalé space, and let \pi:E \to X be the natural projection. Consider the
overcategory In mathematics, specifically category theory, an overcategory (and undercategory) is a distinguished class of categories used in multiple contexts, such as with covering spaces (espace etale). They were introduced as a mechanism for keeping track ...
\text/X of topological spaces over X, that is, the category of topological spaces together with fixed continuous maps to X. Every object of this category is a continuous map f:Y\to X, and a morphism from Y\to X to Z\to X is a continuous map Y\to Z that commutes with the two maps to X. There is a functor
\Gamma:\text/X \to \text
sending an object f:Y\to X to f^ F(Y). For example, if i: U \hookrightarrow X is the inclusion of an open subset, then
\Gamma(i) = f^ F(U) = F(U) = \Gamma(F, U)
and for the inclusion of a point i : \\hookrightarrow X, then
\Gamma(i) = f^ F(\) = F, _x
is the stalk of F at x. There is a natural isomorphism
(f^F)(Y) \cong \operatorname_(f, \pi),
which shows that \pi: E \to X (for the étalé space) represents the functor \Gamma. ''E'' is constructed so that the projection map ''\pi'' is a covering map. In algebraic geometry, the natural analog of a covering map is called an
étale morphism In algebraic geometry, an étale morphism () is a morphism of schemes that is formally étale and locally of finite presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy t ...
. Despite its similarity to "étalé", the word étale has a different meaning in French. It is possible to turn E into a
scheme A scheme is a systematic plan for the implementation of a certain idea. Scheme or schemer may refer to: Arts and entertainment * ''The Scheme'' (TV series), a BBC Scotland documentary series * The Scheme (band), an English pop band * ''The Schem ...
and ''\pi'' into a morphism of schemes in such a way that ''\pi'' retains the same universal property, but ''\pi'' is ''not'' in general an étale morphism because it is not quasi-finite. It is, however, formally étale. The definition of sheaves by étalé spaces is older than the definition given earlier in the article. It is still common in some areas of mathematics such as
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
.


Sheaf cohomology

In contexts, where the open set U is fixed, and the sheaf is regarded as a variable, the set F(U) is also often denoted \Gamma(U, F). As was noted above, this functor does not preserve epimorphisms. Instead, an epimorphism of sheaves \mathcal F \to \mathcal G is a map with the following property: for any section g \in \mathcal G(U) there is a covering \mathcal = \_ where
U = \bigcup_ U_i
of open subsets, such that the restriction g, _ are in the image of \mathcal F(U_i). However, g itself need not be in the image of \mathcal F(U). A concrete example of this phenomenon is the exponential map :\mathcal O \stackrel \to \mathcal O^\times between the sheaf of
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
s and non-zero holomorphic functions. This map is an epimorphism, which amounts to saying that any non-zero holomorphic function g (on some open subset in \C, say), admits a
complex logarithm In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
''locally'', i.e., after restricting g to appropriate open subsets. However, g need not have a logarithm globally. Sheaf cohomology captures this phenomenon. More precisely, for an
exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context o ...
of sheaves of abelian groups :0 \to \mathcal F_1 \to \mathcal F_2 \to \mathcal F_3 \to 0, (i.e., an epimorphism \mathcal F_2 \to \mathcal F_3 whose kernel is \mathcal F_1), there is a long exact sequence0 \to \Gamma(U, \mathcal F_1) \to \Gamma(U, \mathcal F_2) \to \Gamma(U, \mathcal F_3) \to H^1(U, \mathcal F_1) \to H^1(U, \mathcal F_2) \to H^1(U, \mathcal F_3) \to H^2(U, \mathcal F_1) \to \dotsBy means of this sequence, the first cohomology group H^1(U, \mathcal F_1) is a measure for the non-surjectivity of the map between sections of \mathcal F_2 and \mathcal F_3. There are several different ways of constructing sheaf cohomology. introduced them by defining sheaf cohomology as the
derived functor In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics. Motivation It was noted in vari ...
of \Gamma. This method is theoretically satisfactory, but, being based on
injective resolution In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution) is an exact sequence of modules (or, more generally, of objects of an abelian category), which is used to defi ...
s, of little use in concrete computations.
Godement resolution The Godement resolution of a sheaf (mathematics), sheaf is a construction in homological algebra that allows one to view global, cohomological information about the sheaf in terms of local information coming from its stalks. It is useful for comput ...
s are another general, but practically inaccessible approach.


Computing sheaf cohomology

Especially in the context of sheaves on manifolds, sheaf cohomology can often be computed using resolutions by soft sheaves, fine sheaves, and flabby sheaves (also known as ''flasque sheaves'' from the French ''flasque'' meaning flabby). For example, a
partition of unity In mathematics, a partition of unity of a topological space is a set of continuous functions from to the unit interval ,1such that for every point x\in X: * there is a neighbourhood of where all but a finite number of the functions of are 0, ...
argument shows that the sheaf of smooth functions on a manifold is soft. The higher cohomology groups H^i(U, \mathcal F) for i > 0 vanish for soft sheaves, which gives a way of computing cohomology of other sheaves. For example, the
de Rham complex In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapte ...
is a resolution of the constant sheaf \underline on any smooth manifold, so the sheaf cohomology of \underline is equal to its
de Rham cohomology In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapte ...
. A different approach is by
Čech cohomology In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open covers of a topological space. It is named for the mathematician Eduard Čech. Motivation Let ''X'' be a topolo ...
. Čech cohomology was the first cohomology theory developed for sheaves and it is well-suited to concrete calculations, such as computing the
coherent sheaf cohomology In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions with specified properties. Many geometric questions can be formulated as questions about the ex ...
of complex projective space \mathbb^n. It relates sections on open subsets of the space to cohomology classes on the space. In most cases, Čech cohomology computes the same cohomology groups as the derived functor cohomology. However, for some pathological spaces, Čech cohomology will give the correct H^1 but incorrect higher cohomology groups. To get around this,
Jean-Louis Verdier Jean-Louis Verdier (; 2 February 1935 – 25 August 1989) was a French mathematician who worked, under the guidance of his doctoral advisor Alexander Grothendieck, on derived categories and Verdier duality. He was a close collaborator of Grothe ...
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 Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Pr ...
's
mixed Hodge structure In algebraic geometry, a mixed Hodge structure is an algebraic structure containing information about the cohomology of general algebraic varieties. It is a generalization of a Hodge structure, which is used to study smooth projective varieties. ...
s. Many other coherent sheaf cohomology groups are found using an embedding i:X \hookrightarrow Y of a space X into a space with known cohomology, such as \mathbb^n, or some
weighted projective space In algebraic geometry, a weighted projective space P(''a''0,...,''a'n'') is the projective variety Proj(''k'' 'x''0,...,''x'n'' associated to the graded ring ''k'' 'x''0,...,''x'n''where the variable ''x'k'' has degree ''a'k''. Prop ...
. In this way, the known sheaf cohomology groups on these ambient spaces can be related to the sheaves i_*\mathcal, giving H^i(Y,i_*\mathcal) \cong H^i(X,\mathcal). For example, computing the coherent sheaf cohomology of projective plane curves is easily found. One big theorem in this space is the
Hodge decomposition In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every coh ...
found using a spectral sequence associated to sheaf cohomology groups, proved by Deligne. Essentially, the E_1-page with terms
E_1^ = H^p(X,\Omega^q_X)
the sheaf cohomology of a
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
projective variety In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables w ...
X, degenerates, meaning E_1 = E_\infty. This gives the canonical Hodge structure on the cohomology groups H^k(X,\mathbb). It was later found these cohomology groups can be easily explicitly computed using Griffiths residues. See
Jacobian ideal In mathematics the Jacobian ideal or gradient ideal is the ideal generated by the Jacobian of a function or function germ. Let \mathcal(x_1,\ldots,x_n) denote the ring of smooth functions in n variables and f a function in the ring. The Jacobi ...
. These kinds of theorems lead to one of the deepest theorems about the cohomology of algebraic varieties, the decomposition theorem, paving the path for
Mixed Hodge module In mathematics, mixed Hodge modules are the culmination of Hodge theory, mixed Hodge structures, intersection cohomology, and the decomposition theorem yielding a coherent framework for discussing variations of degenerating mixed Hodge structures ...
s. Another clean approach to the computation of some cohomology groups is the Borel–Bott–Weil theorem, which identifies the cohomology groups of some
line bundle In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organisin ...
s on
flag manifold In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space ''V'' over a field F. When F is the real or complex numbers, a generalized flag variety is a smoot ...
s with
irreducible representation In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W,W ...
s of
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
s. This theorem can be used, for example, to easily compute the cohomology groups of all line bundles on projective space and
grassmann manifold In mathematics, the Grassmannian is a space that parameterizes all - dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the project ...
s. In many cases there is a duality theory for sheaves that generalizes
Poincaré duality In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if ''M'' is an ''n''-dimensional oriented closed manifold (compact ...
. See Grothendieck duality and
Verdier duality In mathematics, Verdier duality is a cohomology, cohomological duality in algebraic topology that generalizes Poincaré duality for manifolds. Verdier duality was introduced in 1965 by as an analog for locally compact space, locally compact topolo ...
.


Derived categories of sheaves

The
derived category In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction proce ...
of the category of sheaves of, say, abelian groups on some space ''X'', denoted here as D(X), is the conceptual haven for sheaf cohomology, by virtue of the following relation: :H^n(X, \mathcal F) = \operatorname_(\mathbf Z, \mathcal F . The adjunction between f^, which is the left adjoint of f_* (already on the level of sheaves of abelian groups) gives rise to an adjunction :f^ : D(Y) \rightleftarrows D(X) : R f_* (for f: X \to Y), where Rf_* is the derived functor. This latter functor encompasses the notion of sheaf cohomology since H^n(X, \mathcal F) = R^n f_* \mathcal F for f: X \to \. Like f_*, the direct image with compact support f_! can also be derived. By virtue of the following isomorphism R f_! F parametrizes the
cohomology with compact support In mathematics, cohomology with compact support refers to certain cohomology theories, usually with some condition requiring that cocycles should have compact support. Singular cohomology with compact support Let X be a topological space. Then :\d ...
of the
fibers Fiber or fibre (from la, fibra, links=no) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often incorporate ...
of f: :(R^i f_! F)_y = H^i_c(f^(y), F). This isomorphism is an example of a base change theorem. There is another adjunction :Rf_! : D(X) \rightleftarrows D(Y) : f^!. Unlike all the functors considered above, the twisted (or exceptional) inverse image functor f^! is in general only defined on the level of
derived categories In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction proce ...
, i.e., the functor is not obtained as the derived functor of some functor between abelian categories. If f: X \to \ and ''X'' is a smooth
orientable manifold In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
of dimension ''n'', then :f^! \underline \mathbf R \cong \underline \mathbf R This computation, and the compatibility of the functors with duality (see
Verdier duality In mathematics, Verdier duality is a cohomology, cohomological duality in algebraic topology that generalizes Poincaré duality for manifolds. Verdier duality was introduced in 1965 by as an analog for locally compact space, locally compact topolo ...
) can be used to obtain a high-brow explanation of
Poincaré duality In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if ''M'' is an ''n''-dimensional oriented closed manifold (compact ...
. In the context of quasi-coherent sheaves on schemes, there is a similar duality known as
coherent duality In mathematics, coherent duality is any of a number of generalisations of Serre duality, applying to coherent sheaves, in algebraic geometry and complex manifold theory, as well as some aspects of commutative algebra that are part of the 'local' the ...
.
Perverse sheaves The mathematical term perverse sheaves refers to a certain abelian category associated to a topological space ''X'', which may be a real or complex manifold, or a more general topologically stratified space, usually singular. This concept was intro ...
are certain objects in D(X), i.e., complexes of sheaves (but not in general sheaves proper). They are an important tool to study the geometry of singularities.


Derived categories of coherent sheaves and the Grothendieck group

Another important application of derived categories of sheaves is with the derived category of
coherent sheaves In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...
on a scheme X denoted D_(X). This was used by Grothendieck in his development of
intersection theory In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theore ...
using
derived categories In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction proce ...
and
K-theory In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, ...
, that the intersection product of subschemes Y_1, Y_2 is represented in
K-theory In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, ...
as
_1cdot _2= mathcal_\otimes_^\mathcal_\in K(\text)
where \mathcal_ are
coherent sheaves In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...
defined by the \mathcal_X-modules given by their structure sheaves.


Sites and topoi

André Weil André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. Th ...
's
Weil conjectures In mathematics, the Weil conjectures were highly influential proposals by . They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory. Th ...
stated that there was a
cohomology theory In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
for
algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
over
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
s that would give an analogue of the
Riemann hypothesis In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in ...
. The cohomology of a complex manifold can be defined as the sheaf cohomology of the locally constant sheaf \underline in the Euclidean topology, which suggests defining a Weil cohomology theory in positive characteristic as the sheaf cohomology of a constant sheaf. But the only classical topology on such a variety is the
Zariski topology In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
, and the Zariski topology has very few open sets, so few that the cohomology of any Zariski-constant sheaf on an irreducible variety vanishes (except in degree zero). Alexandre Grothendieck solved this problem by introducing
Grothendieck topologies In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category ''C'' that makes the objects of ''C'' act like the open sets of a topological space. A category together with a choice of Grothendieck topology is cal ...
, 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 In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjecture ...
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 Francis William Lawvere (; born February 9, 1937) is a mathematician known for his work in category theory, topos theory and the philosophy of mathematics. Biography Lawvere studied continuum mechanics as an undergraduate with Clifford Truesdell ...
and Miles Tierney to define an
elementary topos In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notion ...
, which has connections to
mathematical logic Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ...
.


History

The first origins of sheaf theory are hard to pin down – they may be co-extensive with the idea of
analytic continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new ...
. It took about 15 years for a recognisable, free-standing theory of sheaves to emerge from the foundational work on
cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
. * 1936
Eduard Čech Eduard Čech (; 29 June 1893 – 15 March 1960) was a Czech mathematician. His research interests included projective differential geometry and topology. He is especially known for the technique known as Stone–Čech compactification (in topo ...
introduces the ''
nerve A nerve is an enclosed, cable-like bundle of nerve fibers (called axons) in the peripheral nervous system. A nerve transmits electrical impulses. It is the basic unit of the peripheral nervous system. A nerve provides a common pathway for the e ...
'' construction, for associating a
simplicial complex In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set ...
to an open covering. * 1938
Hassler Whitney Hassler Whitney (March 23, 1907 – May 10, 1989) was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, characteristic classes, and geometric integration t ...
gives a 'modern' definition of cohomology, summarizing the work since J. W. Alexander and
Kolmogorov Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...
first defined ''
cochain In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of t ...
s''. * 1943
Norman Steenrod Norman Earl Steenrod (April 22, 1910October 14, 1971) was an American mathematician most widely known for his contributions to the field of algebraic topology. Life He was born in Dayton, Ohio, and educated at Miami University and University of ...
publishes on homology ''with
local coefficients In mathematics, a local system (or a system of local coefficients) on a topological space ''X'' is a tool from algebraic topology which interpolates between cohomology with coefficients in a fixed abelian group ''A'', and general sheaf cohomolog ...
''. * 1945
Jean Leray Jean Leray (; 7 November 1906 – 10 November 1998) was a French mathematician, who worked on both partial differential equations and algebraic topology. Life and career He was born in Chantenay-sur-Loire (today part of Nantes). He studied at Éc ...
publishes work carried out as a
prisoner of war A prisoner of war (POW) is a person who is held captive by a belligerent power during or immediately after an armed conflict. The earliest recorded usage of the phrase "prisoner of war" dates back to 1610. Belligerents hold prisoners of wa ...
, motivated by proving fixed-point theorems for application to PDE theory; it is the start of sheaf theory and
spectral sequence In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by , they hav ...
s. * 1947
Henri Cartan Henri Paul Cartan (; 8 July 1904 – 13 August 2008) was a French mathematician who made substantial contributions to algebraic topology. He was the son of the mathematician Élie Cartan, nephew of mathematician Anna Cartan, oldest brother of co ...
reproves the
de Rham theorem In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly ada ...
by sheaf methods, in correspondence with
André Weil André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. Th ...
(see
De Rham–Weil theorem In algebraic topology, the De Rham–Weil theorem allows computation of sheaf cohomology using an acyclic resolution of the sheaf in question. Let \mathcal F be a sheaf on a topological space X and \mathcal F^\bullet a resolution of \mathcal ...
). 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 In mathematics, a sheaf 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 ...
(''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 was a Japanese mathematician who did fundamental work in the theory of several complex variables. Biography Oka was born in Osaka. He went to Kyoto Imperial University in 1919, turning to mathematics in 1923 and graduating in 1924. He was in ...
introduces an idea (adjacent to that) of a sheaf of ideals, in
several complex variables The theory of functions of several complex variables is the branch of mathematics dealing with complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several complex variable ...
. * 1951 The Cartan seminar proves theorems A and B, based on Oka's work. * 1953 The finiteness theorem for
coherent sheaves In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...
in the analytic theory is proved by Cartan and
Jean-Pierre Serre Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ina ...
, as is
Serre duality In algebraic geometry, a branch of mathematics, Serre duality is a duality for the coherent sheaf cohomology of algebraic varieties, proved by Jean-Pierre Serre. The basic version applies to vector bundles on a smooth projective variety, but Alexa ...
. * 1954 Serre's paper ''
Faisceaux algébriques cohérents This is a list of important publications in mathematics, organized by field. Some reasons why a particular publication might be regarded as important: *Topic creator – A publication that created a new topic *Breakthrough – A public ...
'' (published in 1955) introduces sheaves into
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
. These ideas are immediately exploited by
Friedrich Hirzebruch Friedrich Ernst Peter Hirzebruch ForMemRS (17 October 1927 – 27 May 2012) was a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation. He has been described as ...
, who writes a major 1956 book on topological methods. * 1955 Alexander Grothendieck in lectures in
Kansas Kansas () is a state in the Midwestern United States. Its capital is Topeka, and its largest city is Wichita. Kansas is a landlocked state bordered by Nebraska to the north; Missouri to the east; Oklahoma to the south; and Colorado to the ...
defines
abelian category In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ab ...
and ''presheaf'', and by using
injective resolution In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution) is an exact sequence of modules (or, more generally, of objects of an abelian category), which is used to defi ...
s allows direct use of sheaf cohomology on all topological spaces, as
derived functor In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics. Motivation It was noted in vari ...
s. * 1956
Oscar Zariski , birth_date = , birth_place = Kobrin, Russian Empire , death_date = , death_place = Brookline, Massachusetts, United States , nationality = American , field = Mathematics , work_institutions = ...
's report '' Algebraic sheaf theory'' * 1957 Grothendieck's ''Tohoku'' paper rewrites
homological algebra Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
; he proves Grothendieck duality (i.e., Serre duality for possibly
singular Singular may refer to: * Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms * Singular homology * SINGULAR, an open source Computer Algebra System (CAS) * Singular or sounder, a group of boar, ...
algebraic varieties). * 1957 onwards: Grothendieck extends sheaf theory in line with the needs of algebraic geometry, introducing: schemes and general sheaves on them,
local cohomology In algebraic geometry, local cohomology is an algebraic analogue of relative cohomology. Alexander Grothendieck introduced it in seminars in Harvard in 1961 written up by , and in 1961-2 at IHES written up as SGA2 - , republished as . Given a fu ...
,
derived categories In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction proce ...
(with Verdier), and
Grothendieck topologies In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category ''C'' that makes the objects of ''C'' act like the open sets of a topological space. A category together with a choice of Grothendieck topology is cal ...
. There emerges also his influential schematic idea of '
six operations In mathematics, Grothendieck's six operations, named after Alexander Grothendieck, is a formalism in homological algebra, also known as the six-functor formalism. It originally sprang from the relations in étale cohomology that arise from a mor ...
' in homological algebra. * 1958
Roger Godement Roger Godement (; 1 October 1921 – 21 July 2016) was a French mathematician, known for his work in functional analysis as well as his expository books. Biography Godement started as a student at the École normale supérieure in 1940, where he ...
's book on sheaf theory is published. At around this time
Mikio Sato is a Japanese mathematician known for founding the fields of algebraic analysis, hyperfunctions, and holonomic quantum fields. He is a professor at the Research Institute for Mathematical Sciences in Kyoto. Education Sato studied at the Unive ...
proposes his
hyperfunction In mathematics, hyperfunctions are generalizations of functions, as a 'jump' from one holomorphic function to another at a boundary, and can be thought of informally as distributions of infinite order. Hyperfunctions were introduced by Mikio Sato ...
s, which will turn out to have sheaf-theoretic nature. At this point sheaves had become a mainstream part of mathematics, with use by no means restricted to
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
. It was later discovered that the logic in categories of sheaves is
intuitionistic logic Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems ...
(this observation is now often referred to as
Kripke–Joyal semantics Kripke semantics (also known as relational semantics or frame semantics, and often confused with possible world semantics) is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke and André Jo ...
, but probably should be attributed to a number of authors).


See also

*
Coherent sheaf In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refe ...
*
Gerbe In mathematics, a gerbe (; ) is a construct in homological algebra and topology. Gerbes were introduced by Jean Giraud (mathematician), Jean Giraud following ideas of Alexandre Grothendieck as a tool for non-commutative cohomology in degree 2. Th ...
*
Stack (mathematics) In mathematics a stack or 2-sheaf is, roughly speaking, a sheaf that takes values in categories rather than sets. Stacks are used to formalise some of the main constructions of descent theory, and to construct fine moduli stacks when fine moduli sp ...
*
Sheaf of spectra In algebraic topology, a presheaf of spectra on a topological space ''X'' is a contravariant functor from the category of open subsets of ''X'', where morphisms are inclusions, to the good category of commutative ring spectra. A theorem of Jardine ...
*
Perverse sheaf The mathematical term perverse sheaves refers to a certain abelian category associated to a topological space ''X'', which may be a real or complex manifold, or a more general topologically stratified space, usually singular. This concept was intro ...
* Presheaf of spaces *
Constructible sheaf In mathematics, a constructible sheaf is a sheaf of abelian groups over some topological space ''X'', such that ''X'' is the union of a finite number of locally closed subsets on each of which the sheaf is a locally constant sheaf. It has its origin ...
* De Rham's theorem


Notes


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 In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction proce ...
and
vanishing cycle In mathematics, vanishing cycles are studied in singularity theory and other parts of algebraic geometry. They are those homology cycles of a smooth fiber in a family which vanish in the singular fiber. For example, in a map from a connected compl ...
s on the most reasonable spaces) * (category theory and toposes emphasised) * * * * * (concise lecture notes) * (pedagogic treatment) * (introductory book with open access) {{Authority control * Topological methods of algebraic geometry Algebraic topology