NationMaster - Encyclopedia: Stalk of a sheaf

In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain a bigger one. A presheaf is similar to a sheaf, but it may not be possible to glue. Sheaves, it turns out, enable one to discuss in a refined way what is a local property, as applied to a function. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...

Introduction

Sheaves are used in topology, algebraic geometry and differential geometry whenever one wants to keep track of algebraic data that vary with every open set of the given geometrical space. They are a global tool to study objects which vary locally (that is depend on the open sets). As such, they are a natural instrument to study the global behaviour of entities which are of local nature, such as open sets, analytic functions, manifolds, and so on. Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In mathematics, an analytic function is one that is locally given by a convergent power series. ... This is the current mathematics collaboration of the week! Please help improve it to featured article standard at manifold/rewrite. ...

For a typical example, consider a topological space X, and for every open set U in X, let F(U) be the set of all continuous functions U → R. If V is an open subset of U, then the functions on U can be restricted to V, and we get a map F(U) → F(V). "Gluing" describes the following process: suppose the U_i are given open sets with union U, and for each i we are given an element f_i ∈ F(U_i), a continuous function f_i : U_i → R. If these functions agree where they overlap, then we can glue them together in a unique way to form a continuous function f : U → R which agrees with all the given f_i. The collection of the sets F(U) together with the restriction maps F(U) → F(V) then form a sheaf of sets on X. Furthermore, each F(U) is a commutative ring and the restriction maps are ring homomorphisms, making F a sheaf of rings on X. In topology and related areas of mathematics a continuous function is a morphism between topological spaces; that is, a mapping which preserves the topological structure. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. ...

For a very similar example, consider a differentiable manifold X, and for every open set U of X, let F(U) be the set of differentiable functions U → R. Here too, gluing works and we obtain a sheaf of rings on X. Another sheaf on X assigns to every open set U of X the vector space of all differentiable vector fields defined on U. Restriction and gluing of vector fields works like that of functions, and we obtain a sheaf of vector spaces on the manifold X. This is the current mathematics collaboration of the week! Please help improve it to featured article standard at manifold/rewrite. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ...

The formal definition

To define sheaves we will proceed in two steps. The first step is to introduce the concept of a presheaf, which captures the idea of associating local information to a topological space. The second step is to introduce an additional axiom, called the gluing axiom or the sheaf axiom, which captures the idea of gluing local information to get global information.

Definition of a presheaf

Suppose X is a topological space, and C is a category (often, this is the category of sets, the category of Abelian groups, the category of commutative rings, or the category of modules over a fixed ring). A presheaf F of objects in C on the space X is given by the following data: In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ... In mathematics, a set can be thought of as any well-defined collection of things considered as a whole. ... In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In abstract algebra, a module is a generalization of a vector space. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ...

This definition can be expressed naturally in terms of category theory. First we define the category of open sets on X to be the category Top_X whose objects are the open sets of X and whose morphisms are inclusions. Top_X is then the category corresponding to the partial order ⊂ on the open sets of X. A C-presheaf on X is then a contravariant functor from Top_X to C. In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... For functors in computer science, see the function object article. ...

If F is a C-valued presheaf on X, and U is an open subset of X, then F(U) is called the sections of F over U. (This is by analogy with sections of fiber bundles; see below) If C is a concrete category, then each element of F(U) is called a section. F(U) is also often denoted Γ(U,F). In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ... In mathematics, a concrete category is a category in which, roughly speaking, all objects are sets possibly carrying some additional structure, all morphisms are functions between those sets, and the composition of morphisms is the composition of functions. ...

The gluing axiom

See main article gluing axiom for a higher-level discussion In mathematics, the gluing axiom is introduced to define what a sheaf F on a topological space X must satisfy, given that it is a presheaf, which is by definition a contravariant functor F: Open(X) → C to a category C which initially one takes to be the category of...

Sheaves are presheaves on which sections over small open sets can be glued to give sections over larger open sets. Here the gluing axiom will be given in a form that requires C to be a concrete category.

Let U be the union of the collection of open sets {U_i}. For each U_i, choose a section f_i on U_i. We say that the f_i are compatible if for any i and j,

Intuitively speaking, if the f_i represent functions, this says that any two compatible functions agree where they overlap. The sheaf axiom says that we can produce from the f_i a unique section f over U whose restriction to each U_i is f_i, i.e., res_{U,U_i}(f)=f_i. Sometimes this is split into two axioms, one guaranteeing existence, and the other guaranteeing uniqueness.

Examples

In addition to the sheaves of continuous functions, differentiable functions and vector fields given in the introduction, sheaves of sections are very important examples. Suppose E and X are topological spaces and π : E → X is a continuous map. For every open set U in X, let F(U) be the set all continuous maps f : U → E such that π(f(x)) = x for all x in U. Such a function f is called a section of π. It is not difficult to check that F is a sheaf of sets on X. In fact, every sheaf of sets on X is essentially of this type, for very special maps π; see below.

Given a sheaf F on X, the elements of F(X) are also called the global sections, a terminology motivated by the previous example.

In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ... In mathematics, a locally ringed space (or local ringed space) is, intuitively speaking, a space together with, for each of its open sets, a commutative ring the elements of which are thought of as functions defined on that open set. ... In mathematics, a locally ringed space (or local ringed space) is, intuitively speaking, a space together with, for each of its open sets, a commutative ring the elements of which are thought of as functions defined on that open set. ... In mathematics, more particularly in abstract algebra, local rings are certain rings that are comparatively simple, and serve to describe the local behavior of functions defined on varieties or manifolds. ... In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...

Morphisms of sheaves

Let F and G be two sheaves on X both with values in the category C. We define a morphism from G to F to be a family of morphisms φ_U : G(U) → F(U) in the category C for all opens U in X which commute with the restriction maps. That is, the following diagram must commute In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ... In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ...

for each pair of open sets U ⊆ V in X. If F and G are considered as contravariant functors from Top_X to C then a morphism between them is nothing more than a natural transformation. With this definition the set of all C-valued sheaves on X forms a category (a functor category). An isomorphism of sheaves on X is just an isomorphism in this category. Wikipedia does not have an article with this exact name. ... For functors in computer science, see the function object article. ... In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ... In category theory, the functors between two given categories can themselves be turned into a category; the morphisms in this functor category are natural transformations between functors. ...

One can generalize this notion to morphisms between sheaves on different spaces. Let f : X → Y be a continuous function between two topological spaces, and let F be a sheaf on X and G a sheaf on Y both with values in C. Then a morphism from G to F relative to f is given by a family of morphisms φ_U : G(U) → F(f⁻¹(U)) for each open set U in Y such that the diagram In topology, a continuous function is generally defined as one for which preimages of open sets are open. ...

commutes for each pair of open sets U ⊆ V in Y. The previous definition is the special case resulting when f is the identity map on X. Wikipedia does not have an article with this exact name. ...

The category theoretical description is slightly more complicated in the general case. Let Top be the contravariant functor from the category of topological spaces Top to the category of small categories Cat which sends each space X to the poset category of its open sets Top_X. Here Top(f) is a covariant functor from Top_Y to Top_X sending each open set to its preimage. Composing F with Top(f) we obtain a contravariant functor from Top_Y to C. A morphism from G to F relative to f is then a natural transformation from G to F ○ Top(f). The category Top has topological spaces as objects and continuous maps as morphisms. ... In mathematics, specifically in category theory, the 2-category of small categories is the 2-category whose objects are small categories, whose arrows are functors and whose 2-arrows are natural transformations. ... In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ...

Note that all of the above makes sense if we are working only with presheaves instead of sheaves.

Stalks of a sheaf at a point and germs of functions

Fix a point x of X. We would like to study the behavior of F near the point x. In analytical terms, we would like to somehow take the limit as we get nearer and nearer to the point x. The corresponding concept is to take the direct limit of F(N) as N runs over the open neighbourhoods of x ordered by inclusion (in categorical terminology, this is an example of a colimit). We denote this limit by F_x and call it the stalk of F at x. If F is a C-valued sheaf on X, then the stalk F_x is an object of C, for C a category such as the category of abelian groups or the category of commutative rings. Analysis is the generic name given to any branch of mathematics which depends upon the concepts of limits and convergence, and studies closely related topics such as continuity, integration, differentiability and transcendental functions. ... In mathematics, the direct limit (also called the inductive limit) is a general method of taking limits of directed families of objects. We will first give the definition for algebraic structures like groups and modules, and then the general definition which can be used in any category. ... In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts of mathematics, like products and inverse limits. ... In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. ... In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. ...

For any open set U containing x there is a morphism from F(U) to F_x. If C is a concrete category, then applying this morphism to an element f in F(U) gives an element of F_x called the germ of f at x.

This corresponds to the notion of germ of a function used elsewhere in mathematics. Intuitively, the germ of the function f at x describes the local behavior of f at the point x; it is a kind of 'ghost' of f, looked at only very near x. See also the detailed example given at local ring. In mathematics, a germ is an equivalence class of continuous functions from one topological space to another (often from the real line to itself), in which one point x0 in the domain has been singled out as privileged. ... In mathematics, more particularly in abstract algebra, local rings are certain rings that are comparatively simple, and serve to describe the local behavior of functions defined on varieties or manifolds. ...

For some sheaves, germs behave well, and can give good local information; the germ of an analytic function around a point determines the function in a small neighbourhood of the point, using its power series expansion. However, some sheaves do not behave well; the germ of a smooth function at any point does not determine the function in any small neighbourhood of the point. As an example, take any bump function. Its local behavior on the interval where it is one is that of a constant function, but knowing that a bump function is the constant one near a given point does not tell you where the function begins to decay; from its local behavior, you cannot even conclude that it is a bump function! In mathematics, an analytic function is one that is locally given by a convergent power series. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ...

The étale space of a sheaf

In early developments of sheaf theory, it was shown that giving a sheaf F on X is as good as giving a certain topological space E together with a continuous map from E to X. More precisely: to every sheaf F of sets on X there exists a local homeomorphism In topology, a local homeomorphism is a map from one topological space to another that respects locally the topological structure of the two spaces. ...

such that F is isomorphic to the sheaf of sections of π that was described in the example section above.

Furthermore, the space E is determined up to homeomorphism by F. It is the space of stalks of F: each stalk is given the discrete topology, and we take the disjoint union of all the stalks, with π mapping all of the stalks F_x to x. The topology on this space of stalks can be chosen so that the sheaf F can be recovered as the sheaf of sections of π. Look up Up to on Wiktionary, the free dictionary In mathematics, the phrase up to xxxx indicates that members of an equivalence class are to be regarded as a single entity for some purpose. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ...

At a higher level of abstraction, we can say that the category of sheaves of sets on X is equivalent to the category of local homeomorphisms to X. (One can also consider such a space in the light of the theory of representable functors; the history shows that this theory developed also in the mid-1950s.) In category theory, an abstract branch of 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 from many areas of mathematics. ... In category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. ...

The space E was called espace étalé in Godement's influential book about homological algebra and sheaf theory (Topologie Algebrique et Theorie des Faisceaux, R. Godement); in that book, sheaves are in fact defined as coming from sections of local homeomorphisms; the functorial approach we gave above came later and is now more common. Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ... Roger Godement is a French mathematician, known for his work in functional analysis, and also his expository books. ...

The above considerations remain true for sheaves of C on X: we can still form the space of stalks, each stalk is an object in C, and the sections naturally become objects in C as well.

Given an arbitrary continuous map g : Z → X, the corresponding sheaf of sections gives rise in the above manner to a space of stalks E and a local homeomorphism π : E → X. In a sense this deals with all the 'ramification' in the map g, in the 'best possible way'. This may be expressed by adjoint functors; but is also important as an intuition about sheaves of sets. This collection of ideas is related to topos theory, but in a sense that more general notion of sheaf moves away from geometric intuition. In mathematics, ramification is a geometric term used for branching out, in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign. ... The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory. ... Sheaves were introduced into mathematics in the 1940s and, a major theme since then has been to study a space by studying sheaves on that space. ...

Generalizations

It is possible to define a cohomology theory for sheaves of abelian groups (sheaf cohomology) that can give much useful, more concrete information. The main issue is the existence of the long exact sequence coming from an exact sequence of sheaves. In applications emphasis was placed on sheaves on spaces that were less well-behaved than finite complexes. For example, in algebraic geometry spaces carrying the Zariski topology are rarely Hausdorff. In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. ... In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F. This is the main step, in numerous areas, from sheaf theory as a description of a geometric... In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ... In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ... Mathematicians (and those in related sciences) very frequently speak of whether a mathematical object -- a number, a function, a set, a space of one sort or another -- is well-behaved or not. ... This article needs to be cleaned up to conform to a higher standard of quality. ... Felix Hausdorff (November 8, 1868 - January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory and functional analysis. ...

The algebraic geometry case was first tackled by Jean-Pierre Serre by developing an analogue of Čech cohomology; this worked, though in general the construction doesn't have such good properties. Then Alexander Grothendieck used derived functors of the global section functor, providing a more definitive solution. Jean-Pierre Serre (born September 15, 1926) is one of the leading mathematicians of the twentieth century, active in algebraic geometry, number theory and topology. ... Čech cohomology is a particular type of cohomology in mathematics. ... Alexander Grothendieck (born March 28, 1928, Berlin) was one of the most important mathematicians active in the 20th century. ... In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. ...

Grothendieck was motivated to develop a cohomology theory for sheaves that would give stronger results, and that would, in particular, allow a proof of the Weil conjectures. By precisely analyzing the properties of X needed to define sheaves, he defined the notion of a Grothendieck topology on a category (this came in a somewhat roundabout fashion — see background and genesis of topos theory). In mathematics, the Weil conjectures, which had become theorems by 1975, were some highly-influential proposals from the late 1940s by Andre Weil on the generating functions (known as local zeta-functions) derived from counting the number of points on algebraic varieties over finite fields. ... In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space. ... This page gives some very general background to the mathematical idea of topos. ...

A category together with a Grothendieck topology is called a site. It is possible to define the notion of a sheaf on any site. The notion of sites later led Lawvere to develop the notion of an elementary topos. Francis William Lawvere is a mathematician who is known for his work in category theory and the philosophy of mathematics. ... For discussion of topoi in literary theory, see literary topos. ...

History

The first origins of sheaf theory are hard to pin down — they may be co-extensive with the idea of analytic continuation. It took about 15 years for a recognisable, free-standing theory of sheaves to emerge from the foundational work on cohomology. In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. ... In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. ...

At this point sheaves had become a mainstream part of mathematics, with use by no means restricted to algebraic topology. It was later discovered that the logic in categories of sheaves is intuitionistic logic (this observation is now often referred to as Kripke-Joyal semantics, but probably should be attributed to a number of authors). This shows that some of the facets of sheaf theory can also be traced back as far as Leibniz. Eduard Čech (June 29, 1893 - March 15, 1960) was a mathematician born in Stracov, Bohemia (then Austria-Hungary now Czech Republic). ... In mathematics, the nerve of an open covering is a construction in topology, of an abstract simplicial complex from an open covering of a topological space X. Given an index set I, and open sets Ui contained in X, the nerve N is the set of finite subsets of I... In mathematics, a simplicial complex is a topological space of a particular kind, built up of points, line segments, triangles, and their n-dimensional counterparts. ... Hassler Whitney (23 March 1907 â€“ 10 May 1989) was an American mathematician who was one of the founders of singularity theory, PhB, Yale University, 1928; MusB, 1929; ScD (Honorary), 1947; PhD, Harvard University, under G.D. Birkhoff, 1932. ... J. W. Alexander James Waddell Alexander II (September 19, 1888 – September 23, 1971) was an important topologist of the pre-WWII era and part of an influential Princeton topology elite, which included Oswald Veblen, Solomon Lefschetz, and others. ... Andrey Kolmogorov Andrey Nikolaevich Kolmogorov (Андре́й Никола́евич Колмого́ров) (kahl-mah-GAW-raff) (April 25, 1903 in Tambov - October 20, 1987 in Moscow) was a Russian mathematician who made major advances in the fields of probability theory and topology. ... Norman Earl Steenrod (April 22, 1910–October 14, 1971) was a leading mathematician, working in the field of topology. ... Jean Leray (7 November 1906-10 November 1998) was a French mathematician, who worked on both partial differential equations and algebraic topology. ... Geneva Convention definition A prisoner of war (POW) is a soldier, sailor, airman, or marine who is imprisoned by an enemy power during or immediately after an armed conflict. ... In mathematics, a fixed point of a function f is an argument x such that f(x) = x; see fixed point (mathematics). ... In mathematics, and in particular analysis, a partial differential equation (PDE) is an equation involving partial derivatives of an unknown function. ... In homological algebra, especially applied to algebraic topology or group cohomology, a spectral sequence is a sequence of differential modules (En,dn) such that En+1 = H(En) = ker dn / im dn is the homology of En. ... Henri Cartan (born July 8, 1904) is a son of Elie Cartan, and is, as his father was, a distinguished and influential mathematician. ... In mathematics, de Rham cohomology 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 adapted to computation and the concrete representation of cohomology classes. ... AndrÃ© Weil (May 6, 1906 - August 6, 1998) was one of the great mathematicians of the 20th century, a founding member of the influential Bourbaki group. ... In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain... The word support has several specialized meanings: In mathematics, see support (mathematics). ... In homological algebra, especially applied to algebraic topology or group cohomology, a spectral sequence is a sequence of differential modules (En,dn) such that En+1 = H(En) = ker dn / im dn is the homology of En. ... Kiyoshi Oka (岡潔, April 19, 1901 – March 1, 1978) was a Japanese mathematician, who did fundamental work in the theory of several complex variables. ... The theory of functions of several complex variables is the branch of mathematics dealing with functions f(z1, z2, ... , zn) on the space Cn of n-tuples of complex numbers. ... In mathematics, Cartans theorems A and B are two results proved by Henri Cartan around 1951, concerning a coherent sheaf F on a Stein manifold X. They are significant both as applied to several complex variables, and in the general development of sheaf cohomology. ... In mathematics, especially in algebraic geometry and the theory of complex manifolds, a coherent sheaf F on a locally ringed space X is a sheaf isomorphic with the cokernel of a morphism of OX_modules OXm → OXn. ... Jean-Pierre Serre (born September 15, 1926) is one of the leading mathematicians of the twentieth century, active in algebraic geometry, number theory and topology. ... In algebraic geometry, Serre duality is a duality present on non-singular projective algebraic varieties V of dimension n (and in greater generality) for vector bundles and the more general coherent sheaves. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... Friedrich E.P. Hirzebruch (born 17 October 1927) is a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation. ... Alexander Grothendieck (born March 28, 1928, Berlin) was one of the most important mathematicians active in the 20th century. ... State nickname: The Sunflower State Other U.S. States Capital Topeka Largest city Wichita Governor Kathleen Sebelius (D) Official languages None Area 82,277 miÂ²; 213,096 kmÂ² (15th) - Land 81,815 miÂ²; 211,900 kmÂ² - Water 462 miÂ²; 1,196 kmÂ² (0. ... 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 nice properties. ... In mathematics, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers. ... In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. ... Oscar Zariski was one of the most influential mathematicians working in the field of algebraic geometry in the twentieth century. ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ... Coherent duality in mathematics refers to 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 theory. ... In mathematics, a singular point of an algebraic variety V is a point P that is special (so, singular), in the geometric sense that V is not locally flat there. ... Roger Godement is a French mathematician, known for his work in functional analysis, and also his expository books. ... Mikio Sato (佐藤幹夫, born April 18, 1928) is a Japanese mathematician, working in what he calls algebraic analysis. ... In mathematics, hyperfunctions are sums of boundary values of holomorphic functions, and can be thought of informally as distributions of infinite order. ... In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ... In mathematics, the derived category D(C) of a category C is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C (which therefore should already be an abelian category). ... In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... Intuitionistic logic, or constructivist logic, is the logic used in mathematical intuitionism and other forms of mathematical constructivism. ... Kripke semantics (also known as possible world semantics, relational semantics, or frame semantics) is a formal semantics for modal logic systems, created in late 1950s and early 1960s by Saul Kripke. ... Gottfried Leibniz Gottfried Wilhelm von Leibniz (July 1, 1646 in Leipzig - November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. ...

Contents

Introduction

The formal definition

Definition of a presheaf

The gluing axiom

Examples

Morphisms of sheaves

Stalks of a sheaf at a point and germs of functions

The étale space of a sheaf

Generalizations

History

See also

References

Contents

Introduction GA_googleFillSlot("encyclopedia_square");

The formal definition

Definition of a presheaf

The gluing axiom

Examples

Morphisms of sheaves

Stalks of a sheaf at a point and germs of functions

The étale space of a sheaf

Generalizations

History

See also

References

Introduction