direct limit

In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to 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 enable one to discuss in a refined way what is a local property, as appl ...

Including:

• Sheaf mathematics - Introduction
• Sheaf mathematics - The formal definition
• Sheaf mathematics - Definition of a presheaf
• Sheaf mathematics - The gluing axiom
• Sheaf mathematics - Examples
• Sheaf mathematics - Morphisms of sheaves
• Sheaf mathematics - Stalks of a sheaf at a point and germs of functions
• Sheaf mathematics - The étale space of a sheaf
• Sheaf mathematics - Generalizations
• Sheaf mathematics - History

Read more here: » Sheaf mathematics: Encyclopedia - Sheaf mathematics

Čech cohomology is a particular type of cohomology in mathematics. It is named for the mathematician Eduard Čech. Čech cohomology - Construction. Let X be a topological space with open cover U={Uα}α∈I, where I is a countable ordered set. We will simplify notation by writing intersections as Uα∩ Uβ = Uαβ, and so on for higher intersections. For every intersection Uα ...

Including:

• ÄŒech cohomology - Construction
• ÄŒech cohomology - Relation to other cohomology theories

Read more here: » ÄŒech cohomology: Encyclopedia - ÄŒech cohomology

A subset of X is closed if and only if its preimage under fi is closed in Yi for each i ∈ I. The final topology on X can be characterized by the following universal property: a function g from X to some space Z is continuous if and only if ...

See also:

Final topology, Final topology - Definition, Final topology - Examples, Final topology - Properties

Read more here: » Final topology: Encyclopedia II - Final topology - Properties

As an abelian group the dyadic rationals are the direct limit of infinite cyclic subgroups 2−nZ for n = 0, 1, 2, ... . In the spirit of Pontryagin duality, there is a dual object, namely the inverse limit of the unit circle group under the repeated squaring map ζ → ζ2. The resulting topological group D is called the dyadic solenoid (see solenoid group). As a t ...

See also:

Dyadic rational, Dyadic rational - Dyadic solenoid

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Inverse limit - Algebraic objects. We start with the definition of an inverse system of groups and homomorphisms. Let (I, ≤) be a directed poset (not all authors require I to be directed). Let (Ai)i∈I be a family of groups and suppose we have a family of homomorphisms fij : Aj → Ai for all i ≤ j (note the order) with the following properties:< ...

See also:

Inverse limit, Inverse limit - Formal definition, Inverse limit - Algebraic objects, Inverse limit - General definition, Inverse limit - Examples, Inverse limit - Related concepts and generalizations

Read more here: » Inverse limit: Encyclopedia II - Inverse limit - Formal definition

Formally, a pro-finite group is a group that is isomorphic to the inverse limit of an inverse system of finite groups. Pro-finite groups are naturally regarded as topological groups: each of the finite groups carries the discrete topology, and since the inverse limit is a subset of the product of these discrete spaces, it inherits a topology which turns it into a topological group. Equivalently, one can define pro-finite groups to be the topological groups that a ...

See also:

Pro-finite group, Pro-finite group - Definition, Pro-finite group - Examples, Pro-finite group - Properties and facts, Pro-finite group - Pro-finite completion, Pro-finite group - Ind-finite groups

Read more here: » Pro-finite group: Encyclopedia II - Pro-finite group - Definition

Universal property - Existence and uniqueness. Defining a quantity does not guarantee its existence. Given a functor U and an object X as above, there may or may not exist a universal morphism from X to U (or from U to X). If, however, a universal morphism (A, φ) does exists then it is unique up to a unique isomorphism. That is, if (A′, φ′) is another such pair then there exists a unique isomorphism g : A → A′ such ...

See also:

Universal property, Universal property - Formal definition, Universal property - Properties, Universal property - Existence and uniqueness, Universal property - Equivalent formulations, Universal property - Relation to adjoint functors, Universal property - Examples, Universal property - Tensor algebras, Universal property - Kernels, Universal property - Limits and colimits, Universal property - What is it good for?, Universal property - History

Read more here: » Universal property: Encyclopedia II - Universal property - Properties

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 op ...

See also:

Sheaf mathematics, Sheaf mathematics - Introduction, Sheaf mathematics - The formal definition, Sheaf mathematics - Definition of a presheaf, Sheaf mathematics - The gluing axiom, Sheaf mathematics - Examples, Sheaf mathematics - Morphisms of sheaves, Sheaf mathematics - Stalks of a sheaf at a point and germs of functions, Sheaf mathematics - The étale space of a sheaf, Sheaf mathematics - Generalizations, Sheaf mathematics - History

Read more here: » Sheaf mathematics: Encyclopedia II - Sheaf mathematics - Introduction

The algebraic geometers of the Italian school had often used the somewhat foggy concept of "generic point" when proving statements about algebraic varieties. What is true for the generic point is true for all points of the variety except a small number of special points. In the 1920s, Emmy Noether had first suggested a way to clarify the concept: start with the coordinate ring of the variety (the ring of all polynomial functions defined on the variety); the maximal ideals of this ring will correspond to ordinary points of the variety (under ...

See also:

Scheme mathematics, Scheme mathematics - History and motivation, Scheme mathematics - Definitions, Scheme mathematics - The category of schemes, Scheme mathematics - Types of schemes, Scheme mathematics - O_X modules

Read more here: » Scheme mathematics: Encyclopedia II - Scheme mathematics - History and motivation

Let X be a topological space with open cover U={Uα}α∈I, where I is a countable ordered set. We will simplify notation by writing intersections as Uα∩ Uβ = Uαβ, and so on for higher intersections. For every intersection Uα0···αn there are n + 1 inclusions defined as follows: ∂i : Uα0···αn→ Uα0···α(i ∠...

See also:

ÄŒech cohomology, ÄŒech cohomology - Construction, ÄŒech cohomology - Relation to other cohomology theories

Read more here: » ÄŒech cohomology: Encyclopedia II - ÄŒech cohomology - Construction

Main article: limit of a function Suppose f(x) is a real function and c is a real number. The expression: means that f(x) can be made to be as close to L as desired by making x sufficiently close to c. In that case, we say that "the limit of f(x), as x approaches c". Note that this statement can be true even if f(c) L. Indeed, the function f(x) need not even b ...

See also:

Limit mathematics, Limit mathematics - Limit of a function, Limit mathematics - Formal definition, Limit mathematics - Limit of a function at infinity, Limit mathematics - Limit of a sequence, Limit mathematics - Topological net, Limit mathematics - Limit in category theory

Read more here: » Limit mathematics: Encyclopedia II - Limit mathematics - Limit of a function

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 carry ...

See also:

Sheaf mathematics, Sheaf mathematics - Introduction, Sheaf mathematics - The formal definition, Sheaf mathematics - Definition of a presheaf, Sheaf mathematics - The gluing axiom, Sheaf mathematics - Examples, Sheaf mathematics - Morphisms of sheaves, Sheaf mathematics - Stalks of a sheaf at a point and germs of functions, Sheaf mathematics - The étale space of a sheaf, Sheaf mathematics - Generalizations, Sheaf mathematics - History

Read more here: » Sheaf mathematics: Encyclopedia II - Sheaf mathematics - Generalizations

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. 1936 Eduard Čech introduces the nerve construction, for associating a simplicial complex to an open covering. 1938 Hassler Whitney gives a 'modern' definition of cohomology, summarizing the work since J. W. Alexander and Kolmogorov first defined cochainsSee also:

Sheaf mathematics, Sheaf mathematics - Introduction, Sheaf mathematics - The formal definition, Sheaf mathematics - Definition of a presheaf, Sheaf mathematics - The gluing axiom, Sheaf mathematics - Examples, Sheaf mathematics - Morphisms of sheaves, Sheaf mathematics - Stalks of a sheaf at a point and germs of functions, Sheaf mathematics - The étale space of a sheaf, Sheaf mathematics - Generalizations, Sheaf mathematics - History

Read more here: » Sheaf mathematics: Encyclopedia II - Sheaf mathematics - History

We give a few worked examples to highlight the general idea. The reader can construct numerous other examples by consulting the articles mentioned in the introduction. Universal property - Tensor algebras. Let C be the category of vector spaces K-Vect over a field K and let D be the category of algebras K-Alg over K (assumed to be unital and associative). Let U be the forgetful functor which assig ...

See also:

Universal property, Universal property - Formal definition, Universal property - Properties, Universal property - Existence and uniqueness, Universal property - Equivalent formulations, Universal property - Relation to adjoint functors, Universal property - Examples, Universal property - Tensor algebras, Universal property - Kernels, Universal property - Limits and colimits, Universal property - What is it good for?, Universal property - History

Read more here: » Universal property: Encyclopedia II - Universal property - Examples

In early developments of sheaf theory, it was shown that giving a sheaf F on X is as good as giving a certain topological space E together with a continuous map from E to X. More precisely: to every sheaf F of sets on X there exists a local homeomorphism π: E → X such that F is isomorphic to the sheaf o ...

See also:

Sheaf mathematics, Sheaf mathematics - Introduction, Sheaf mathematics - The formal definition, Sheaf mathematics - Definition of a presheaf, Sheaf mathematics - The gluing axiom, Sheaf mathematics - Examples, Sheaf mathematics - Morphisms of sheaves, Sheaf mathematics - Stalks of a sheaf at a point and germs of functions, Sheaf mathematics - The étale space of a sheaf, Sheaf mathematics - Generalizations, Sheaf mathematics - History

Read more here: » Sheaf mathematics: Encyclopedia II - Sheaf mathematics - The étale space of a sheaf

Every product of (arbitrarily many) pro-finite groups is pro-finite; the topology arising from the pro-finiteness agrees with the product topology. Every closed subgroup of a pro-finite group is itself pro-finite; the topology arising from the pro-finiteness agrees with the subspace topology. If N is a closed normal subgroup of a pro-finite group G, then the factor group G/N is pro-finite; the topology arising f ...

See also:

Pro-finite group, Pro-finite group - Definition, Pro-finite group - Examples, Pro-finite group - Properties and facts, Pro-finite group - Pro-finite completion, Pro-finite group - Ind-finite groups

Read more here: » Pro-finite group: Encyclopedia II - Pro-finite group - Properties and facts

Let U : D → C be a functor from a category D to a category C, and let X be an object of C. A universal morphism from X to U consists of a pair (A, φ) where A is an object of D and φ : X → U(A) is a morphism in C, such that the following universal property is satisfied: Whenever Y is an object of D and f : X → U(Y) is a morphism in C, then there exists a unique morphism g : A → < ...

See also:

Universal property, Universal property - Formal definition, Universal property - Properties, Universal property - Existence and uniqueness, Universal property - Equivalent formulations, Universal property - Relation to adjoint functors, Universal property - Examples, Universal property - Tensor algebras, Universal property - Kernels, Universal property - Limits and colimits, Universal property - What is it good for?, Universal property - History

Read more here: » Universal property: Encyclopedia II - Universal property - Formal definition

Given a set X and a family of topological spaces Yi with functions the final topology Ï„ on X is the finest topology such that each is continuous. Explicitly, the final topology may be described as follows: a subset U of X is open if and only if is open in Y< ...

See also:

Final topology, Final topology - Definition, Final topology - Examples, Final topology - Properties

Read more here: » Final topology: Encyclopedia II - Final topology - Definition

Fix a point x of X. We would like to study the behavior of F near the point x. In analytical terms, we would like to somehow take the limit as we get nearer and nearer to the point x. The corresponding concept is to take the direct limit of F(N) as N runs over the open neighbourhoods of x ordered by inclusion (in categorical terminology, this is an example of a colimit). We denote this limit by Fx and call it the stalk of F at x. If F ...

See also:

Sheaf mathematics, Sheaf mathematics - Introduction, Sheaf mathematics - The formal definition, Sheaf mathematics - Definition of a presheaf, Sheaf mathematics - The gluing axiom, Sheaf mathematics - Examples, Sheaf mathematics - Morphisms of sheaves, Sheaf mathematics - Stalks of a sheaf at a point and germs of functions, Sheaf mathematics - The étale space of a sheaf, Sheaf mathematics - Generalizations, Sheaf mathematics - History

Read more here: » Sheaf mathematics: Encyclopedia II - Sheaf mathematics - Stalks of a sheaf at a point and germs of functions

There are many ways one can qualify a scheme. According to a basic idea of Grothendieck, conditions should be applied to a morphism of schemes. Any scheme S has a unique morphism to Spec(Z), so this attitude, part of the relative point of view, doesn't lose anything. For detail on the development of scheme theory, which quickly becomes technically demanding, see first glossary of scheme theory. ...

See also:

Scheme mathematics, Scheme mathematics - History and motivation, Scheme mathematics - Definitions, Scheme mathematics - The category of schemes, Scheme mathematics - Types of schemes, Scheme mathematics - O_X modules

Read more here: » Scheme mathematics: Encyclopedia II - Scheme mathematics - Types of schemes