Encyclopedia - Abelian Category: Encyclopedia - 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 an...   » Read the article

Encyclopedia - Category Mathematics: Encyclopedia - Category Mathematics
In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. Categories...   » Read the article

Encyclopedia - Category Of Abelian Groups: Encyclopedia - Category Of Abelian Groups
In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abeli...   » Read the article

Encyclopedia - Sheaf Mathematics: Encyclopedia - Sheaf Mathematics
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 ope...   » Read the article

Encyclopedia - Abelian: Encyclopedia - Abelian
In mathematics, the term abelian is used in many different definitions: Abelian - In group theory. An abelian group is a group in which...   » Read the article

Encyclopedia - Abelian Group: Encyclopedia - Abelian Group
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. In o...   » Read the article

Encyclopedia - Group Theory: Encyclopedia Ii - Group Theory - History
There are three historical roots of group theory: the theory of algebraic equations, number theory and geometry. Euler, Gauss, Lagrange, ...   » Read the article

Encyclopedia - Functor Category: Encyclopedia Ii - Functor Category - Definition
Suppose C is a small category (i.e. the objects form a set rather than a proper class) and D is an arbitrary category. The category of fu...   » Read the article

Encyclopedia - Ringed Space: Encyclopedia Ii - Ringed Space - Morphisms
A morphism of ringed spaces is simply a morphism of sheaves. Explicitly, a morphism from (X, OX) to (Y, OY) is given by the following dat...   » Read the article

Encyclopedia - Flatness: Encyclopedia Ii - Flatness - Flatness In Mechanical Engineering
Joseph Whitworth popularized the first practical method of making accurate flat surfaces during the 1830s, using engineer's blue and scra...   » Read the article

Encyclopedia - Normal Morphism: Encyclopedia Ii - Normal Morphism - Definition
A category C must have zero morphisms for the concept of normality to make complete sense. In that case, we say that a monomorphism is no...   » Read the article

Encyclopedia - Universal Enveloping Algebra: Encyclopedia Ii - Universal Enveloping Algebra - Direct Construction
For general reasons having to do with universal properties, we can say that if a Lie algebra has a universal enveloping algebra, then thi...   » Read the article

Encyclopedia - Yoneda Lemma: Encyclopedia Ii - Yoneda Lemma - Formal Statement
Yoneda lemma - General version. Yoneda's lemma concerns functors from a fixed category C to the category of sets, Set. If C is a locall...   » Read the article

Encyclopedia - Injective Cogenerator: Encyclopedia Ii - Injective Cogenerator - General Theory
In topological language, we try to find covers of unfamiliar objects. Finding a generator of an abelian category allows one to express ev...   » Read the article

Encyclopedia - Group Homomorphism: Encyclopedia Ii - Group Homomorphism - Image And Kernel
We define the kernel of h to be ker(h) = { u in G : h(u) = eH } and the image of h to be im(h) = { h(u) : u in G }. The ker...   » Read the article

Encyclopedia - Injective Sheaf: Encyclopedia Ii - Injective Sheaf - Injective Sheaves
An injective sheaf F is just a sheaf that is an injective element of the category of abelian sheaves; in other words, homomorphisms from ...   » Read the article

Encyclopedia - Equivalence Of Categories: Encyclopedia Ii - Equivalence Of Categories - Equivalent Characterizations
One can show that a functor F : C -> D yields an equivalence of categories if and only if has all of the following three properti...   » Read the article

Encyclopedia - Grothendieck Group: Encyclopedia Ii - Grothendieck Group - Explicit Construction
To construct the Grothendieck group of a commutative monoid M, one forms the Cartesian product M×M. The two coordinates are meant to r...   » Read the article

Encyclopedia - Sheaf Cohomology: Encyclopedia Ii - Sheaf Cohomology - Definitions
The first version of sheaf cohomology to be defined was that based on Čech cohomology, in which the relatively small change was made of ...   » Read the article

Encyclopedia - Module Mathematics: Encyclopedia Ii - Module Mathematics - Motivation
In a vector space, the set of scalars forms a field and acts on the vectors by scalar multiplication, subject to certain formal laws such...   » Read the article

Encyclopedia - Abelian Group: Encyclopedia Ii - Abelian Group - Properties
If n is a natural number and x is an element of an abelian group G written additively, then nx can be defined as x + x + ... + x (n summa...   » Read the article

Encyclopedia - Category Mathematics: Encyclopedia Ii - Category Mathematics - Definition
A category C consists of a class ob(C) of objects: a class hom(C) of morphisms. Each morphism f has a unique source object a and target ...   » Read the article

Encyclopedia - Sheaf Mathematics: Encyclopedia Ii - Sheaf Mathematics - Introduction
Sheaves are used in topology, algebraic geometry and differential geometry whenever one wants to keep track of algebraic data that vary w...   » Read the article

Encyclopedia - Sheaf Mathematics: Encyclopedia Ii - Sheaf Mathematics - Generalizations
It is possible to define a cohomology theory for sheaves of abelian groups (sheaf cohomology) that can give much useful, more concrete in...   » Read the article

Encyclopedia - Grothendieck Group: Encyclopedia Ii - Grothendieck Group - Generalization
To apply the Grothendieck group to purely algebraic settings, it is useful to generalize it to the case of an essentially small abelian c...   » Read the article

Encyclopedia - Sheaf Mathematics: Encyclopedia Ii - Sheaf Mathematics - 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 togethe...   » Read the article

Encyclopedia - Flatness: Encyclopedia Ii - Flatness - Flatness In Cosmology
In cosmology, the concept of "curvature of space" is considered. A space without curvature is called a "flat space" or Euclidean space. A...   » Read the article

Encyclopedia - Injective Sheaf: Encyclopedia Ii - Injective Sheaf - Soft Sheaves
A soft sheaf F over X is one such that any section over any closed subset of X can be extended to a global section. Soft sheaves are acyc...   » Read the article

Encyclopedia - Injective Sheaf: Encyclopedia Ii - Injective Sheaf - Flasque Or Flabby Sheaves
A flasque sheaf (also called a flabby sheaf) is a sheaf with the following property: if X is the base topological space on which the she...   » Read the article

Encyclopedia - Flatness: Encyclopedia Ii - Flatness - Flatness In Mathematics
The flatness of a surface is the degree to which it approximates a mathematical plane. The term is generalized for higher-dimensional man...   » Read the article

Encyclopedia - Grothendieck Group: Encyclopedia Ii - Grothendieck Group - Splitting Principle
The relationship between K0 of a commutative monoid and K0 of an abelian category comes from the splitting principle. According to the sp...   » Read the article

Encyclopedia - Group Homomorphism: Encyclopedia Ii - Group Homomorphism - Isomorphisms Endomorphisms And Automorphisms
If the homomorphism h is a bijection, then one can show that its inverse is also a group homomorphism, and h is called a group isomorphis...   » Read the article

Encyclopedia - Sheaf Mathematics: Encyclopedia Ii - Sheaf Mathematics - 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 asso...   » Read the article

Encyclopedia - Universal Enveloping Algebra: Encyclopedia Ii - Universal Enveloping Algebra - Examples In Particular Cases
If L is abelian (that is, the bracket is always 0), then U(L) is commutative; if a basis of the vector space L has been chosen, then U(L)...   » Read the article

Encyclopedia - Yoneda Lemma: Encyclopedia Ii - Yoneda Lemma - Generalities
The Yoneda lemma suggests that instead of studying the (small) category C, one should study the category of all functors of C into Set (t...   » Read the article

Encyclopedia - Universal Enveloping Algebra: Encyclopedia Ii - Universal Enveloping Algebra - Universal Property
Let L be any Lie algebra over K. Given a unital associative K-algebra U and a Lie algebra homomorphism h: L → UL, (notation as above)...   » Read the article

Encyclopedia - Sheaf Mathematics: Encyclopedia Ii - Sheaf Mathematics - Examples
In addition to the sheaves of continuous functions, differentiable functions and vector fields given in the introduction, sheaves of sect...   » Read the article

Encyclopedia - Injective Sheaf: Encyclopedia Ii - Injective Sheaf - Fine Sheaves
A fine sheaf over X is one with "partitions of unity"; more precisely for any open cover of the space X we can find a family of homomorph...   » Read the article

Encyclopedia - Injective Cogenerator: Encyclopedia Ii - Injective Cogenerator - The Abelian Group Case
Assuming one has a category like that of abelian groups, one can in fact form direct sums of copies of G until the morphism f:Sum(G) ->...   » Read the article

Encyclopedia - Sheaf Mathematics: Encyclopedia Ii - Sheaf Mathematics - 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 φU ...   » Read the article

Encyclopedia - Sheaf Mathematics: Encyclopedia Ii - Sheaf Mathematics - 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 lim...   » Read the article

Encyclopedia - Sheaf Cohomology: Encyclopedia Ii - Sheaf Cohomology - Applications
Subsequently there were further technical extensions (for example in Godement's book), and areas of application. For example, sheaves wer...   » Read the article

Encyclopedia - Ringed Space: Encyclopedia Ii - Ringed Space - Tangent Spaces
Locally ringed spaces have just enough structure to allow the meaningful definition of tangent spaces. Let X be locally ringed space with...   » Read the article

Encyclopedia - Abelian Group: Encyclopedia Ii - Abelian Group - Relation To Other Mathematical Topics
The abelian group, together with group homomorphisms, form a category, the prototype of an abelian category. In this encyclopedia, we den...   » Read the article

Encyclopedia - Module Mathematics: Encyclopedia Ii - Module Mathematics - Definition
Specifically, a left module over the ring R consists of an abelian group (M, +) and an operation R × M → M (called scalar multiplicati...   » Read the article

Encyclopedia - Ringed Space: Encyclopedia Ii - Ringed Space - Examples
An arbitrary topological space X can be considered a locally ringed space by taking OX to be the sheaf of real-valued (or complex-valued)...   » Read the article

Encyclopedia - Ringed Space: Encyclopedia Ii - Ringed Space - Definition
Formally, a ringed space is a topological space X together with a sheaf of commutative rings OX on X. The sheaf OX is called the structur...   » Read the article

Encyclopedia - Category Mathematics: Encyclopedia Ii - Category Mathematics - Types Of Morphisms
A morphism f : a → b is called a monomorphism (or monic) if fg1 = fg2 implies g1 = g2 for all morphisms g1, g2 : x → a. an...   » Read the article

Encyclopedia - Group Theory: Encyclopedia Ii - Group Theory - Elementary Introduction
Groups are used throughout mathematics and the sciences, often to capture the internal symmetry of other structures, in the form of autom...   » Read the article

Encyclopedia - Group Theory: Encyclopedia Ii - Group Theory - Generalizations
In abstract algebra, we get some related structures which are similar to groups by relaxing some of the axioms given at the top of the ar...   » Read the article

Encyclopedia - Module Mathematics: Encyclopedia Ii - Module Mathematics - Submodules And Homomorphisms
Suppose M is a left R-module and N is a subgroup of M. Then N is a submodule (or R-submodule, to be more explicit) if, for any n in N and...   » Read the article

Encyclopedia - Module Mathematics: Encyclopedia Ii - Module Mathematics - Types Of Modules
Finitely generated. A module M is finitely generated if there exist finitely many elements x1,...,xn in M such that every element of M is...   » Read the article

Encyclopedia - Abelian Group: Encyclopedia Ii - Abelian Group - Examples
Every cyclic group G is abelian, because if x, y are in G, then xy = aman = am + n = an + m = anam = yx. In particular, the integers Z fo...   » Read the article

Encyclopedia - Abelian Group: Encyclopedia Ii - Abelian Group - Notation
There are two main notational conventions for abelian groups -- additive and multiplicative. The multiplicative notation is the usual not...   » Read the article

Encyclopedia - Equivalence Of Categories: Encyclopedia Ii - Equivalence Of Categories - Definition
Formally, given two categories C and D, an equivalence of categories consists of a functor F : C -> D, a functor G : D ->...   » Read the article

Encyclopedia - Abelian Group: Encyclopedia Ii - Abelian Group - Multiplication Table
To verify that a certain finite group is indeed abelian, a table (matrix) can be drawn up in the similar fashion to a multiplication tabl...   » Read the article

Encyclopedia - Abelian Group: Encyclopedia Ii - Abelian Group - Finite Abelian Groups
The fundamental theorem of finite abelian groups states that every finite abelian group can be expressed as the direct sum of cyclic subg...   » Read the article

Encyclopedia - Module Mathematics: Encyclopedia Ii - Module Mathematics - Relation To Representation Theory
If M is a left R-module, then the action of an element r in R is defined to be the map M → M that sends each x to rx (or xr in the case...   » Read the article

Encyclopedia - Module Mathematics: Encyclopedia Ii - Module Mathematics - Generalizations
Any ring R can be viewed as a preadditive category with a single object. With this understanding, a left R-module is nothing but a (covar...   » Read the article

Encyclopedia - Abelian Group: Encyclopedia Ii - Abelian Group - List Of Small Abelian Groups
Extracted from the list of small groups is the following table of small abelian groups. Note that e.g. "3 × Z2" means that there are 3 s...   » Read the article

Encyclopedia - Sheaf Mathematics: Encyclopedia Ii - Sheaf Mathematics - 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...   » Read the article