Encyclopedia - Zero Morphism: Encyclopedia - Zero Morphism
In category theory, a zero morphism is a special kind of "trivial" morphism. Suppose C is a category, and for any two objects X and Y in ...   » Read the article

Encyclopedia - Coherent Sheaf: Encyclopedia - Coherent Sheaf
In mathematics, especially in algebraic geometry and the theory of complex manifolds, a coherent sheaf F on a locally ringed space X is a...   » Read the article

Encyclopedia - Commensurability Mathematics: Encyclopedia - Commensurability Mathematics
Commensurability mathematics - Commensurability in general. Generally, two quantities are commensurable if both can be measured in the ...   » 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 - 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 - Kernel Category Theory: Encyclopedia Ii - Kernel Category Theory - Definition
Let C be a category. In order to define a kernel in the general category-theoretical sense, C needs to have zero morphisms. In that case,...   » Read the article

Encyclopedia - Preadditive Category: Encyclopedia Ii - Preadditive Category - Elementary Properties
Because every hom-set Hom(A,B) is an abelian group, it has a zero element 0. This is the zero morphism from A to B. Because composition o...   » Read the article

Encyclopedia - Coequalizer: Encyclopedia Ii - Coequalizer - Definition
The coequalizer is a special kind of colimit in category theory. Specifically it is the colimit of the diagram consisting of two objects ...   » Read the article

Encyclopedia - Initial Object: Encyclopedia Ii - Initial Object - Properties
Not all categories have initial or terminal objects, as will be seen below. Directly from the definition, one can show however that if an...   » Read the article

Encyclopedia - Atiyah–singer Index Theorem: Encyclopedia Ii - Atiyah–singer Index Theorem - An Example On The Circle
We start by considering complex-valued functions on the circle that are "square integrable"(i.e., elements of L2) and have no Fourier coe...   » 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 - Additive Category: Encyclopedia Ii - Additive Category - Elementary Properties
Every additive category is of course a preadditive category, and many basic properties of these categories are described under that subje...   » Read the article

Encyclopedia - Abelian Category: Encyclopedia Ii - Abelian Category - Definitions
A category is abelian if it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal. By...   » 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 - Universal Property: Encyclopedia Ii - Universal Property - Properties
Universal property - Existence and uniqueness. Defining a quantity does not guarantee its existence. Given a functor U and an object X ...   » Read the article

Encyclopedia - Model Category: Encyclopedia Ii - Model Category - Formal Definition
The definition given initially by Quillen was that of a closed model category, the assumptions of which seemed strong at the time, motiva...   » Read the article

Encyclopedia - Model Category: Encyclopedia Ii - Model Category - Some Constructions
Every closed model category has a terminal object by the completeness axiom and an initial object by the cocompleteness axiom since these...   » Read the article

Encyclopedia - Coequalizer: Encyclopedia Ii - Coequalizer - Special Cases
In categories with zero morphisms, one can define a cokernel of a morphism f as the coequalizer of f and the parallel zero morphism. In p...   » Read the article

Encyclopedia - Universal Property: Encyclopedia Ii - Universal Property - Examples
We give a few worked examples to highlight the general idea. The reader can construct numerous other examples by consulting the articles ...   » Read the article

Encyclopedia - Kernel Category Theory: Encyclopedia Ii - Kernel Category Theory - Relation To Other Categorical Concepts
The dual concept to that of kernel is that of cokernel. That is, the kernel of a morphism is its cokernel in the opposite category, and v...   » Read the article

Encyclopedia - Kernel Category Theory: Encyclopedia Ii - Kernel Category Theory - Examples
Kernels are familiar in many categories from abstract algebra, such as the category of groups or the category of (left) modules over a fi...   » Read the article

Encyclopedia - Universal Property: Encyclopedia Ii - Universal Property - Formal Definition
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 consis...   » Read the article

Encyclopedia - Preadditive Category: Encyclopedia Ii - Preadditive Category - Kernels And Cokernels
Because the hom-sets in a preadditive category have zero morphisms, the notion of kernel and cokernel make sense. That is, if f: A&#...   » Read the article

Encyclopedia - Additive Category: Encyclopedia Ii - Additive Category - Additive Functors
Recall that a functor F: C → D between preadditive categories is additive if it is an Abelian group homomorphism on each hom-set in C. ...   » Read the article

Encyclopedia - Additive Category: Encyclopedia Ii - Additive Category - Examples
The original example of an additive category is the category Ab of Abelian groups with group homomorphisms. Ab is preadditive because it ...   » 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 - Abelian Category: Encyclopedia Ii - Abelian Category - Elementary Properties
Given any pair A, B of objects in an abelian category, there is a special zero morphism from A to B. This can be defined as the zero elem...   » Read the article

Encyclopedia - Preadditive Category: Encyclopedia Ii - Preadditive Category - Examples
The most obvious example of a preadditive category is the category Ab itself. More precisely, Ab is a closed monoidal category. (Note tha...   » Read the article

Encyclopedia - Preadditive Category: Encyclopedia Ii - Preadditive Category - Additive Functors
If C and D are preadditive categories, then a functor F: C → D is additive if it too is enriched over the category Ab. T...   » Read the article

Encyclopedia - Atiyah–singer Index Theorem: Encyclopedia Ii - Atiyah–singer Index Theorem - Proof Techniques
The index theorem has been proved, and reproved, as a general statement. Atiyah-Singer comment that the initial proof was based on that o...   » Read the article

Encyclopedia - Atiyah–singer Index Theorem: Encyclopedia Ii - Atiyah–singer Index Theorem - History
The theorem came at the end of more than 100 years' development on the theory of elliptic operators (such as Laplacians), going back to t...   » Read the article

Encyclopedia - Atiyah–singer Index Theorem: Encyclopedia Ii - Atiyah–singer Index Theorem - More Formal Statement
We start with a compact smooth manifold M (without boundary), a vector bundle, "E" on M and an elliptic operator D on M. Here "D" is a di...   » Read the article

Encyclopedia - Preadditive Category: Encyclopedia Ii - Preadditive Category - Biproducts
Any finite product in a preadditive category must also be a coproduct, and conversely. In fact, finite products and coproducts in additiv...   » Read the article

Encyclopedia - Model Category: Encyclopedia Ii - Model Category - Motivation
Model categories can provide a natural setting for homotopy theory: the category of topological spaces is a model category, with the homo...   » Read the article