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 C we are given a morphism 0XY : X → Y with the following property: for any two morphism f : R → S and g : U → V we obtain a commutative diagram: Then the morphisms 0XY are c ...

Including:

• Zero morphism - Examples

Read more here: » Zero morphism: Encyclopedia - Zero morphism

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. Here OX is the structure sheaf of local rings, given by definition on X. The form of the definition is a global (on X) way of carrying across the idea of a finitely-presented mo ...

Including:

• Coherent sheaf - Coherent cohomology

Read more here: » Coherent sheaf: Encyclopedia - Coherent sheaf

Commensurability mathematics - Commensurability in general. Generally, two quantities are commensurable if both can be measured in the same units. For example, a distance measured in miles and a quantity of water measured in gallons are incommensurable. A time measured in weeks and a time measured in minutes are commensurable because a week is a constant number of minutes (10080), so that one can convert between the two units by multiplying or dividing by 10080. Commensurabilit ...

Including:

• Commensurability mathematics - Commensurability in general
• Commensurability mathematics - Commensurability in mathematics

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

In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. Categories appear in virtually every branch of modern mathematics and are a central unifying notion. The study of categories in their own right is known as category theory. For more extensive motivational background and historical notes, see category theory and the list of category theory topics. Category mathematics - Definition. A category C consists of Including:

• Category mathematics - Definition
• Category mathematics - Examples
• Category mathematics - Types of morphisms
• Category mathematics - Types of categories

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

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. The motivating prototype example of an abelian category is the category of abelian groups, Ab. 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 a theorem of Pe ...

Including:

• Abelian category - Definitions
• Abelian category - Examples
• Abelian category - Elementary properties
• Abelian category - Related concepts
• Abelian category - History

Read more here: » Abelian category: Encyclopedia - Abelian category

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, if f : X → Y is an arbitrary morphism in C, then a kernel of f is an equaliser of f and the zero morphism from X to Y. In symbols: ker(f) = eq(f, 0XY) To be more explicit, the following universal property can be used. A kernel of f is any morphism k : See also:

Kernel category theory, Kernel category theory - Definition, Kernel category theory - Examples, Kernel category theory - Relation to other categorical concepts, Kernel category theory - Relationship to algebraic kernels

Read more here: » Kernel category theory: Encyclopedia II - Kernel category theory - Definition

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 of morphisms is bilinear, the composition of a zero morphism and any other morphism (on either side) must be another zero morphism. If you think of composition as analogous to multiplication, then this says that multiplication by zero always results in a product of zero, which is a familiar intuition. Extending this analogy, the fact that composition is bilinear in gener ...

See also:

Preadditive category, Preadditive category - Examples, Preadditive category - Elementary properties, Preadditive category - Additive functors, Preadditive category - Biproducts, Preadditive category - Kernels and cokernels, Preadditive category - Special cases

Read more here: » Preadditive category: Encyclopedia II - Preadditive category - Elementary properties

The coequalizer is a special kind of colimit in category theory. Specifically it is the colimit of the diagram consisting of two objects X and Y and two parallel morphisms f, g : X → Y. More explicity, the coequalizer can be defined as an object Q and a morphism q : Y → Q such that q O f = q O g. Moreover, the pair (Q, q) must be universal in the sense that given any other such pair (Q′, q′) there exists a unique morphism u : Q → Q′ ...

See also:

Coequalizer, Coequalizer - Definition, Coequalizer - Examples, Coequalizer - Special cases

Read more here: » Coequalizer: Encyclopedia II - Coequalizer - Definition

Not all categories have initial or terminal objects, as will be seen below. Directly from the definition, one can show however that if an initial object exists, then it is unique up to a unique isomorphism. The same is true for terminal objects. The automorphism group of an initial (or terminal) object I is trivial. Aut(I) = Hom(I,I) = { idI }. ...

See also:

Initial object, Initial object - Properties, Initial object - Examples

Read more here: » Initial object: Encyclopedia II - Initial object - Properties

We start by considering complex-valued functions on the circle that are "square integrable"(i.e., elements of L2) and have no Fourier coefficients with negative phase (equivalently, that extend to be holomorphic in the disk). We want an operator that takes one of these functions and gives us back another one. Given a continuous function f, let Tf be the operator that multiplies by f and then kills off the negative Fourier coefficients of the resulting function. For ...

See also:

Atiyah–Singer index theorem, Atiyah–Singer index theorem - An example on the circle, Atiyah–Singer index theorem - More formal statement, Atiyah–Singer index theorem - History, Atiyah–Singer index theorem - Proof techniques, Atiyah–Singer index theorem - Further developments

Read more here: » Atiyah–Singer index theorem: Encyclopedia II - Atiyah–Singer index theorem - An example on the circle

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 normal if it is the kernel of some morphism, and an epimorphism is normal (or conormal) if it is the cokernel of some morphism. C itself is normal if every monomorphism is normal. C is conormal if every epimorphism is normal. Finally, C is binormal if it's both normal and conormal. But note that some authors will use only the word "normal" t ...

See also:

Normal morphism, Normal morphism - Definition, Normal morphism - Examples

Read more here: » Normal morphism: Encyclopedia II - Normal morphism - Definition

Every additive category is of course a preadditive category, and many basic properties of these categories are described under that subject. This article concerns itself with the properties that exist specifically because of the existence of biproducts. First note that because nullary biproducts exist, every additive category has a zero object, commonly denoted simply "0". Given objects A and B in an additive category, we can use matrices to study the biproducts of A and B with themselves. Specifical ...

See also:

Additive category, Additive category - Examples, Additive category - Elementary properties, Additive category - Additive functors, Additive category - Special cases, Additive category - Sources

Read more here: » Additive category: Encyclopedia II - Additive category - Elementary properties

A category is abelian if it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal. By a theorem of Peter Freyd, this definition is equivalent to the following "piecemeal" definition: A category is preadditive if it is enriched over the monoidal category Ab of abelian groups. This means that all hom-sets are abelian groups and the composition of morphisms is bilinear. A preadditive category is ...

See also:

Abelian category, Abelian category - Definitions, Abelian category - Examples, Abelian category - Elementary properties, Abelian category - Related concepts, Abelian category - History

Read more here: » Abelian category: Encyclopedia II - Abelian category - Definitions

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 object b where a and b are in ob(C). We write f: a → b, and we say "f is a morphism from a to b". We write hom(a, b) (or homC(a, b)) to denote the hom-class of all morphisms from a to b. (Some a ...

See also:

Category mathematics, Category mathematics - Definition, Category mathematics - Examples, Category mathematics - Types of morphisms, Category mathematics - Types of categories

Read more here: » Category mathematics: Encyclopedia II - Category mathematics - 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

The definition given initially by Quillen was that of a closed model category, the assumptions of which seemed strong at the time, motivating others to weaken some of the assumptions to define a model category. In practice the distinction has not proven significant and most recent authors work with closed model categories and simply drop the adjective 'closed'. The definition has been separated to that of a model structure on a category and then further categorical conditions on that category, the necessity of which may seem unmotivated at first but becomes important later. T ...

See also:

Model category, Model category - Motivation, Model category - Formal definition, Model category - Example: Topological spaces, Model category - Example: Chain complexes of R-modules, Model category - Some constructions, Model category - Homotopy and the homotopy category

Read more here: » Model category: Encyclopedia II - Model category - Formal definition

Every closed model category has a terminal object by the completeness axiom and an initial object by the cocompleteness axiom since these objects are the limit and colimit, respectively, of the empty diagram. Given an object X in the model category, if the unique map from the initial object to X is a cofibration, then X is said to be cofibrant. Analogously, if the unique map from X to the terminal object is a ...

See also:

Model category, Model category - Motivation, Model category - Formal definition, Model category - Example: Topological spaces, Model category - Example: Chain complexes of R-modules, Model category - Some constructions, Model category - Homotopy and the homotopy category

Read more here: » Model category: Encyclopedia II - Model category - Some constructions

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 preadditive categories it makes sense to add and subtract morphisms (the hom-sets actually form abelian groups). In such categories, one can define the coequalizer of two morphisms f and g as the cokernel of their difference: coeq(f, < ...

See also:

Coequalizer, Coequalizer - Definition, Coequalizer - Examples, Coequalizer - Special cases

Read more here: » Coequalizer: Encyclopedia II - Coequalizer - Special cases

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

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 vice versa. As mentioned above, a kernel is a type of binary equaliser, or difference kernel. Conversely, in a preadditive category, every binary equaliser can be constructed as a kernel. To be specific, the equaliser of the morphisms f and g is the kernel of the difference g − f. In s ...

See also:

Kernel category theory, Kernel category theory - Definition, Kernel category theory - Examples, Kernel category theory - Relation to other categorical concepts, Kernel category theory - Relationship to algebraic kernels

Read more here: » Kernel category theory: Encyclopedia II - Kernel category theory - Relation to other categorical concepts