category of groups

In category theory, a functor is a special type of mapping between categories. Functors can be thought of as morphisms in the category of small categories. Functors were first considered in algebraic topology, where algebraic objects (like the fundamental group) are associated to topological spaces, and algebraic homomorphisms are associated to continuous maps. Nowadays, functors are used throughout modern mathematics to relate various categories. Functor - Definition. Let C and D be ca ...

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• Functor - Definition
• Functor - Covariance and contravariance
• Functor - Examples
• Functor - Properties
• Functor - Relation to other categorical concepts

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In category theory, the coproduct, or categorical sum, is the dual notion to the categorical product. Basically, this means the definition is the same as the product but with all arrows reversed. Despite this innocuous-looking change in the name and notation, coproducts can be dramatically different from products. The formal definition is as follows: Let C be a category and let {Xj | j ∈ J} be a indexed family of objects in C. The coproduct of the set {Xj} is an object ...

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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. Accordingly, the dual notion of a colimit, generalizes disjoint unions and direct sums. Limits and colimits have strong relationships to the categorial concepts of universal morphisms and adjoint functors. Limit category theory - Definition. Before defining limits, it is useful to defin ...

Including:

• Limit category theory - Definition
• Limit category theory - Examples
• Limit category theory - Complete categories
• Limit category theory - Continuous functors
• Limit category theory - Colimits
• Limit category theory - Creation of Limits and Co-Limits

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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 ...

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• Zero morphism - Examples

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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′ ...

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

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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 ...

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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

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The definition of limits is general enough to subsume several constructions useful in practical settings. In the following we will consider the limit (L, φX) of a functor F : J → C. Terminal objects. If J is the empty category, then the above definitions imply that every object of C is a cone of F. The limit of F is any object that has a unique factorization through any other object. This is just the definition of a terminal object.See also:

Limit category theory, Limit category theory - Definition, Limit category theory - Examples, Limit category theory - Complete categories, Limit category theory - Continuous functors, Limit category theory - Colimits, Limit category theory - Creation of Limits and Co-Limits

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According to Yoneda's lemma, natural transformations from Hom(A,–) to F are in one-to-one correspondence with the elements of F(A). Given a natural transformation Φ : Hom(A,–) → F the corresponding element of u ∈ F(A) is given by Conversely, given any element u ∈ F(A) we may define a natural transformation Φ
...

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Representable functor, Representable functor - Definition, Representable functor - Universal elements, Representable functor - Uniqueness, Representable functor - Examples, Representable functor - Relation to universal morphisms and adjoints

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The units of R form a group U(R) under multiplication, the group of units of R. The group of units U(R) is sometimes also denoted R* or R×. In a commutative unital ring R, the group of units U(R) acts on R via multiplication. The orbits of this action are called sets of associates; in other words, there is an equivalence relation ~ on R called associatedness such that r ~ s means that ...

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Unit ring theory, Unit ring theory - Group of units, Unit ring theory - Examples

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Constant functor: A very boring functor C → D is one which maps every object of C to a fixed object X in D and every morphism in C to the identity morphism on X. Such a functor is called a constant or selection functor. Power sets: The power set functor P : Set → Set maps each set to its power set and each function to the map which sends to its image . One can also consider the contravariant power set functor which sends f to the map which sends USee also:

Functor, Functor - Definition, Functor - Covariance and contravariance, Functor - Examples, Functor - Properties, Functor - Relation to other categorical concepts

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Here are some examples of pushouts in familiar categories. Note that in each case, we are only providing a construction of an object in the isomorphism class of pushouts; as mentioned above, there may be other ways to construct it, but they are all equivalent. 1. Suppose that X and Y as above are sets. Then if we write Z for their intersection, there are morphisms f : Z → X and g : Z → Y given by inclusion. The pushout of f and g is the union of X and Y t ...

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Pushout category theory, Pushout category theory - Universal property, Pushout category theory - Examples of pushouts, Pushout category theory - Construction via coproducts and coequalizers, Pushout category theory - Application: The Seifert-van Kampen theorem

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A simplicial set is a categorical (that is, purely algebraic) model capturing those topological spaces which can be built up (or faithfully represented up to homotopy) from simplices and their incidence relations. This is similar to the approach of CW complexes to modeling topological spaces, with the crucial difference that simplicial sets are purely algebraic and do not carry any actual topology (this ...

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Simplicial set, Simplicial set - Motivation, Simplicial set - Formal definition, Simplicial set - Face and degeneracy maps, Simplicial set - The standard n-simplex and the simplex category, Simplicial set - Geometric realization, Simplicial set - Singular set for a space, Simplicial set - Homotopy theory of simplicial sets, Simplicial set - Simplicial objects, Simplicial set - Reference

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Empty product - Frequent examples. Two often-seen instances are a0 = 1 (any number raised to the zeroth power is one) and 0! = 1 (the factorial of zero is one). It can also be motivated by the fact that if all factors of the numerator or the denominator in a fraction cancel (as would 2 and 3 in the following example), the remaining value is 1, The numerator becomes here a "pro ...

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Empty product, Empty product - Nullary arithmetic product, Empty product - Frequent examples, Empty product - Conceptual justification, Empty product - Technical justification, Empty product - 0 raised to the 0th power, Empty product - Nullary intersection, Empty product - Nullary categorical product, Empty product - In computer programming, Empty product - Quote

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Using the language of category theory, a simplicial set X is a contravariant functor X: Δop → Set where Δ denotes the simplicial category whose objects are finite strings of ordinal numbers of the form 0 → 1 → ... → n (or in other words totally ordered finite sets) and whose morphisms are order-preserving functions between ...

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Simplicial set, Simplicial set - Motivation, Simplicial set - Formal definition, Simplicial set - Face and degeneracy maps, Simplicial set - The standard n-simplex and the simplex category, Simplicial set - Geometric realization, Simplicial set - Singular set for a space, Simplicial set - Homotopy theory of simplicial sets, Simplicial set - Simplicial objects, Simplicial set - Reference

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In Δop, there are two particularly important classes of maps called face maps and degeneracy maps which capture the underlying combinatorial structure of simplicial sets. The face maps di : n → n − 1 are given by di (0 → … → n) = (0 → … → i − 1 → i + 1 → … → n). The degeneracy maps si : n → n + 1 ...

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Simplicial set, Simplicial set - Motivation, Simplicial set - Formal definition, Simplicial set - Face and degeneracy maps, Simplicial set - The standard n-simplex and the simplex category, Simplicial set - Geometric realization, Simplicial set - Singular set for a space, Simplicial set - Homotopy theory of simplicial sets, Simplicial set - Simplicial objects, Simplicial set - Reference

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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 ...

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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

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In the category of simplicial sets one can define fibrations to be Kan fibrations. A map of simplicial sets is defined to be a weak equivalence if the geometric realization is a weak equivalence of spaces. A map of simplicial sets is defined to be a cofibration if it is a monomorphism of simplicial sets. It is a difficult theorem of Daniel Quillen that the category of simplicial sets with these classes of morphism ...

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Simplicial set, Simplicial set - Motivation, Simplicial set - Formal definition, Simplicial set - Face and degeneracy maps, Simplicial set - The standard n-simplex and the simplex category, Simplicial set - Geometric realization, Simplicial set - Singular set for a space, Simplicial set - Homotopy theory of simplicial sets, Simplicial set - Simplicial objects, Simplicial set - Reference

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A simplicial object X in a category C is a contravariant functor X: Δop → C. When C is the category of sets, we are just talking about simplicial sets. Letting C be the category of groups or category of abelian groups, we obtain the categories sGrp of simplicial groups and sAb of simplicial abelian groups, respectively. Simplicial groups and simplicial abelian groups also carry closed model structur ...

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Simplicial set, Simplicial set - Motivation, Simplicial set - Formal definition, Simplicial set - Face and degeneracy maps, Simplicial set - The standard n-simplex and the simplex category, Simplicial set - Geometric realization, Simplicial set - Singular set for a space, Simplicial set - Homotopy theory of simplicial sets, Simplicial set - Simplicial objects, Simplicial set - Reference

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The singular set of a topological space Y is the simplicial set defined by S(Y): n → hom(|Δn|, Y) for each object n ∈ Δ, with the obvious functoriality condition on the morphisms. This definition is analogous to a standard idea in singular homology of "probing" a target topological space with standard topological n-simplices. Furthermore, the singular functor S is right adjoint to the geometric realization functor described above, i.e.: homTop(|X|, Y< ...

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Simplicial set, Simplicial set - Motivation, Simplicial set - Formal definition, Simplicial set - Face and degeneracy maps, Simplicial set - The standard n-simplex and the simplex category, Simplicial set - Geometric realization, Simplicial set - Singular set for a space, Simplicial set - Homotopy theory of simplicial sets, Simplicial set - Simplicial objects, Simplicial set - Reference

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There is a functor |•|: S → CGHaus called the geometric realization taking a simplicial set X to its corresponding realization in the category of compactly-generated Hausdorff topological spaces. This larger category is used as the target of the functor because, in particular, a product of simplicial sets is realized as a product of the corresponding topological spaces, where denotes the Kelley space product. To define the realization funct ...

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Simplicial set, Simplicial set - Motivation, Simplicial set - Formal definition, Simplicial set - Face and degeneracy maps, Simplicial set - The standard n-simplex and the simplex category, Simplicial set - Geometric realization, Simplicial set - Singular set for a space, Simplicial set - Homotopy theory of simplicial sets, Simplicial set - Simplicial objects, Simplicial set - Reference

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