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Waldhausen Category
In mathematics, a Waldhausen category is a category ''C'' equipped with some additional data, which makes it possible to construct the K-theory spectrum of ''C'' using a so-called S-construction. It's named after Friedhelm Waldhausen, who introduced this notion (under the term category with cofibrations and weak equivalences) to extend the methods of algebraic K-theory to categories not necessarily of algebraic origin, for example the category of topological spaces. Definition Let ''C'' be a category, co(''C'') and we(''C'') two classes of morphisms in ''C'', called cofibrations and weak equivalences respectively. The triple (''C'', co(''C''), we(''C'')) is called a Waldhausen category if it satisfies the following axioms, motivated by the similar properties for the notions of cofibrations and weak homotopy equivalences of topological spaces: * ''C'' has a zero object, denoted by 0; * isomorphisms are included in both co(''C'') and we(''C''); * co(''C'') and we(''C'') are closed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Theory
In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topology, the theory has also been used in other areas of mathematics such as algebraic geometry (e.g., A1 homotopy theory) and category theory (specifically the study of higher categories). Concepts Spaces and maps In homotopy theory and algebraic topology, the word "space" denotes a topological space. In order to avoid pathologies, one rarely works with arbitrary spaces; instead, one requires spaces to meet extra constraints, such as being compactly generated, or Hausdorff, or a CW complex. In the same vein as above, a "map" is a continuous function, possibly with some extra constraints. Often, one works with a pointed space -- that is, a space with a "distinguished point", called a basepoint. A pointed map is then a map which preserv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Segal Space
In mathematics, a Segal space is a simplicial space satisfying some pullback conditions, making it look like a homotopical version of a category. More precisely, a simplicial set, considered as a simplicial discrete space, satisfies the Segal conditions iff it is the nerve A nerve is an enclosed, cable-like bundle of nerve fibers (called axons) in the peripheral nervous system. A nerve transmits electrical impulses. It is the basic unit of the peripheral nervous system. A nerve provides a common pathway for the e ... of a category. The condition for Segal spaces is a homotopical version of this. Complete Segal spaces were introduced by as models for (∞, 1)-categories. References * External links * *{{nlab, id=complete+Segal+space, title=Complete Segal space Category theory Simplicial sets ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Opposite Category
In category theory, a branch of mathematics, the opposite category or dual category ''C''op of a given category ''C'' is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields the original category, so the opposite of an opposite category is the original category itself. In symbols, (C^)^ = C. Examples * An example comes from reversing the direction of inequalities in a partial order. So if ''X'' is a set and ≤ a partial order relation, we can define a new partial order relation ≤op by :: ''x'' ≤op ''y'' if and only if ''y'' ≤ ''x''. : The new order is commonly called dual order of ≤, and is mostly denoted by ≥. Therefore, duality plays an important role in order theory and every purely order theoretic concept has a dual. For example, there are opposite pairs child/parent, descendant/ancestor, infimum/supremum, down-set/ up-set, ideal/filter etc. This order theoretic duality is in turn a special c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quillen's Q-construction
In algebra, Quillen's Q-construction associates to an exact category (e.g., an abelian category) an algebraic K-theory. More precisely, given an exact category ''C'', the construction creates a topological space B^+C so that \pi_0 (B^+C) is the Grothendieck group of ''C'' and, when ''C'' is the category of finitely generated projective modules over a ring ''R'', for i = 0, 1, 2, \pi_i (B^+C) is the ''i''-th K-group of ''R'' in the classical sense. (The notation "+" is meant to suggest the construction adds more to the classifying space ''BC''.) One puts :K_i(C) = \pi_i(B^+C) and call it the ''i''-th K-group of ''C''. Similarly, the ''i''-th K-group of ''C'' with coefficients in a group ''G'' is defined as the homotopy group with coefficients: :K_i(C; G) = \pi_i(B^+ C; G). The construction is widely applicable and is used to define an algebraic K-theory in a non-classical context. For example, one can define equivariant algebraic K-theory as \pi_* of B^+ of the category of equivari ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set and an Binary operation, operation that combines any two Element (mathematics), elements of the set to produce a third element of the set, in such a way that the operation is Associative property, associative, an identity element exists and every element has an Inverse element, inverse. These three axioms hold for Number#Main classification, number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The concept of a group and the axioms that define it were elaborated for handling, in a unified way, essential structural properties of very different mathematical entities such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry groups arise naturally in the study of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kan Complex
In mathematics, Kan complexes and Kan fibrations are part of the theory of simplicial sets. Kan fibrations are the fibrations of the standard model category structure on simplicial sets and are therefore of fundamental importance. Kan complexes are the fibrant objects in this model category. The name is in honor of Daniel Kan. Definitions Definition of the standard n-simplex For each ''n'' ≥ 0, recall that the standard n-simplex, \Delta^n, is the representable simplicial set :\Delta^n(i) = \mathrm_ ( Applying the geometric realization functor to this simplicial set gives a space homeomorphic to the topological standard n-simplex: the convex subspace of ℝn+1 consisting of all points (t_0,\dots,t_n) such that the coordinates are non-negative and sum to 1. Definition of a horn For each ''k'' ≤ ''n'', this has a subcomplex \Lambda^n_k, the ''k''-th horn inside \Delta^n, corresponding to the boundary of the ''n''-simplex, with the ''k''-th face r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Model Category
In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows') called ' weak equivalences', ' fibrations' and 'cofibrations' satisfying certain axioms relating them. These abstract from the category of topological spaces or of chain complexes (derived category theory). The concept was introduced by . In recent decades, the language of model categories has been used in some parts of algebraic ''K''-theory and algebraic geometry, where homotopy-theoretic approaches led to deep results. Motivation Model categories can provide a natural setting for homotopy theory: the category of topological spaces is a model category, with the homotopy corresponding to the usual theory. Similarly, objects that are thought of as spaces often admit a model category structure, such as the category of simplicial sets. Another model category is the category of chain complexes of ''R''-modules for a commutative ring ''R''. Homotopy th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
S-construction
In mathematics, a Waldhausen category is a category ''C'' equipped with some additional data, which makes it possible to construct the K-theory spectrum of ''C'' using a so-called S-construction. It's named after Friedhelm Waldhausen, who introduced this notion (under the term category with cofibrations and weak equivalences) to extend the methods of algebraic K-theory to categories not necessarily of algebraic origin, for example the category of topological spaces. Definition Let ''C'' be a category, co(''C'') and we(''C'') two classes of morphisms in ''C'', called cofibrations and weak equivalences respectively. The triple (''C'', co(''C''), we(''C'')) is called a Waldhausen category if it satisfies the following axioms, motivated by the similar properties for the notions of cofibrations and weak homotopy equivalences of topological spaces: * ''C'' has a zero object, denoted by 0; * isomorphisms are included in both co(''C'') and we(''C''); * co(''C'') and we(''C'') are closed un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-construction
In algebra, Quillen's Q-construction associates to an exact category (e.g., an abelian category) an algebraic K-theory. More precisely, given an exact category ''C'', the construction creates a topological space B^+C so that \pi_0 (B^+C) is the Grothendieck group of ''C'' and, when ''C'' is the category of finitely generated projective modules over a ring ''R'', for i = 0, 1, 2, \pi_i (B^+C) is the ''i''-th K-group of ''R'' in the classical sense. (The notation "+" is meant to suggest the construction adds more to the classifying space ''BC''.) One puts :K_i(C) = \pi_i(B^+C) and call it the ''i''-th K-group of ''C''. Similarly, the ''i''-th K-group of ''C'' with coefficients in a group ''G'' is defined as the homotopy group with coefficients: :K_i(C; G) = \pi_i(B^+ C; G). The construction is widely applicable and is used to define an algebraic K-theory in a non-classical context. For example, one can define equivariant algebraic K-theory as \pi_* of B^+ of the category of equivar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exact Category
In mathematics, an exact category is a concept of category theory due to Daniel Quillen which is designed to encapsulate the properties of short exact sequences in abelian categories without requiring that morphisms actually possess kernels and cokernels, which is necessary for the usual definition of such a sequence. Definition An exact category E is an additive category possessing a class ''E'' of "short exact sequences": triples of objects connected by arrows : M' \to M \to M''\ satisfying the following axioms inspired by the properties of short exact sequences in an abelian category: * ''E'' is closed under isomorphisms and contains the canonical ("split exact") sequences: :: M' \to M' \oplus M''\to M''; * Suppose M \to M'' occurs as the second arrow of a sequence in ''E'' (it is an admissible epimorphism) and N \to M'' is any arrow in E. Then their pullback exists and its projection to N is also an admissible epimorphism. Dually, if M' \to M occurs as the first arrow of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
