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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] |
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] |
K-theory Of A Category
In algebraic ''K''-theory, the ''K''-theory of a category ''C'' (usually equipped with some kind of additional data) is a sequence of abelian groups ''K''i(''C'') associated to it. If ''C'' is an abelian category, there is no need for extra data, but in general it only makes sense to speak of K-theory after specifying on ''C'' a structure of an exact category, or of a Waldhausen category, or of a dg-category, or possibly some other variants. Thus, there are several constructions of those groups, corresponding to various kinds of structures put on ''C''. Traditionally, the ''K''-theory of ''C'' is ''defined'' to be the result of a suitable construction, but in some contexts there are more conceptual definitions. For instance, the ''K''-theory is a 'universal additive invariant' of dg-categories and small stable ∞-categories. The motivation for this notion comes from algebraic K-theory of rings. For a ring ''R'' Daniel Quillen in introduced two equivalent ways to find the h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Friedhelm Waldhausen
Friedhelm Waldhausen (born 1938 in Millich, Hückelhoven, Rhine Province) is a German mathematician known for his work in algebraic topology. He made fundamental contributions in the fields of 3-manifolds and (algebraic) K-theory. Career Waldhausen studied mathematics at the universities of Göttingen, Munich and Bonn. He obtained his Ph.D. in 1966 from the University of Bonn; his advisor was Friedrich Hirzebruch and his thesis was entitled "Eine Klasse von 3-dimensionalen Mannigfaltigkeiten" (A class of 3-dimensional manifolds). After visits to Princeton University, the University of Illinois and the University of Michigan he moved in 1968 to the University of Kiel, where he completed his habilitation (qualified to assume a professorship). In 1969, he was appointed professor at the Ruhr University Bochum before in 1971 becoming a professor at Bielefeld University, an appointment he held until his retirement in 2004. Academic work His early work was mainly on the theory of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic K-theory
Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense of abstract algebra. They contain detailed information about the original object but are notoriously difficult to compute; for example, an important outstanding problem is to compute the ''K''-groups of the integers. ''K''-theory was discovered in the late 1950s by Alexander Grothendieck in his study of intersection theory on algebraic varieties. In the modern language, Grothendieck defined only ''K''0, the zeroth ''K''-group, but even this single group has plenty of applications, such as the Grothendieck–Riemann–Roch theorem. Intersection theory is still a motivating force in the development of (higher) algebraic ''K''-theory through its links with motivic cohomology and specifically Chow groups. The subject also includes classical ... [...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] |
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] |
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] |
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] |
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] |
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] |
