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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] |
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] |
Daniel Quillen
Daniel Gray "Dan" Quillen (June 22, 1940 – April 30, 2011) was an American mathematician. He is known for being the "prime architect" of higher algebraic ''K''-theory, for which he was awarded the Cole Prize in 1975 and the Fields Medal in 1978. From 1984 to 2006, he was the Waynflete Professor of Pure Mathematics at Magdalen College, Oxford. Education and career Quillen was born in Orange, New Jersey, and attended Newark Academy. He entered Harvard University, where he earned both his AB, in 1961, and his PhD in 1964; the latter completed under the supervision of Raoul Bott, with a thesis in partial differential equations. He was a Putnam Fellow in 1959. Quillen obtained a position at the Massachusetts Institute of Technology after completing his doctorate. He also spent a number of years at several other universities. He visited France twice: first as a Sloan Fellow in Paris, during the academic year 1968–69, where he was greatly influenced by Grothendieck, and the ... [...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] |
Quillen's Theorem B
In topology, a branch of mathematics, Quillen's Theorem A gives a sufficient condition for the classifying spaces of two categories to be homotopy equivalent. Quillen's Theorem B gives a sufficient condition for a square consisting of classifying spaces of categories to be homotopy Cartesian. The two theorems play central roles in Quillen's Q-construction in algebraic K-theory and are named after Daniel Quillen Daniel Gray "Dan" Quillen (June 22, 1940 – April 30, 2011) was an American mathematician. He is known for being the "prime architect" of higher algebraic ''K''-theory, for which he was awarded the Cole Prize in 1975 and the Fields Medal in 197 .... The precise statements of the theorems are as follows. In general, the homotopy fiber of Bf: BC \to BD is not naturally the classifying space of a category: there is no natural category Ff such that FBf = BFf. Theorem B constructs Ff in a case when f is especially nice. References * * * * Theorems in topology ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Module Spectrum
In algebra, a module spectrum is a spectrum (topology), spectrum with an action of a ring spectrum; it generalizes a module (mathematics), module in abstract algebra. The ∞-category of (say right) module spectra is stable ∞-category, stable; hence, it can be considered as either analog or generalization of the derived category of modules over a ring. K-theory Lurie defines the K-theory of a ring spectrum ''R'' to be the K-theory of the ∞-category of perfect modules over ''R'' (a perfect module being defined as a compact object in the ∞-category of module spectra.) See also *G-spectrum References *J. LurieLecture 19: Algebraic K-theory of Ring Spectra {{algebra-stub Homotopy theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivariant Algebraic K-theory
In mathematics, the equivariant algebraic K-theory is an algebraic K-theory associated to the category \operatorname^G(X) of equivariant coherent sheaves on an algebraic scheme ''X'' with action of a linear algebraic group ''G'', via Quillen's Q-construction; thus, by definition, :K_i^G(X) = \pi_i(B^+ \operatorname^G(X)). In particular, K_0^G(C) is the Grothendieck group of \operatorname^G(X). The theory was developed by R. W. Thomason in 1980s. Specifically, he proved equivariant analogs of fundamental theorems such as the localization theorem. Equivalently, K_i^G(X) may be defined as the K_i of the category of coherent sheaves on the quotient stack /G/math>. (Hence, the equivariant K-theory is a specific case of the K-theory of a stack.) A version of the Lefschetz fixed-point theorem holds in the setting of equivariant (algebraic) K-theory. Fundamental theorems Let ''X'' be an equivariant algebraic scheme. Examples One of the fundamental examples of equivariant K-theo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibered Category
Fibred categories (or fibered categories) are abstract entities in mathematics used to provide a general framework for descent theory. They formalise the various situations in geometry and algebra in which ''inverse images'' (or ''pull-backs'') of objects such as vector bundles can be defined. As an example, for each topological space there is the category of vector bundles on the space, and for every continuous map from a topological space ''X'' to another topological space ''Y'' is associated the pullback functor taking bundles on ''Y'' to bundles on ''X''. Fibred categories formalise the system consisting of these categories and inverse image functors. Similar setups appear in various guises in mathematics, in particular in algebraic geometry, which is the context in which fibred categories originally appeared. Fibered categories are used to define stacks, which are fibered categories (over a site) with "descent". Fibrations also play an important role in categorical semantics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Fiber
In mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber)Joseph J. Rotman, ''An Introduction to Algebraic Topology'' (1988) Springer-Verlag ''(See Chapter 11 for construction.)'' is part of a construction that associates a fibration to an arbitrary continuous function of topological spaces f:A \to B. It acts as a homotopy theoretic kernel of a mapping of topological spaces due to the fact it yields a long exact sequence of homotopy groups\cdots \to \pi_(B) \to \pi_n(\text(f)) \to \pi_n(A) \to \pi_n(B) \to \cdotsMoreover, the homotopy fiber can be found in other contexts, such as homological algebra, where the distinguished triangleC(f)_\bullet 1\to A_\bullet \to B_\bullet \xrightarrowgives a long exact sequence analogous to the long exact sequence of homotopy groups. There is a dual construction called the homotopy cofiber. Construction The homotopy fiber has a simple description for a continuous map f:A \to B. If we replace f by a fibra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Pullback
In mathematics, especially in algebraic topology, the homotopy limit and colimitpg 52 are variants of the notions of limit and colimit extended to the homotopy category \text(\textbf). The main idea is this: if we have a diagramF: I \to \textbfconsidered as an object in the homotopy category of diagrams F \in \text(\textbf^I), (where the homotopy equivalence of diagrams is considered pointwise), then the homotopy limit and colimits then correspond to the cone and cocone\begin \underset(F)&: * \to \textbf\\ \underset(F)&: * \to \textbf \endwhich are objects in the homotopy category \text(\textbf^*), where * is the category with one object and one morphism. Note this category is equivalent to the standard homotopy category \text(\textbf) since the latter homotopy functor category has functors which picks out an object in \text and a natural transformation corresponds to a continuous function of topological spaces. Note this construction can be generalized to model categories, which g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eilenberg–MacLane Space
In mathematics, specifically algebraic topology, an Eilenberg–MacLane spaceSaunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name. (See e.g. ) In this context it is therefore conventional to write the name without a space. is a topological space with a single nontrivial homotopy group. Let ''G'' be a group and ''n'' a positive integer. A connected topological space ''X'' is called an Eilenberg–MacLane space of type K(G,n), if it has ''n''-th homotopy group \pi_n(X) isomorphic to ''G'' and all other homotopy groups trivial. If n > 1 then ''G'' must be abelian. Such a space exists, is a CW-complex, and is unique up to a weak homotopy equivalence, therefore any such space is often just called K(G,n). The name is derived from Samuel Eilenberg and Saunders Mac Lane, who introduced such spaces in the late 1940s. As such, an Eilenberg–MacLane space is a special k ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nerve (category Theory)
In category theory, a discipline within mathematics, the nerve ''N''(''C'') of a small category ''C'' is a simplicial set constructed from the objects and morphisms of ''C''. The geometric realization of this simplicial set is a topological space, called the classifying space of the category ''C''. These closely related objects can provide information about some familiar and useful categories using algebraic topology, most often homotopy theory. Motivation The nerve of a category is often used to construct topological versions of moduli spaces. If ''X'' is an object of ''C'', its moduli space should somehow encode all objects isomorphic to ''X'' and keep track of the various isomorphisms between all of these objects in that category. This can become rather complicated, especially if the objects have many non-identity automorphisms. The nerve provides a combinatorial way of organizing this data. Since simplicial sets have a good homotopy theory, one can ask questions about the mean ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
