Author: Angélica María Osorno
Publisher:
ISBN:
Category :
Languages : en
Pages : 64
Book Description
In recent work of Baas-Dundas-Richter-Rognes, the authors introduce the notion of the K- theory of a bimonoidal category R, and show that it is equivalent to the algebraic K-theory space of the ring spectrum KR. In this thesis we show that K(R) is the group completion of the classifying space of the 2-category ModR of modules over R, and show that ModR is a symmetric monoidal 2-category. We explain how to use this symmetric monoidal structure to produce a [Gamma]-(2-category), which gives an infinite loop space structure on K(R). We show that the equivalence mentioned above is an equivalence of infinite loop spaces.
An Infinite Loop Space Structure for K-theory of Bimonoidal Categories
Author: Angélica María Osorno
Publisher:
ISBN:
Category :
Languages : en
Pages : 64
Book Description
In recent work of Baas-Dundas-Richter-Rognes, the authors introduce the notion of the K- theory of a bimonoidal category R, and show that it is equivalent to the algebraic K-theory space of the ring spectrum KR. In this thesis we show that K(R) is the group completion of the classifying space of the 2-category ModR of modules over R, and show that ModR is a symmetric monoidal 2-category. We explain how to use this symmetric monoidal structure to produce a [Gamma]-(2-category), which gives an infinite loop space structure on K(R). We show that the equivalence mentioned above is an equivalence of infinite loop spaces.
Publisher:
ISBN:
Category :
Languages : en
Pages : 64
Book Description
In recent work of Baas-Dundas-Richter-Rognes, the authors introduce the notion of the K- theory of a bimonoidal category R, and show that it is equivalent to the algebraic K-theory space of the ring spectrum KR. In this thesis we show that K(R) is the group completion of the classifying space of the 2-category ModR of modules over R, and show that ModR is a symmetric monoidal 2-category. We explain how to use this symmetric monoidal structure to produce a [Gamma]-(2-category), which gives an infinite loop space structure on K(R). We show that the equivalence mentioned above is an equivalence of infinite loop spaces.
Infinite Loop Spaces (AM-90), Volume 90
Author: John Frank Adams
Publisher: Princeton University Press
ISBN: 1400821258
Category : Mathematics
Languages : en
Pages : 230
Book Description
The theory of infinite loop spaces has been the center of much recent activity in algebraic topology. Frank Adams surveys this extensive work for researchers and students. Among the major topics covered are generalized cohomology theories and spectra; infinite-loop space machines in the sense of Boadman-Vogt, May, and Segal; localization and group completion; the transfer; the Adams conjecture and several proofs of it; and the recent theories of Adams and Priddy and of Madsen, Snaith, and Tornehave.
Publisher: Princeton University Press
ISBN: 1400821258
Category : Mathematics
Languages : en
Pages : 230
Book Description
The theory of infinite loop spaces has been the center of much recent activity in algebraic topology. Frank Adams surveys this extensive work for researchers and students. Among the major topics covered are generalized cohomology theories and spectra; infinite-loop space machines in the sense of Boadman-Vogt, May, and Segal; localization and group completion; the transfer; the Adams conjecture and several proofs of it; and the recent theories of Adams and Priddy and of Madsen, Snaith, and Tornehave.
Infinite Loop Structures on the Algebraic K-theory of Spaces
The Geometry of Iterated Loop Spaces
Author: J.P. May
Publisher: Springer
ISBN: 3540376038
Category : Mathematics
Languages : en
Pages : 184
Book Description
Publisher: Springer
ISBN: 3540376038
Category : Mathematics
Languages : en
Pages : 184
Book Description
The Local Structure of Algebraic K-Theory
Author: Bjørn Ian Dundas
Publisher: Springer Science & Business Media
ISBN: 1447143930
Category : Mathematics
Languages : en
Pages : 447
Book Description
Algebraic K-theory encodes important invariants for several mathematical disciplines, spanning from geometric topology and functional analysis to number theory and algebraic geometry. As is commonly encountered, this powerful mathematical object is very hard to calculate. Apart from Quillen's calculations of finite fields and Suslin's calculation of algebraically closed fields, few complete calculations were available before the discovery of homological invariants offered by motivic cohomology and topological cyclic homology. This book covers the connection between algebraic K-theory and Bökstedt, Hsiang and Madsen's topological cyclic homology and proves that the difference between the theories are ‘locally constant’. The usefulness of this theorem stems from being more accessible for calculations than K-theory, and hence a single calculation of K-theory can be used with homological calculations to obtain a host of ‘nearby’ calculations in K-theory. For instance, Quillen's calculation of the K-theory of finite fields gives rise to Hesselholt and Madsen's calculations for local fields, and Voevodsky's calculations for the integers give insight into the diffeomorphisms of manifolds. In addition to the proof of the full integral version of the local correspondence between K-theory and topological cyclic homology, the book provides an introduction to the necessary background in algebraic K-theory and highly structured homotopy theory; collecting all necessary tools into one common framework. It relies on simplicial techniques, and contains an appendix summarizing the methods widely used in the field. The book is intended for graduate students and scientists interested in algebraic K-theory, and presupposes a basic knowledge of algebraic topology.
Publisher: Springer Science & Business Media
ISBN: 1447143930
Category : Mathematics
Languages : en
Pages : 447
Book Description
Algebraic K-theory encodes important invariants for several mathematical disciplines, spanning from geometric topology and functional analysis to number theory and algebraic geometry. As is commonly encountered, this powerful mathematical object is very hard to calculate. Apart from Quillen's calculations of finite fields and Suslin's calculation of algebraically closed fields, few complete calculations were available before the discovery of homological invariants offered by motivic cohomology and topological cyclic homology. This book covers the connection between algebraic K-theory and Bökstedt, Hsiang and Madsen's topological cyclic homology and proves that the difference between the theories are ‘locally constant’. The usefulness of this theorem stems from being more accessible for calculations than K-theory, and hence a single calculation of K-theory can be used with homological calculations to obtain a host of ‘nearby’ calculations in K-theory. For instance, Quillen's calculation of the K-theory of finite fields gives rise to Hesselholt and Madsen's calculations for local fields, and Voevodsky's calculations for the integers give insight into the diffeomorphisms of manifolds. In addition to the proof of the full integral version of the local correspondence between K-theory and topological cyclic homology, the book provides an introduction to the necessary background in algebraic K-theory and highly structured homotopy theory; collecting all necessary tools into one common framework. It relies on simplicial techniques, and contains an appendix summarizing the methods widely used in the field. The book is intended for graduate students and scientists interested in algebraic K-theory, and presupposes a basic knowledge of algebraic topology.
Infinite Loop Spaces
Author: John Frank Adams
Publisher: Princeton University Press
ISBN: 9780691082066
Category : Mathematics
Languages : en
Pages : 232
Book Description
The theory of infinite loop spaces has been the center of much recent activity in algebraic topology. Frank Adams surveys this extensive work for researchers and students. Among the major topics covered are generalized cohomology theories and spectra; infinite-loop space machines in the sense of Boadman-Vogt, May, and Segal; localization and group completion; the transfer; the Adams conjecture and several proofs of it; and the recent theories of Adams and Priddy and of Madsen, Snaith, and Tornehave.
Publisher: Princeton University Press
ISBN: 9780691082066
Category : Mathematics
Languages : en
Pages : 232
Book Description
The theory of infinite loop spaces has been the center of much recent activity in algebraic topology. Frank Adams surveys this extensive work for researchers and students. Among the major topics covered are generalized cohomology theories and spectra; infinite-loop space machines in the sense of Boadman-Vogt, May, and Segal; localization and group completion; the transfer; the Adams conjecture and several proofs of it; and the recent theories of Adams and Priddy and of Madsen, Snaith, and Tornehave.
Homotopy Invariant Algebraic Structures on Topological Spaces
Author: J. M. Boardman
Publisher: Springer
ISBN: 3540377999
Category : Mathematics
Languages : en
Pages : 268
Book Description
Publisher: Springer
ISBN: 3540377999
Category : Mathematics
Languages : en
Pages : 268
Book Description
The $K$-book
Author: Charles A. Weibel
Publisher: American Mathematical Soc.
ISBN: 0821891324
Category : Mathematics
Languages : en
Pages : 634
Book Description
Informally, $K$-theory is a tool for probing the structure of a mathematical object such as a ring or a topological space in terms of suitably parameterized vector spaces and producing important intrinsic invariants which are useful in the study of algebr
Publisher: American Mathematical Soc.
ISBN: 0821891324
Category : Mathematics
Languages : en
Pages : 634
Book Description
Informally, $K$-theory is a tool for probing the structure of a mathematical object such as a ring or a topological space in terms of suitably parameterized vector spaces and producing important intrinsic invariants which are useful in the study of algebr
Noncommutative Motives
Author: Gonçalo Tabuada
Publisher: American Mathematical Soc.
ISBN: 1470423979
Category : Mathematics
Languages : en
Pages : 127
Book Description
The theory of motives began in the early 1960s when Grothendieck envisioned the existence of a "universal cohomology theory of algebraic varieties". The theory of noncommutative motives is more recent. It began in the 1980s when the Moscow school (Beilinson, Bondal, Kapranov, Manin, and others) began the study of algebraic varieties via their derived categories of coherent sheaves, and continued in the 2000s when Kontsevich conjectured the existence of a "universal invariant of noncommutative algebraic varieties". This book, prefaced by Yuri I. Manin, gives a rigorous overview of some of the main advances in the theory of noncommutative motives. It is divided into three main parts. The first part, which is of independent interest, is devoted to the study of DG categories from a homotopical viewpoint. The second part, written with an emphasis on examples and applications, covers the theory of noncommutative pure motives, noncommutative standard conjectures, noncommutative motivic Galois groups, and also the relations between these notions and their commutative counterparts. The last part is devoted to the theory of noncommutative mixed motives. The rigorous formalization of this latter theory requires the language of Grothendieck derivators, which, for the reader's convenience, is revised in a brief appendix.
Publisher: American Mathematical Soc.
ISBN: 1470423979
Category : Mathematics
Languages : en
Pages : 127
Book Description
The theory of motives began in the early 1960s when Grothendieck envisioned the existence of a "universal cohomology theory of algebraic varieties". The theory of noncommutative motives is more recent. It began in the 1980s when the Moscow school (Beilinson, Bondal, Kapranov, Manin, and others) began the study of algebraic varieties via their derived categories of coherent sheaves, and continued in the 2000s when Kontsevich conjectured the existence of a "universal invariant of noncommutative algebraic varieties". This book, prefaced by Yuri I. Manin, gives a rigorous overview of some of the main advances in the theory of noncommutative motives. It is divided into three main parts. The first part, which is of independent interest, is devoted to the study of DG categories from a homotopical viewpoint. The second part, written with an emphasis on examples and applications, covers the theory of noncommutative pure motives, noncommutative standard conjectures, noncommutative motivic Galois groups, and also the relations between these notions and their commutative counterparts. The last part is devoted to the theory of noncommutative mixed motives. The rigorous formalization of this latter theory requires the language of Grothendieck derivators, which, for the reader's convenience, is revised in a brief appendix.
The Homology of Iterated Loop Spaces
Author: F. R. Cohen
Publisher: Springer
ISBN: 3540379851
Category : Mathematics
Languages : en
Pages : 501
Book Description
Publisher: Springer
ISBN: 3540379851
Category : Mathematics
Languages : en
Pages : 501
Book Description