Abstract Ktheory

General Introduction
This page is supposed to be a place where we collect various notions of "abstract" Ktheory, such as Ktheory of various kinds of categories, and various abstract constructions used in definitions of Ktheory.
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Search results
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Online References
Perhaps Weibel's online book.
For Ktheory of DGcategories, see Toen: Lecture on DGcategories. File Toen web unpubl swisk.pdf. Treats basic theory, localization, relation to model cats, functorial cones, Ktheory and Hochschild cohomology, and descent problems.
Excellent reference: Toen and Vezzosi: A remark on Ktheory and Scats. File Toen web publ kthe.pdf.
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Paper References
Waldhausen: Algebraic Ktheory of spaces. LNM 1126.
Neeman: Ktheory of triangulated categories.
Maybe Schlichting's notes.
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Definition
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Properties
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Standard theorems
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Open Problems
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Connections to Number Theory
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Computations and Examples
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History and Applications
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Some Research Articles
arXiv:1111.3335 Categorical Foundations for KTheory fra arXiv Front: math.CT av Nicolas Michel Recall that the definition of the $K$theory of an object C (e.g., a ring or a space) has the following pattern. One first associates to the object C a category AC that has a suitable structure (exact, Waldhausen, symmetric monoidal, ...). One then applies to the category AC a "$K$theory machine", which provides an infinite loop space that is the $K$theory K(C) of the object C.
We study the first step of this process. What are the kinds of objects to be studied via $K$theory? Given these types of objects, what structured categories should one associate to an object to obtain $K$theoretic information about it? And how should the morphisms of these objects interact with this correspondence?
We propose a unified, conceptual framework for a number of important examples of objects studied in $K$theory. The structured categories associated to an object C are typically categories of modules in a monoidal (op)fibred category. The modules considered are "locally trivial" with respect to a given class of trivial modules and a given Grothendieck topology on the object C's category.
arXiv:1002.3622 An Inverse KTheory Functor from arXiv Front: math.KT by Michael A. Mandell Thomason showed that the Ktheory of symmetric monoidal categories models all connective spectra. This paper describes a new construction of a permutative category from a Gammaspace, which is then used to reprove Thomason's theorem and a noncompleted variant.
arXiv:1001.2282 A universal characterization of higher algebraic Ktheory from arXiv Front: math.KT by Andrew J. Blumberg, David Gepner, Goncalo Tabuada In this paper we establish a universal characterization of higher algebraic Ktheory in the setting of stable infinity categories. Specifically, we prove that connective algebraic Ktheory is the universal additive invariant, i.e., the universal functor with values in a stable presentable infinity category which inverts Morita equivalences, preserves filtered colimits, and satisfies Waldhausen's additivity theorem. Similarly, we prove that nonconnective algebraic Ktheory is the universal localizing invariant, i.e., the universal functor that moreover satisfies the "ThomasonTrobaughNeeman" localization theorem. In addition, by adapting the standard cosimplicial affine space to the setting of stable categories, we generalize the classical notion of algebraic homotopy invariance and prove that KaroubiVillamayor's algebraic Ktheory is the universal additive homotopy invariant and that Weibel's homotopy Ktheory is the universal localizing homotopy invariant.
To prove these results, we construct and study various stable symmetric monoidal infinity categories of "noncommutative motives". In these infinity categories, Waldhausen's S. construction corresponds to the suspension functor and the various algebraic Ktheory spectra becomes corepresentable by the unit object. Moreover, these infinity categories are enriched over Waldhausen's Atheory of a point and the homotopy Ktheory of the sphere spectrum, respectively.
We give several applications of our theory. We obtain a complete classification of all natural transformations from higher algebraic Ktheory to THH and TC. Notably, we obtain a canonical construction and universal description of the cyclotomic trace map. We also exhibit a lax symmetric monoidal structure on the different algebraic Ktheory functors, implying in particular that En ring spectra give rise to E{n1} Ktheory spectra.
arXiv:1103.5936 The fundamental theorem via derived Morita invariance, localization, and A^1homotopy invariance from arXiv Front: math.AG by Goncalo Tabuada We prove that every functor defined on dg categories, which is derived Morita invariant, localizing, and A^1homotopy invariant, satisfies the fundamental theorem. As an application, we recover in a unified and conceptual way, Weibel and Kassel's fundamental theorems in homotopy algebraic Ktheory, and periodic cyclic homology, respectively.
arXiv:1103.3923 Uniqueness of the multiplicative cyclotomic trace from arXiv Front: math.AG by Andrew J. Blumberg, David Gepner, Goncalo Tabuada Using Lurie's theory of infinite operads, we construct a symmetric monoidal structure on the infinite category of all functors (from small stable infinite categories to spectra) that satisfy additivity. The unit of this symmetric monoidal structure is the algebraic Ktheory functor and (Einfinite) algebras correspond to the lax (symmetric) monoidal functors. As applications we show that the space of multiplicative structures on the algebraic Ktheory functor is contractible, and that the cyclotomic trace can be characterized as the unique multiplicative natural transformation from Ktheory to THH.
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Other Information
arXiv:1304.0520 Parametrized KTheory fra arXiv Front: math.AG av Nicolas Michel In nature, one observes that a Ktheory of an object is defined in two steps. First a "structured" category is associated to the object. Second, a Ktheory machine is applied to the latter category to produce an infinite loop space. We develop a general framework that deals with the first step of this process. The Ktheory of an object is defined via a category of "locally trivial" objects with respect to a pretopology. We study conditions ensuring an exact structure on such categories. We also consider morphisms in Ktheory that such contexts naturally provide. We end by defining various Ktheories of schemes and morphisms between them.
Weibel's Thomason obituary describes infinite loop spaces briefly. The input is a space with some addition properties/data, for example the geometric realization of a symmetric monoidal category. As a special case, we can get Quillen's Ktheory spectrum, and in general we call the output the Ktheory spectrum of the symmetric monoidal category. Thomason also showed that symmetric monoidal categories model all connective spectra.
The same obituary also sketches some other things, such as the Ktheory of schemes in versions of Quillen, Bass, and Waldhausen.
http://mathoverflow.net/questions/74237/thevanishingofnonconnectivektheoryinnegativedegrees
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