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AG (Algebraic geometry), CT (Category theory), KT (K-theory)
Kth
See Neeman in K-theory handbook
Negative K-theory of Derived Categories, by Marco Schlichting: http://www.math.uiuc.edu/K-theory/0636
On the algebraic K-theory of model categories, by Steffen Sagave: http://www.math.uiuc.edu/K-theory/0655
Rings, modules, and algebras in infinite loop space theory, by Anthony D. Elmendorf and Michael A. Mandell http://www.math.uiuc.edu/K-theory/0748
The 1-type of a Waldhausen K-theory spectrum, by Fernando Muro and Andrew Tonks: http://www.math.uiuc.edu/K-theory/0775
Muro and Tonks on K_1
Abelian subcategories closed under extensions K-theory and decompositions, by Sunil K. Chebolu http://www.math.uiuc.edu/K-theory/0776
Feliu describes three ways of defining K-theory of an exact cat: Quillen’s Q-construction, Waldhausen K-theory, and the complex of cubes. I think the latter only works for rational K-groups; it was constructed in the thesis of Wang. Furthermore, she describes K-theory as a generalized cohomology theory, and introduces some interesting related notions (stable K-theory, unstable, both rep by a simplicial sheaf). There is also much more material here, with comparison to Quillen’s K-theory, and notion of a space being K-coherent. Also
Waldhausen: Algebraic K-theory of spaces. LNM 1126.
Neeman: K-theory of triangulated categories.
Maybe Schlichting’s notes.
Perhaps Weibel’s online book.
For K-theory of DG-categories, see Toen: Lecture on DG-categories. File Toen web unpubl swisk.pdf. Treats basic theory, localization, relation to model cats, functorial cones, K-theory and Hochschild cohomology, and descent problems.
Excellent reference: Toen and Vezzosi: A remark on K-theory and S-cats. File Toen web publ k-the.pdf.
This page is supposed to be a place where we collect various notions of “abstract” K-theory, such as K-theory of various kinds of categories, and various abstract constructions used in definitions of K-theory.
arXiv:1111.3335 Categorical Foundations for K-Theory fra arXiv Front: math.CT av Nicolas Michel Recall that the definition of the -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 A_C that has a suitable structure (exact, Waldhausen, symmetric monoidal, …). One then applies to the category A_C a “-theory machine”, which provides an infinite loop space that is the -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 -theory? Given these types of objects, what structured categories should one associate to an object to obtain -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 -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 K-Theory Functor from arXiv Front: math.KT by Michael A. Mandell Thomason showed that the K-theory of symmetric monoidal categories models all connective spectra. This paper describes a new construction of a permutative category from a Gamma-space, which is then used to re-prove Thomason’s theorem and a non-completed variant.
arXiv:1001.2282 A universal characterization of higher algebraic K-theory from arXiv Front: math.KT by Andrew J. Blumberg, David Gepner, Goncalo Tabuada In this paper we establish a universal characterization of higher algebraic K-theory in the setting of stable infinity categories. Specifically, we prove that connective algebraic K-theory 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 non-connective algebraic K-theory is the universal localizing invariant, i.e., the universal functor that moreover satisfies the “Thomason-Trobaugh-Neeman” 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 Karoubi-Villamayor’s algebraic K-theory is the universal additive homotopy invariant and that Weibel’s homotopy K-theory is the universal localizing homotopy invariant
To prove these results, we construct and study various stable symmetric monoidal infinity categories of “non-commutative motives”. In these infinity categories, Waldhausen’s S. construction corresponds to the suspension functor and the various algebraic K-theory spectra becomes corepresentable by the unit object. Moreover, these infinity categories are enriched over Waldhausen’s A-theory of a point and the homotopy K-theory of the sphere spectrum, respectively
We give several applications of our theory. We obtain a complete classification of all natural transformations from higher algebraic K-theory 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 K-theory functors, implying in particular that E_n ring spectra give rise to E_{n-1} K-theory spectra.
arXiv:1103.5936 The fundamental theorem via derived Morita invariance, localization, and A^1-homotopy 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^1-homotopy 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 K-theory, 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 K-theory functor and (E-infinite) algebras correspond to the lax (symmetric) monoidal functors. As applications we show that the space of multiplicative structures on the algebraic K-theory functor is contractible, and that the cyclotomic trace can be characterized as the unique multiplicative natural transformation from K-theory to THH.
arXiv:1304.0520 Parametrized K-Theory fra arXiv Front: math.AG av Nicolas Michel In nature, one observes that a K-theory of an object is defined in two steps. First a “structured” category is associated to the object. Second, a K-theory 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 K-theory 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 K-theory that such contexts naturally provide. We end by defining various K-theories 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 K-theory spectrum, and in general we call the output the K-theory 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 K-theory of schemes in versions of Quillen, Bass, and Waldhausen.
http://mathoverflow.net/questions/74237/the-vanishing-of-non-connective-k-theory-in-negative-degrees
nLab page on Abstract K-theory