# Algebraic K-theory and abstract homotopy theory

@article{Blumberg2007AlgebraicKA, title={Algebraic K-theory and abstract homotopy theory}, author={Andrew J. Blumberg and Michael A. Mandell}, journal={Advances in Mathematics}, year={2007}, volume={226}, pages={3760-3812} }

Abstract We decompose the K -theory space of a Waldhausen category in terms of its Dwyer–Kan simplicial localization. This leads to a criterion for functors to induce equivalences of K -theory spectra that generalizes and explains many of the criteria appearing in the literature. We show that under mild hypotheses, a weakly exact functor that induces an equivalence of homotopy categories induces an equivalence of K -theory spectra.

#### 33 Citations

On the algebraic K-theory of higher categories

- Mathematics
- 2012

We prove that Waldhausen K-theory, when extended to a very general class of quasicategories, can be described as a Goodwillie differential. In particular, K-theory spaces admit canonical (connective)… Expand

Derived Koszul duality and involutions in the algebraic K-theory of spaces

- Mathematics
- 2011

We interpret different constructions of the algebraic K-theory of spaces as an instance of derived Koszul (or bar) duality and also as an instance of Morita equivalence. We relate the interplay… Expand

Theorems in Higher Category Theory and Applications

- Mathematics
- 2019

In this thesis we construct the universal coCartesian fibration , which (strictly) classifies coCartesian fibrations of simplicial sets with small fibers. To this end, we develop a general framework… Expand

Invariance de la K-théorie par équivalences dérivées

- Mathematics
- 2010

The aim of these notes is to prove that any right exact functor between reasonable Waldhausen categories, that induces an equivalence at the level of homotopy categories, gives rise to a homotopy… Expand

Split Injectivity of A-Theoretic Assembly Maps

- Mathematics
- International Mathematics Research Notices
- 2019

We construct an equivariant coarse homology theory arising from the algebraic $K$-theory of spherical group rings and use this theory to derive split injectivity results for associated assembly… Expand

Workshop on the homotopy theory of homotopy theories

- Mathematics
- 2011

These notes are from a series of lectures given at the Workshop on the Homotopy Theory of Homotopy Theories which took place in Caesarea, Israel, in May 2010. The workshop was organized by David… Expand

Algebraic K-theory of finitely projective modules on $\mathbb{E}_{\infty}$-rings

- Mathematics
- 2016

In this paper, we study the K-theory on higher modules in spectral algebraic geometry. We relate the K-theory of an $\infty$-category of finitely generated projective modules on certain… Expand

Higher stacks as diagrams

- Mathematics
- 2021

Several possible presentations for the homotopy theory of (non-hypercomplete) ∞-stacks on a classical site S are discussed. In particular, it is shown that an elegant combinatorial description in… Expand

A universal characterization of higher algebraic K-theory

- Mathematics
- 2010

In this paper we establish a universal characterization of higher algebraic K-theory in the setting of small stable infinity categories. Specifically, we prove that connective algebraic K-theory is… Expand

#### References

SHOWING 1-10 OF 38 REFERENCES

A note on K-theory and triangulated categories

- Mathematics
- 2002

Abstract.We provide an example of two closed model categories having equivalent homotopy categories but different Waldhausen K-theories. We also show that there cannot exist a functor from small… Expand

A remark on K-theory and S-categories

- Mathematics
- 2002

Abstract It is now well known that the K-theory of a Waldhausen category depends on more than just its (triangulated) homotopy category (Invent. Math. 150 (2002) 111). The purpose of this note is to… Expand

The localization sequence for the algebraic K-theory of topological K-theory

- Mathematics
- 2006

We verify a conjecture of Rognes by establishing a localization cofiber sequence of spectra $K(\mathbb{Z})\to K(ku)\to K(KU) \to\Sigma K(\mathbb{Z})$ for the algebraic K-theory of topological… Expand

K-THEORY AND DERIVED EQUIVALENCES

- Mathematics
- 2002

We show that if two rings have equivalent derived categories then they have the same algebraic K-theory. Similar results are given for G-theory, and for a large class of abelian categories.

Function complexes in homotopical algebra

- Mathematics
- 1980

1 .l Summary IN [l] QUILLEN introduced the notion of a model category (a category together with three classes of maps: weak equivalences, fibrations and cofibrations, satisfying certain axioms (1.4… Expand

ALGEBRAIC K-THEORY OF SPACES I

- Mathematics
- 1978

The algebraic K–theory of spaces is a variant, invented by F. Waldhausen in the late 1970’s, of the standard algebraic K–theory of rings. Until that time, applications of algebraic K–theory to… Expand

Negative K-theory of derived categories

- Mathematics
- 2006

We define negative K-groups for exact categories and for ``derived categories'' in the framework of Frobenius pairs, generalizing definitions of Bass, Karoubi, Carter, Pedersen-Weibel and Thomason.… Expand

Homotopy Limit Functors on Model Categories and Homotopical Categories

- Mathematics
- 2005

Model categories: An overview Model categories and their homotopy categories Quillen functors Homotopical cocompleteness and completeness of model categories Homotopical categories: Summary of part… Expand

Equivalence of simplicial localizations of closed model categories

- Mathematics
- 1999

Abstract We determine a necessary and sufficient condition for a functor between closed model categories to induce an equivalence of Dwyer–Kan simplicial localizations.

Higher Algebraic K-Theory of Schemes and of Derived Categories

- Mathematics
- 1990

In this paper we prove a localization theorem for the K-theory of commutative rings and of schemes, Theorem 7.4, relating the K-groups of a scheme, of an open subscheme, and of the category of those… Expand