Geometry of a category of complexes and algebraic K-theory
type: Item
instance of: scholarly article
title: Geometry of a category of complexes and algebraic K-theory
zbMATH Open document ID: 0574.55020
DOI: 10.1215/S0012-7094-85-05220-2
published in: Duke Mathematical Journal
publication date: 1985-01-01T00:00:00Z
review text: The Grothendieck group of an abelian category A has a natural description in terms of the derived category of A, but it is wellknown that an analogous description of higher K-functors, even of \(K_ 1\), is not possible. In this paper the authors provide a new description of higher algebraic K-theory as the homotopy groups of a suitably defined classifying space for the category of bounded chain complexes in A. They construct this classifying space using a new notion of ''pseudo-functor'', and their main theorem asserts that the geometric realizations of six different constructions, among them Quillen's Q-construction and Waldhausen's S-construction, are homotopy equivalent. In an appendix, their methods are applied to classifying spaces of monoidal categories.
Mathematics Subject Classification ID: 55U99
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 55R35
zbMATH DE Number: 3917266
zbMATH Keywords: Grothendieck group of an abelian category
zbMATH Keywords: derived category
zbMATH Keywords: higher K- functors
zbMATH Keywords: geometric realizations
zbMATH Keywords: classifying spaces of monoidal categories
author: Vladimir Hinich
author: Vadim Schechtman
reviewed by: Michael R. Stein
MaRDI profile type: Publication
cites work: Q5580926
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description: scientific article; zbMATH DE number 3917266