Multiplicative infinite loop structure in algebraic K-theory
type: Item
instance of: scholarly article
title: Multiplicative infinite loop structure in algebraic K-theory
zbMATH Open document ID: 0643.55009
DOI: 10.1007/BF01157032
published in: Mathematical Notes
publication date: 1986-01-01T00:00:00Z
review text: In a series of important papers, \textit{J. P. May} has initiated a systematic study of infinite loop spaces, specialized ring spectra, etc. [see for example J. Pure Appl. Algebra 26, 1-69 (1982; Zbl 0532.55013), or Geom. Appl. Homotopy Theory II; Proc. Conf. Evanston 1977, Lect. Notes Math. 658, 240-315 (1978; Zbl 0425.18014)]. This work unifies and clarifies significant amounts of mathematics. Starting with an \(A_{\infty}\)-ring X, one may form \(KX=Z\times (BGLX)\) \(+\), getting the theory of Quillen in the discrete case as well as the algebraic K-theory of a space of \textit{F. Waldhausen} [Algebr. Geom. Topol., Stanford/Calif. 1976, Proc. Symp. Pure Math., Vol. 32, Part 1, 35-60 (1978; Zbl 0414.18010)]. The present work has two sections. The first - which is rather categorical - compares various constructions for arbitrary, simplicial or topological categories (small). For example, the author studies the adjointness between product and Hom in a very abstract setting, and also produces a left adjoint for \(\Delta\) 0 in a suitable functor category, where \(\Delta\) 0C refers to the category of simplicial objects over the category C, i.e. the functors \(\Delta\) \(0\to C.\) The second section in more geometrical. The principal result involves a diagram of functors which relate \(E_{\infty}\) rings with various types of operad action. The principal corollary asserts that if X is an \(E_{\infty}\)-ring, then so is KX. Unfortunately, the details of this densely written note are too technical to discuss in a review.
Mathematics Subject Classification ID: 55P47
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 55R40
Mathematics Subject Classification ID: 55P42
Mathematics Subject Classification ID: 55N15
Mathematics Subject Classification ID: 18A25
zbMATH DE Number: 4048661
zbMATH Keywords: infinite loop spaces
zbMATH Keywords: \(A_{\infty }\)-ring
zbMATH Keywords: algebraic K-theory of a space
zbMATH Keywords: product
zbMATH Keywords: Hom
zbMATH Keywords: category of simplicial objects over the category
zbMATH Keywords: \(E_{\infty }\) rings
zbMATH Keywords: operad action
author: R. Krasauskas
reviewed by: Donald W. Kahn
MaRDI profile type: Publication
cites work: Q3048986
cites work: Infinite loop structures on the algebraicK-theory of spaces
cites work: Multiplicative infinite loop space theory
cites work: Categories and cohomology theories
cites work: Pairings of categories and spectra
full work available at URL: https://doi.org/10.1007/bf01157032
OpenAlex ID: W2050804879
description: scientific article; zbMATH DE number 4048661