Multiplicative infinite loop structure in algebraic K-theory
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.
category of simplicial objects over the category
scientific article; zbMATH DE number 4048661