Multiplicative infinite loop space theory

Abstract

In [22], Thomason and I gave a synthesis which combined the different existing infinite loop space theories into a single coherent whole. In particular, we proved that, up to equivalence, there is only one sensible way to pass from space level input data to spectrum level output. In [21], I elaborated the additive theory by showing how to incorporate into it a theory of pairings. This explained how to pass from space level pairing data to pairings of spectra. The input data there was as general as would be likely to find use, and the output, while deduced using one particular infinite loop space machine, automatically applied to all machines by virtue of the uniqueness theorem. We shall here obtain a comparably complete multiplicative infinite loop space theory, The idea is to start with input data consisting of ring spaces up to all possible htgher coherence homotopies and to obtain output consisting of ring spectra with enriched internal structure. Applications of such internal structure abound, both in infinite loop space theory and its applications to geometric topology [5,14, IS] and in stable homotopy theory [3,16]. We shall explain the notion of a ‘category of ring operators’ I in Section 1 and the notion of a F-space in Section 2. We shall see that this notion includes as special cases both the (V, Y)-spaces that were the input of the E, ring theory in [14] and the SLY&paces that provide the simplest input for a Segal style development of multiplicative infinite loop space theory. On a technical note, we shall define 8_spaces with a cofibration condition, but we shall see in Appendix C that the cofibration condition results in little loss of generality. We shall also see that if f and .X are categories of ring operators which are equivalent in a suitable sense, then the categories of T-spaces and of .X-spaces are equivalent. All of this is precisely parallel to the additive theory in (221. The main applications start with categories with products @ and @ which satisfy the axioms for a commutative ring up to coherent natural isomorphism. Making as many diagrams as possible commute strictly, one arrives at the notion of a bipermutative category. We shall show in Section 3 that bipermutative categories functorially