Pairings of categories and spectra

Abstract

One flourishing branch of category theory, namely coherence theory, lies at the heart of algebraic K-theory. Coherence theory was initiated in MacLane’s paper [13]. There is an analogous coherence theory of higher homotopies, and the classifying space construction transports categorical coherence to homotopical coherence. When applied to interesting discrete categories, this process leads to the products and pairings (and deeper internal structure) of algebraic K-theory. In much of the literature on algebraic K-theory, the underlying coherence theory is tacitly assumed (as indeed it is throughout mathematics). However, the details of coherence theory are crucial for rigor. For one thin g, they explain which diagrams can, and which cannot, be made simultaneously to commute. For example, a symmetric monoidal category is one with a coherently unital, associative, and commutative product. It can be replaced by an equivalent permutative category, namely one with a strictly unital and associative but coherently commutative product. One cannot achieve strict commutativity except in trivial cases. Thomason [26] has given an amusing illustration of the sort of mistake that can arise from a too cavalier attitude towards this kind of categorical distinction when studying pairings of categories, and one of my concerns is to correct a similar mistake of my own. In [ 171, I developed a coherence theory of higher homotopies for ring spaces up to homotopy and for pairings of H-spaces. That theory is entirely correct. I also discussed the analogous categorical coherence, proving some results and asserting others. That theory too is entirely correct, my unproven assertions having been carefully proven by Laplaza [unpublished]. However, my translations from the categorical to the homotopical theories in [17], that of course being the part I thought to be obvious, are quite wrong. The moral is that to treat the transition from categorical coherence to homotopical coherence smoothly and rigorously, one should take advantage of the definitional framework established by the category theorists. Given the work of MacLane, Kelly, Street, Laplaza, and others [9, 10,241, this transition is really quite easy. One can handle the simplest coherence situations satisfactorily without it, as in Segal’s original passage from permutative categories to f-spaces [22] or my original