Symmetric Monoidal Structure Research Articles

Invertible Spectra in the E (n )-Local Stable Homotopy Category

Suppose C is a category with a symmetric monoidal structure, which we will refer to as the smash product. Then the Picard category is the full subcategory of objects which have an inverse under the smash product in C, and the Picard group Pic(C) is the collection of isomorphism classes of such invertible objects. The Picard group need not be a set in general, but if it is then it is an abelian group canonically associated with C. There are many examples of symmetric monoidal categories in stable homotopy theory. In particular, one could take the whole stable homotopy category S. In this case, it was proved by Hopkins that the Picard group is just Z, where a representative for n can be taken to be simply the n-sphere Sn [8, 19]. It is more interesting to consider Picard groups of the E-local category, for various spectra E (all of which will be p-local for some fixed prime p in this paper). Here the smash product of two E-local spectra need not be E-local, so one must relocalize the result by applying the Bousfield localization functor LE. The best-known case is E=K(n), the nth Morava K-theory, considered in [8]. In this paper we study the case E=E(n), where E(n) is the Johnson–Wilson spectrum. In this case the E-localization functor is universally denoted Ln, and we denote the category of E-local spectra by L. Our main theorem is the following result.

Function spectra

Boardman's stable category (see [5]) is a closed category ([4], VII·7), and in the best of all possible worlds, the category of spectra underlying the stable category would be closed as well; this would make life considerably easier for those doing calculations in stable homotopy theory. Unfortunately none of the categories of spectra introduced to date are closed; only S, the category introduced in [2], is even symmetric monoidal. The problem with making S closed is that it comes equipped with an augmentation to I, the category of universes and linear isometries (called Un in [2]), which preserves the symmetric monoidal structure. Since I is not closed, this makes it difficult to see how S might be closed.

Universal topological algebra needs closed topological categories

It is shown that a development of universal topological algebra, based in the obvious way on the category of topological spaces, leads in general to a pathological situation. The pathology disappears when the base category is changed to a cartesian closed topological category or to a topological category endowed with a compatible closed symmetric monoidal structure, provided that in the latter case, the algebraic operations are expressed in terms of monoidal powers rather than the usual cartesian powers. With such base categories, universal topological algebra becomes virtually as well-behaved as ordinary (setbased) universal algebra.

Stabilizing tensor products

Let C C be a symmetric monoidal category with a suspension, and let SC be the resulting stable category. We shall give necessary and sufficient conditions for extending the symmetric monoidal structure to a monoidal structure on SC. These imply that the usual smash product on finite pointed CW complexes cannot be extended to a smash product (with S 0 {S^0} as unit) on finite spectra, hence not on Boardman spectra. This confirms a conjecture of Alex Heller.

A smash product for spectra

We shall show that the smash product for pointed CW complexes induces a smash product A on the homotopy category of Adams's stable category with the following properties, A is coherently homotopy unitary (S°), associative, and commutative, A commutes with suspension up to homotopy, and A satisfies a Kunneth formula. Introduction. Precisely, we shall show that the homotopy category of a technical variant of Adams's stable category [1], a fraction category of CW prespectra equivalent to that of Boardman [3], [8], admits a symmetric monoidal structure in the sense of [4]. Whitehead's pairings of prespectra [9] and Kan and Whitehead's nonassociative smash product for simplicial spectra [6] were the first attempts at a smash product. Boardman gave the first homotopy associative, commutative, and unitary smash product in his stable category [3]. Adams has recently obtained a similar construction [2], We acknowledge with thanks many helpful conversations with Professors Alex Heller and John C. Moore. Details and applications will appear elsewhere. 1. The interchange problem. A S^-prespectrum X consists of a sequence of pointed spaces {Xn | n ^ 0}, together with inclusions Xn A S k -+ Xn + l. Consider S = {Sn = S } as a ring with respect to the smash product of spaces. Then X is a right S-module. Construction of a homotopy associative smash product for S-prespectra requires permutations n of S A • • • A S. Since S is not strictly commutative, but only graded homotopy commutative, this requires defining suitably canonical maps of degree — 1 (for k odd) and homotopies % ~ ±id. We avoid sign problems by using S-prespectra and define canonical homotopies Hn as follows. Make the standard identifications S s S A • • • A S £ J /a/ A • • • A 7 /5/ s I^/dl* s D/S*-K Then n simply permutes factors of C x • • • x C. Hence n e SU(2k). AMS {MOS) subject classifications (1970). Primary 18D45, 55J99; Secondary 55J10, 55J25.

Homotopy theory of $dg$ categories via localizing pairs and Drinfeld’s $dg$ quotient

Using localizing pairs and Drinfeld's dg quotient we construct a new Quillen model for the homotopy theory of dg categories. We prove that, in contrast with the original model, this new Quillen model carries a natural closed symmetric monoidal structure. As an application, we obtain a simple construction of the internal Hom-objects and a conceptual characterization of Toen's previous work. Making use of this new Quillen model, Lowen has recently developed a derived deformation theory.