Stable model categories are categories of modules

Abstract

A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for deciding when two stable model categories represent 'the same homotopy theory'. We show that stable model categories with a single compact generator are equivalent to modules over a ring spectrum. More generally stable model categories with a set of generators are characterized as modules over a 'ring spectrum with several objects', i.e., as spectrum valued diagram categories. We also prove a Morita theorem which shows how equivalences between module categories over ring spectra can be realized by smashing with a pair of bimodules. Finally, we characterize stable model categories which represent the derived category of a ring. This is a slight generalization of Rickard's work on derived equivalent rings. We also include a proof of the model category equivalence of modules over the Eilenberg-Mac Lane spectrum HR and (unbounded) chain complexes of R-modules for a ring R.

A curious example of triangulated-equivalent model categories which are not Quillen equivalent

“…Let T and T denote the homotopy endomorphism dgas of k as computed in M and M , respectively; see Theorem 3.5 and Corollary 4.4. By results from Dugger [3], DuggerShipley [6], Schwede-Shipley [19] and Shipley [20], it follows that M and M are Quillen equivalent to the model categories Mod-T and Mod-T , respectively. In fact, in this case it is quite easy to construct the Quillen equivalences directly without referring to the cited work above.…”

Galois descent for real spectra

“…Then the map f is of effective descent for modules (Section 2.6, Thm.2.6). In particular, there is an equivalence of stable ∞-categories [20].…”

Homotopy theory of modules over diagrams of rings

“…Categories of modules over diagrams of rings have created useful new models; see for example [8,18]. These examples use two underlying contexts: differential graded modules over differential graded algebras (DGAs) and module spectra over ring spectra.…”