Localization of Enriched Categories and Cubical Sets

The invertibility hypothesis for a monoidal model category S asks that localizing an S-enriched category with respect to an equivalence results in an weakly equivalent enriched category. This is the most technical among the axioms for S to be an excellent model category in the sense of Lurie, who showed that the category CatS of S-enriched categories then has a model structure with characterizable fibrant objects. We use a universal property of cubical sets, as a monoidal model category, to show that the invertibility hypothesis is a consequence of the other axioms. Topological categories, simplicial categories, and differential graded categories are special types of enriched categories: the enriching category has a notion of weak equivalence and its own homotopy theory. These have played a prominent role a diverse array of subjects. Getting control over the homotopy theory of some of these enriched categories and homotopical constructions in them (such as pushouts, pullbacks, and other derived limit and colimit constructions) is easier in the presence of model structures. If S is a monoidal model category, Lurie gave conditions for the existence of a model structure with many useful properties on the collection CatS of S-enriched categories [Lur09, A.3.2.4]. (In the terminology of [BM13], this allows Lurie to assert that the canonical model structure exists.) The cofibrations and weak equivalences in CatS have a relatively straightforward description (see §2), but in order to get a useful characterization of the fibrations more assumptions are required. With this goal, Lurie defined an excellent model category as a model category S, with a symmetric monoidal structure, satisfying additional axioms labeled (A1) through (A5). The first four of these axioms are all relatively standard concepts or are straightforward to verify. Axiom (A5) is called the invertibility hypothesis. It is more technical—it roughly asserts that inverting a weak equivalence results in a weakly equivalent enriched category—and is more difficult to verify in practice. The fact that the category Set∆ of simplicial sets satisfies the invertibility hypothesis is an important result of Dwyer and Kan [DK80, 10.4]. The invertibility hypothesis for differential graded categories is a consequence of work of Toën [Toë07, 8.7], and for enrichment in simplicial model categories it is a theorem of Dundas [Dun01, 0.9]. Our main result is the following. Partially supported by NSF grant DMS–1206008. c © Tyler Lawson, 2016. Permission to copy for private use granted.