Morita homotopy theory for (∞,1)(\infty,1)-categories and ∞\infty-operads

We prove the existence of Morita model structures on the categories of small simplicial categories, simplicial sets, simplicial operads and dendroidal sets, modelling the Morita homotopy theory of $(\infty,1)$-categories and $\infty$-operads. We give a characterization of the weak equivalences in terms of simplicial presheaves, simplicial algebras and slice categories. In the case of the Morita model structure for simplicial categories and simplicial operads, we also show that each of these model structures can be obtained as an explicit left Bousfield localization of the Bergner model structure on simplicial categories and the Cisinski--Moerdijk model structure on simplicial operads, respectively.Comment: 30 pages. v2: Sections 4 and 5 about the relation between Morita equivalences of algebraic theories and operads have been removed and made a separate note (arXiv:arXiv:1808.09045). The paper has been rewritten accordingly with minor modifications. v3: Final versio

A generalization of Ohkawa's theorem

A theorem due to Ohkawa states that the collection of Bousfield equivalence classes of spectra is a set. We extend this result to arbitrary combinatorial model categories.Comment: 13 pages; consequences in motivic homotopy theory have been adde