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

TL;DR

This paper establishes Morita model structures for various categories modeling $( abla,1)$-categories and $ abla$-operads, providing new characterizations of weak equivalences and explicit localizations of existing models.

Contribution

It introduces Morita model structures on categories of simplicial categories, sets, operads, and dendroidal sets, advancing the homotopy theory of $( abla,1)$-categories and $ abla$-operads.

Findings

Morita model structures exist on categories of small simplicial categories, sets, operads, and dendroidal sets.

Weak equivalences are characterized via presheaves, algebras, and slice categories.

Model structures are obtained as explicit Bousfield localizations of known models.

Abstract

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.