Lax-idempotent monads in homotopy theory

Abstract

We extend the theory of lax-idempotent monads to the setting of (∞,2)-categories. In particular, we prove an analogue of Kock and Zöberlein's characterization of the category of algebras over a lax-idempotent monad. We then apply this theory to many examples that arise naturally in homotopy theory. For example, we show that (∞,1)-toposes are monadic over presentable (∞,1)-categories, with left adjoint given by the symmetric (∞,1)-topos. We also show that ∞-operads are comonadic over symmetric monoidal (∞,1)-categories through the monoidal envelope construction, obtaining along the way a new proof of Barkan–Steinebrunner's result that the envelope functor is a faithful inclusion. We furthermore use this theory to simplify Ramzi's proof that dualizable stable (∞,1)-categories are comonadic over ℵ₁-compactly generated stable (∞,1)-categories.