Large condensation in enriched $\infty$-categories

Abstract

We demonstrate that many algebraic features of higher condensation theory are really features of higher algebra. We work in the context of monads on the unit of a monoidal 2- category. Assuming that 2-categorical unstraightening is monoidal, we show that the functor which takes such a monad to its Eilenberg-Moore object is lax monoidal. We use this to introduce an analog of condensation for codimension-i symmetries, without assuming dualizability. We give sufficient conditions for Morita equivalence of endomorphisms of condensates. We relate higher centralizers to iterated categories of modules. Along the way, we prove results pertaining to functoriality of Day convolution for enriched infinity-categories, which may be of independent interest.

arXiv commentV2: Revised for correctness and readability. Main theorem weakened; a necessary 2-categorical hypothesis made explicit and proof simplified. For Morita equivalence, an additional generation criterion added and proof simplified. Condensation procedure modified to avoid set-theoretic issues and simplify notation. Introduction and abstract rewritten; article rearranged for narrative clarity