Large condensation in enriched $\infty$-categories

Abstract

Using the language of enriched $\infty$-categories, we formalize and generalize the definition of fusion n-category, and an analogue of iterative condensation of $E_i$-algebras. The former was introduced by Johnson-Freyd, and the latter by Kong, Zhang, Zhao, and Zheng. This extends categorical condensation beyond fusion n-categories to all enriched monoidal $\infty$-categories with certain colimits. The resulting theory is capable of treating symmetries of arbitrary dimension and codimension that are enriched, continuous, derived, non-semisimple and non-separable. Additionally, we consider a truncated variant of the notion of condensation introduced by Gaiotto and Johnson-Freyd, and show that iterative condensation of monoidal monads and $E_i$-algebras provide examples. In doing so, we prove results on functoriality of Day convolution for enriched $\infty$-categories, and monoidality of two versions of the Eilenberg-Moore functor, which may be of independent interest.