Brane actions for coherent $\infty$-operads

Abstract

We prove that Mann-Robalo's construction of the brane action extends to general coherent $\infty$-operads, with possibly multiple colors and non-contractible spaces of unary operations. This requires to establish two results regarding spaces of extensions that were left unproven in the aforementioned construction. First, we show that Lurie's and Mann-Robalo's models for such spaces are equivalent. Second, we prove that the space of extensions in the sense of Lurie is not in general equivalent to the homotopy fiber of the associated forgetful morphism, but rather to its homotopy quotient by the $\infty$-groupoid of unary operations, correcting an oversight in existing literature. As an application, we obtain that the $\infty$-operads of $B$-framed little disks are coherent and therefore yield new operations on spaces of branes of perfect derived stacks.