The higher Morita category of $\mathbb{E}_{n}$–algebras

Abstract

We introduce simple models for associative algebras and bimodules in the context of nonsymmetric $\infty$–operads, and use these to construct an $\left(\infty ,2\right)$–category of associative algebras, bimodules and bimodule homomorphisms in a monoidal $\infty$–category. By working with $\infty$–operads over ${\Delta }^{n,op}$ we iterate these definitions and generalize our construction to get an $\left(\infty ,n+1\right)$–category of ${\mathbb{E}}_n$–algebras and iterated bimodules in an ${\mathbb{E}}_n$–monoidal $\infty$–category. Moreover, we show that if $\mathcal{C}$ is an ${\mathbb{E}}_{n+k}$–monoidal $\infty$–category then the $\left(\infty ,n+1\right)$–category of ${\mathbb{E}}_n$–algebras in $\mathcal{C}$ has a natural ${\mathbb{E}}_k$–monoidal structure. We also identify the mapping $\left(\infty ,n\right)$–categories between two ${\mathbb{E}}_n$–algebras, which allows us to define interesting nonconnective deloopings of the Brauer space of a commutative ring spectrum.