The higher Morita category of $E_n$-algebras

Abstract

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