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.

103 pages, v2: accepted version, v3: fixed some TeX issues

References in corpus (1)

• (Op)lax natural transformations, twisted quantum field theories, and "even higher" Morita categories

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