Volume 21, issue 3 (2017)

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

Rune Haugseng

Geometry & Topology 21 (2017) 1631–1730
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 ${\Delta }^{\phantom{\rule{0.3em}{0ex}}n,op}$ 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 $\mathsc{C}$ is an ${\mathbb{E}}_{n+k}$–monoidal $\infty$–category then the $\left(\infty ,n+1\right)$–category of ${\mathbb{E}}_{n}$–algebras in $\mathsc{C}$ 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.

Keywords
iterated bimodules, \mathbbE_n–algebras, higher Morita category
Mathematical Subject Classification 2010
Primary: 18D50, 55U35
Secondary: 16D20