The higher Morita category of 𝔼n–algebras

Haugseng, Rune

doi:10.2140/gt.2017.21.1631

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

Rune Haugseng

Geometry & Topology 21 (2017) 1631–1730

DOI: 10.2140/gt.2017.21.1631

Abstract

We introduce simple models for associative algebras and bimodules in the context of nonsymmetric $\infty$ –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 $Δ^{n, op}$ 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 $C$ is an $E_{n + k}$ –monoidal $\infty$ –category then the $(\infty, n + 1)$ –category of $E_{n}$ –algebras in $C$ 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 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

References

Publication

Received: 18 February 2015

Revised: 7 April 2016

Accepted: 8 May 2016

Published: 10 May 2017

Proposed: Mark Behrens

Seconded: Peter Teichner, Jesper Grodal

Authors

Rune Haugseng
Department of Mathematical Sciences University of Copenhagen Universitetsparken 5 DK-2100 København Ø Denmark
http://sites.google.com/site/runehaugseng

Volume 21, issue 3 (2017)