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

Rune Haugseng

Geometry & Topology 21 (2017) 1631–1730

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

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iterated bimodules, \mathbbE_n–algebras, higher Morita category
Mathematical Subject Classification 2010
Primary: 18D50, 55U35
Secondary: 16D20
Received: 18 February 2015
Revised: 7 April 2016
Accepted: 8 May 2016
Published: 10 May 2017
Proposed: Mark Behrens
Seconded: Peter Teichner, Jesper Grodal
Rune Haugseng
Department of Mathematical Sciences
University of Copenhagen
Universitetsparken 5
DK-2100 København Ø