Incomplete Tambara functors

Blumberg, Andrew; Hill, Michael

doi:10.2140/agt.2018.18.723

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Incomplete Tambara functors

Andrew J Blumberg and Michael A Hill

Algebraic & Geometric Topology 18 (2018) 723–766

DOI: 10.2140/agt.2018.18.723

Abstract

For a “genuine” equivariant commutative ring spectrum $R$ , $π_{0} (R)$ admits a rich algebraic structure known as a Tambara functor. This algebraic structure mirrors the structure on $R$ arising from the existence of multiplicative norm maps. Motivated by the surprising fact that Bousfield localization can destroy some of the norm maps, in previous work we studied equivariant commutative ring structures parametrized by $N_{\infty}$ operads. In a precise sense, these interpolate between “naive” and “genuine” equivariant ring structures.

In this paper, we describe the algebraic analogue of $N_{\infty}$ ring structures. We introduce and study categories of incomplete Tambara functors, described in terms of certain categories of bispans. Incomplete Tambara functors arise as $π_{0}$ of $N_{\infty}$ algebras, and interpolate between Green functors and Tambara functors. We classify all incomplete Tambara functors in terms of a basic structural result about polynomial functors. This classification gives a conceptual justification for our prior description of $N_{\infty}$ operads and also allows us to easily describe the properties of the category of incomplete Tambara functors.

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Keywords

Tambara functor, Mackey functor, equivariant homotopy

Mathematical Subject Classification 2010

Primary: 55P91, 55N91

Secondary: 18B99, 19A22

References

Publication

Received: 14 March 2016

Revised: 20 June 2017

Accepted: 27 July 2017

Published: 12 March 2018

Authors

Andrew J Blumberg
Department of Mathematics University of Texas Austin, TX United States

Michael A Hill
Department of Mathematics University of California Los Angeles, CA United States
http://math.ucla.edu/~mikehill

Volume 18, issue 2 (2018)