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CATEGORIFYING RATIONALIZATION

Part of: Higher algebraic $K$-theory Categories and geometry

Published online by Cambridge University Press:  09 January 2020

CLARK BARWICK ,
SAUL GLASMAN ,
MARC HOYOIS ,
DENIS NARDIN  and
JAY SHAH
CLARK BARWICK
Affiliation:
School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK; clarkbar@gmail.com
MARC HOYOIS
Affiliation:
Department of Mathematics, University of Southern California, 3620 South Vermont Avenue, Los Angeles, CA 90089, USA; hoyois@usc.edu
DENIS NARDIN
Affiliation:
Département de Mathématiques, Institut Galilée, Université Paris 13, 99 av. J.B. Clément, 93430 Villetaneuse, France; nardin@math.univ-paris13.fr
JAY SHAH
Affiliation:
Department of Mathematics, University of Notre Dame, 255 Hurley, Notre Dame, IN 46556, USA; jshah3@nd.edu

Abstract

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We construct, for any set of primes $S$, a triangulated category (in fact a stable $\infty$-category) whose Grothendieck group is $S^{-1}\mathbf{Z}$. More generally, for any exact $\infty$-category $E$, we construct an exact $\infty$-category $S^{-1}E$ of equivariant sheaves on the Cantor space with respect to an action of a dense subgroup of the circle. We show that this $\infty$-category is precisely the result of categorifying division by the primes in $S$. In particular, $K_{n}(S^{-1}E)\cong S^{-1}K_{n}(E)$.

MSC classification

Primary: 19D99: None of the above, but in this section
Secondary: 18F25: Algebraic $K$-theory and $L$-theory
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 7 , 2019 , e42
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2019

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