C∗–algebraic drawings of dendroidal sets

Mahanta, Snigdhayan

doi:10.2140/agt.2019.19.1171

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$C^*$–algebraic drawings of dendroidal sets

Snigdhayan Mahanta

Algebraic & Geometric Topology 19 (2019) 1171–1206

DOI: 10.2140/agt.2019.19.1171

Abstract

In recent years the theory of dendroidal sets has emerged as an important framework for higher algebra. We introduce the concept of a $C^{*}$ –algebraic drawing of a dendroidal set. It depicts a dendroidal set as an object in the category of presheaves on $C^{*}$ –algebras. We show that the construction is functorial and, in fact, it is the left adjoint of a Quillen adjunction between combinatorial model categories. We use this construction to produce a bridge between the two prominent paradigms of noncommutative geometry via adjunctions of presentable $\infty$ –categories, which is the primary motivation behind this article. As a consequence we obtain a single mechanism to construct bivariant homology theories in both paradigms. We propose a (conjectural) roadmap to harmonize algebraic and analytic (or topological) bivariant $K$ –theory. Finally, a method to analyze graph algebras in terms of trees is sketched.

Keywords

$C^*$–algebras, graph algebras, noncommutative spaces, dendroidal sets, simplicial sets, infinity operads, infinity categories

Mathematical Subject Classification 2010

Primary: 46L85, 55P48

Secondary: 18D50, 46L87, 55U10

References

Publication

Received: 14 January 2017

Revised: 27 June 2018

Accepted: 14 October 2018

Published: 21 May 2019

Authors

Snigdhayan Mahanta
Fakultät für Mathematik Universität Regensburg Regensburg Germany

Volume 19, issue 3 (2019)