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$C^*$–algebraic drawings of dendroidal sets

Snigdhayan Mahanta

Algebraic & Geometric Topology 19 (2019) 1171–1206

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 –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.

$C^*$–algebras, graph algebras, noncommutative spaces, dendroidal sets, simplicial sets, infinity operads, infinity categories
Mathematical Subject Classification 2010
Primary: 46L85, 55P48
Secondary: 18D50, 46L87, 55U10
Received: 14 January 2017
Revised: 27 June 2018
Accepted: 14 October 2018
Published: 21 May 2019
Snigdhayan Mahanta
Fakultät für Mathematik
Universität Regensburg