C∗–algebraic drawings of dendroidal sets

Mahanta, Snigdhayan¹

¹ Fakultät für Mathematik, Universität Regensburg, Regensburg, Germany

Algebraic and Geometric Topology, Tome 19 (2019) no. 3, pp. 1171-1206

Résumé

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.

Détail

DOI : 10.2140/agt.2019.19.1171

Classification : 46L85, 55P48, 18D50, 46L87, 55U10
Keywords: $C^*$–algebras, graph algebras, noncommutative spaces, dendroidal sets, simplicial sets, infinity operads, infinity categories