$C^*$–algebraic drawings of dendroidal sets

Abstract

In recent years the theory of dendroidal sets has emerged as an important framework for higher algebra. We introduce the concept of a $C^{\ast }$–algebraic drawing of a dendroidal set. It depicts a dendroidal set as an object in the category of presheaves on $C^{\ast }$–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.