$C^*$-algebraic drawings of dendroidal sets

TL;DR

This paper introduces a $C^*$-algebraic drawing of dendroidal sets, creating a functorial bridge between higher algebra and noncommutative geometry, enabling new constructions in bivariant homology and K-theory.

Contribution

It defines a new $C^*$-algebraic representation of dendroidal sets and establishes an adjunction that connects algebraic and topological noncommutative geometry paradigms.

Findings

Constructs a functorial $C^*$-algebraic drawing of dendroidal sets.

Establishes an adjunction between combinatorial model categories.

Proposes a framework for harmonizing algebraic and topological bivariant K-theory.

Abstract

In recent years the theory of dendroidal sets has emerged as an important framework for higher algebra. In this article 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 analyse graph algebras in terms of trees is sketched.