$C^*$-algebraic drawings of dendroidal sets

In recent years the theory of dendroidal sets has emerged as an important framework for combinatorial topology. 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 model categories. We use this construction to produce a bridge between the two prominent paradigms of noncommutative geometry via adjunctions of presentable $\infty$-categories. As a consequence we obtain a new homotopy theory for $C^*$-algebras that is well-adapted to the notion of weak operadic equivalences. Finally, a method to analyse graph algebras in terms of trees is sketched.