In recent years the theory of dendroidal sets has emerged as an
important framework for higher algebra. We introduce the concept of a
–algebraicdrawing of a dendroidal set. It depicts a dendroidal set as an object in the category of presheaves
on
–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
–theory.
Finally, a method to analyze graph algebras in terms of trees is sketched.
