Quantizing homotopy types

Kontsevich (90’s) proposed a topological quantization of (sigma-models into) finite homotopy types to top dimensions (d, d+1). Its enhancement to a `fully extended’ TQFT was described later (Freed, Hopkins, Lurie and the speaker) in the target category of iterated algebras. Independently, Chas and Sullivan constructed a (partially defined) 2-dimensional TQFT (d=1) with target compact oriented manifolds.
I will briefly review the features of the finite homotopy theory and its boundary conditions, with particular interest in Dirichlet conditions; their analogue in Chas-Sullivan theory (older work by Blumberg, Cohen and the speaker). Finally, I propose a generalization combining these to a higher-dimensional Chas-Sullivan theory.