Tensor schemes "with relations"

Question

In "The geometry of tensor calculus I," Joyal and Street introduce the notion of a tensor scheme, a set of abstract objects, together with formal morphisms between words in this set. They then prove that there is a "free monoidal category" on this data, given by certain classes of decorated graphs up to deformation, as well as free monoidal categories subject to various structures, such as symmetric or braided.

From my perspective, a natural variation on this construction is to add "relations," i.e. to declare that the composition of two morphisms in the tensor scheme must be a 3rd one. This does not seem to be possible in Joyal and Street's formalism, but to me looks like a small tweak to it. In fancy modern terminology, we would like of this as adding higher morphisms to the tensor scheme (and then considering a monoidal category as an $(\infty,2)$-category with one object).

Is there any good reference for these variations of Joyal and Street's construction?

Answer 1

I guess you are essentially after the notion of computad (also called polygraph) which allows for generators in arbitrary dimensions. In particular, those generators can be seen as relations starting from a particular dimension, leading to a reasonable notion of presentation for higher categories (which includes monoidal categories as a particular case). Those have been most extensively developed for strict higher categories, although weak variants have also been defined, e.g. by Batanin (if that's what you're after).

Answer 2

In "A graphical calculus for semi-groupal categories", we have proposed a theory of PI-monoidal categories, which can be viewed as a categorification of the theory of PI-algebras. The key point in our approach to add relations on tensor schemes is the fact that the unit convention can be naturally generalized into $\Omega$-conventions.