Approximate groupoids again

This post is for future (google) reference for my project of relating approximate groups with emergent algebras. I would appreciate any constructive comment which could validate (or invalidate) this path of research.

Here is the path I would like to pursue further. The notion of approximate groupoid (see here for the definition) is not complete, because it is flattened, i.e. the set of arrows K should be seen as a set of variables. What I think is that the correct notion of approximate groupoid is a polynomial functor over groupoids (precisely a specific family of such functors). The category Grpd is cartesian closed,  so it has an associated model of (typed) lambda calculus. By using this observation I could apply emergent algebra techniques (under the form of my graphic lambda calculus, which was developed with — and partially funded by —  this application in mind) to approximate groupoids and hope  to obtain streamlined proofs of Breuillard-Green-Tao type results.

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