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Polynomial functors are a categorical generalization of the usual notion ofpolynomial, which has found many applications in higher categories and typetheory: those are generated by polynomials consisting a set of monomials builtfrom sets of variables. They can be organized into a cartesian bicategory,which unfortunately fails to be closed for essentially two reasons, which weaddress here by suitably modifying the model. Firstly, a naive closure is toolarge to be well-defined, which can be overcome by restricting to polynomialswhich are finitary. Secondly, the resulting putative closure fails to properlytake the 2-categorical structure in account. We advocate here that this can beaddressed by considering polynomials in groupoids, instead of sets. For those,the constructions involved into composition have to be performed up tohomotopy, which is conveniently handled in the setting of homotopy type theory:we use it here to formally perform the constructions required to build ourcartesian bicategory, in Agda. Notably, this requires us introducing anaxiomatization in a small universe of the type of finite types, as anappropriate higher inductive type of natural numbers and bijections.

A Cartesian Bicategory of Polynomial Functors in Homotopy Type Theory