A Cartesian Bicategory of Polynomial Functors in Homotopy Type Theory

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