On polynomial functors and polynomial comonads over infinity groupoids
Kun Chen
TL;DR
This work develops polynomial functors over the $\infty$-category $\mathcal{S}$ of $\infty$-groupoids, showing that such single-variable polynomials are colimits of representables indexed by $\infty$-groupoids and establishing the bicompleteness of $Poly_{\mathcal{S}}$. It introduces a monoidal structure on $Poly_{\mathcal{S}}$ via composition and defines polynomial comonads as comonoid objects, aiming to connect these to complete Segal spaces and to generalize the classical Ahman-Uustalu correspondence to the $\infty$-categorical setting. The paper provides explicit constructions for the composition and coclosure, analyzes mapping spaces between polynomials, and develops a décalage-like functor to build augmented cosimplicial structures from comonoids. A partial but substantive result shows that comonoids yield augmented cosimplicial spaces with Segal-type properties, offering a concrete path toward a higher-categorical analogue of the Ahman-Uustalu theorem and a potential bridge to complete Segal spaces.
Abstract
We show that single-variable polynomial functors over the category $\mathcal{S}$ of infinity groupoids, as defined by Gepner-Haugseng-Kock, are exactly colimits of representable copresheaves indexed by infinity groupoid. This allows us to establish certain categorical properties of the $\infty$-category $Poly_{\mathcal{S}}$, in parallel with the case of the ordinary category $Poly$. We define the notion of polynomial comonad under the monoidal structure of $Poly_{\mathcal{S}}$ induced by composition of polynomials, and describe a construction toward exploring the connection between polynomial comonads and complete Segal spaces. This construction partially generalizes the classical one given in the proof of a theorem of Ahman-Uustalu.