On polynomial functors and polynomial comonads over infinity groupoids

TL;DR

This paper characterizes polynomial functors over infinity groupoids as colimits of representables, explores their categorical properties, and introduces polynomial comonads, linking them to complete Segal spaces and generalizing classical results.

Contribution

It establishes a categorical framework for polynomial functors over infinity groupoids and introduces polynomial comonads, extending classical polynomial theory to higher categorical contexts.

Findings

Polynomial functors over infinity groupoids are colimits of representable copresheaves.

The paper defines polynomial comonads within this setting.

A construction linking polynomial comonads to complete Segal spaces is presented.

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.