On polynomial functors and polynomial comonads over infinity groupoids

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.

💡 Research Summary

The paper develops a theory of single‑variable polynomial functors in the ∞‑category 𝒮 of ∞‑groupoids, following the definition introduced by Gepner‑Haugseng‑Kock (GHK). The authors first observe that a polynomial functor can be described either as a composite \