Polynomial monads and delooping of mapping spaces

TL;DR

This paper extends classical homotopy theory results to polynomial monads and their algebras, providing new categorical proofs and applications to the delooping of spaces of long knots.

Contribution

It generalizes key theorems from small category homotopy theory to polynomial monads, offering novel categorical proofs and applications.

Findings

Generalized classical theorems to polynomial monads

Provided categorical proofs for delooping results

Applied to spaces of long knots

Abstract

We extend some classical results - such as Quillen's Theorem A, the Grothendieck construction, Thomason's Theorem and the characterisation of homotopically cofinal functors - from the homotopy theory of small categories to polynomial monads and their algebras. As an application we give a categorical proof of the Dwyer-Hess and Turchin results concerning the explicit double delooping of spaces of long knots.