Cofinal morphism of polynomial monads and double delooping

TL;DR

This paper constructs a homotopically cofinal morphism of polynomial monads using internal algebra classifiers, providing a categorical double delooping proof for the Turchin-Dwyer-Hess theorem on long knots.

Contribution

It introduces a new homotopically cofinal morphism of polynomial monads and applies it to a categorical proof of double delooping in knot space topology.

Findings

Constructed a homotopically cofinal morphism of polynomial monads.

Provided a categorical double delooping proof for the Turchin-Dwyer-Hess theorem.

Connected algebraic monad theory with topological knot space results.

Abstract

Using the theory of internal algebras classifiers developed by Batanin and Berger, we construct a morphism of polynomial monads which we prove is homotopically cofinal. We then describe how this result constitutes the main conceptual argument for a categorical direct double delooping proof of the Turchin-Dwyer-Hess theorem concerning the explicit double delooping of spaces of long knots.