Polynomial monads and delooping of mapping spaces

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.