Grothendieck homotopy theory and polynomial monads

Grothendieck in Pursuing Stacks has developed a beautiful axiomatic approach to homotopy theory as theory of localisations of the category of small categories Cat. This theory was further deepened by Multsiniotis and Cisinski. The category of finitary polynomial monads is equivalent to the category of Σ-free coloured symmetric operads in Set and contains Cat as a full subcategory of "linear" monads. In this talk I will show that many fundamental constructions of Grothendieck theory can be extended from small categories and presheaves over them to polynomial monads and their algebras. This includes: Grothendieck construction, Quillen Theorem A, Thomason theorem, homotopy left Kan extensions formula, theory of final functors, and so on. Some new phenomena also show up in this extended theory. If time permits I will briefly mention two applications:

1. A theory of delooping of mapping spaces between algebras of polynomial monads developed by Batanin and Florian De Leger. A seminal theorem of Dwyer-Hess-Turchin on double delooping of space of long knots is a consequence.

2. A theory of locally constant algebras of polynomial monads constructed by Batanin and David White which generalises Cisinski theory of locally constant functors. A stabilisation theorem for higher braided operads is one of the applications. This theorem, in its turn, implies Baez-Dolan stabilisation for higher categories and higher groupoids and classical Freudenthal theorem.