Grothendieck homotopy theory and polynomial monads

Abstract: Grothendieck developed in Pursuing Stacks [5] a beautiful axiomatic approach to homotopy theory as a theory of localisations of the category of small categories Cat. This theory was further deepened by Cisinski in [3, 4].

The category of polynomial monads PolyMon contains Cat as a full subcategory of "linear" monads. It is natural to ask how the homotopy theory of Grothendieck, or at least some parts of it, can be extended to PolyMon.

In this talk we show that many fundamental constructions of Grothendieck theory have their analogues in the word of polynomial monads. This includes: Quillen Theorem A, Grothendieck constructions, Thomason theorem, homotopy left Kan extensions, the theory of final functors, Cisinski localisations etc.

As illustration we consider two applications:

A theory of delooping of mapping spaces between algebras of polynomial monads developed in [1]. A seminal theorem of Dwyer–Hess–Turchin of double delooping of space of long knots will be a consequence.

A theory of locally constant algebras of polynomial monads from [2] which generalises Cisinski theory of locally constant functors [3, 4]. If time permits a sketch of a proof of stabilisation theorem for higher braided operads which, in its turn, implies Baez–Dolan stabilisation for higher categories, will be provided.

References:

[1] Batanin M. and De Leger F., Polynomial monads and delooping of mapping spaces, J. Noncommut. Geom. 13 (2019), 1521–1576.

[2] Batanin M. and White D., Homotopy theory of algebras of substitudes and their localisation, arXiv:2001.05432.

[3] Cisinski D.-C., Les préfaisceaux comme modèles des types d'homotopie, Astérisque 308, 2006.

[4] Cisinski D.-C., Locally constant functors, Math. Proc. Cambridge Philos. Soc. 147 (2009), 593–614.

[5] Grothendieck A., Pursuing Stacks, Manuscript, 1983.

Mathematics

Audience: researchers in the topic

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Opening Workshop (IRP Higher Homotopy Structures 2021, CRM-Bellaterra)

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