# PhD Defense Daria Poliakova

Title: Homotopical algebra and combinatorics of polytopes

In this thesis we present several connected results in homotopical algebra, aimed at understanding strong homotopy monoidality. Polytopal methods are used.

We begin from obtaining an explicit model for a homotopy limit of a cosimplicial DG-category, associated to a (derived) algebraic group via BG-construction. The answer is given in terms of Maurer-Cartan elements in the Cobar-construction of the corresponding coalgebra, or, equivalently, in terms of A-comodules over this coalgebra. These can be viewed as homotopy characters of the corresponding group in the sense of Abad-Crainic-Dherin. To arrive at this answer, we fill in some details for Holstein simplicial resolutions, which are used in Arkhipov-Orsted formula for cosimplicial homotopy limits of DG-categories.

Polytopal methods come into play when approaching the conjectural weakly monoidal structure on the DG-category described above. For this, we formulate the notion of operadic pairs, and identify the families of polytopes behind the 2-colored operadic pair governing DG-algebras with A-modules over them. These are freehedra of Saneblidze, which were not previously known to have an operadic meaning.

We then direct our attention to the general machinery giving rise to polytopal diagonals such as Saneblidze-Umble one. To each directed polytope we associate a colored operad, and the diagonals are reinterpreted as components in the Poincare-Hilbert series of such operads. For some polytopes (simplices, polygons and product thereof) we prove Koszulity and quadratic self-duality of the corresponding operads, which equips the cellular complexes of respective polytopes with a structure of what we call "integrated A-coalgebra".

As our last input, we construct a bi-family of embedded polytopes, constrainahedra, that we expect to govern the (non-existent) theory of strong homotopy duoids.

Link to  hybrid defense: will be announced later

Link to Thesis: will be announced later

###### Assessment Committee:

Prof. Nathalie Wahl, (chair) University of Copehagen

Prof. Vladimir Dotsenko, Université de Strasbourg

Prof. Vladimir Baranovsky, University of Illinois, Chicago