A famous Turchin-Dwyer-Hess theorem states that for a reduced multiplicative nonsymmetric operad X the totalisation of associated cosimplicial object is homotopy equivalent to the double loop space over mapping space of associativity operad to X. This theorem implies Kontsevich’s conjecture on space of long knots and has many other applications and generalisations. In 2019 paper Batanin and De Leger have developed an extension of Grothendieck homotopy theory to the category of polynomial monads and demonstrated that the TDH theorem can be proved within the framework of this homotopy theory. From the point of view of polynomial monad theory the monad for nonsymmetric operad is a double Baez-Dolan +-construction over the identity monad on Set. The aim of the project is to generalise the TDH theorem to the multiple iterations of +-construction with a view of future applications to new models of embedding spaces of manifolds.
Seznam odborné literatury
1) Batanin, De Leger: Polynomial monads and delooping of mapping spaces, Journal of Noncommutative Geometry 13 (4), 1521-1576 (2019), arXiv preprint arXiv:1712.00904
2) Dwyer W., Hess K., Long knots and maps between operads, Geometry & Topology 16 (2012), 919–955
3) J.Kock, A.Joyal, M.Batanin and J.-F. Mascari, Polynomial functors and opetopes, Adv. Math., 224 no. 6, 2690-2737 (2010).
Předběžná náplň práce
Cílem práce je zobecnění Turchinovy-Dwyerovy-Hessové věty na vícenásobné iterace +-konstrukce s využitím rozšíření Grothendieckovy homotopické teorie na kategorii polynomiálních monád.
Předběžná náplň práce v anglickém jazyce
A famous Turchin-Dwyer-Hess theorem states that for a reduced multiplicative nonsymmetric operad X the totalisation of associated cosimplicial object is homotopy equivalent to the double loop space over mapping space of associativity operad to X. This theorem implies Kontsevich’s conjecture on space of long knots and has many other applications and generalisations. In 2019 paper Batanin and De Leger have developed an extension of Grothendieck homotopy theory to the category of polynomial monads and demonstrated that the TDH theorem can be proved within the framework of this homotopy theory. From the point of view of polynomial monad theory the monad for nonsymmetric operad is a double Baez-Dolan +-construction over the identity monad on Set. The aim of the project is to generalise the TDH theorem to the multiple iterations of +-construction with a view of future applications to new models of embedding spaces of manifolds.
- zadáno a potvrzeno stud. odd.