Université du Luxembourg, campus Kirchberg
Mathematics Research Unit
6 rue Richard Coudenhove-Kalergi
|I am currently assistant chercheur (postdoc) at Luxembourg University with Sergei Merkulov, in the team Geometry and the Mathematical theory of Quantization.|
| Research interests
The homotopy theory of bialgebras over pairs of operads, Journal of Pure and Applied Algebra, Volume 218, Issue 6, June 2014, Pages 973-991 (Version arXiv).
Abstract. We endow the category of bialgebras over a pair of operads in distribution with a cofibrantly generated model category structure. We work in the category of chain complexes over a field of characteristic zero. We split our construction in two steps. In the first step, we equip coalgebras over an operad with a cofibrantly generated model category structure. In the second one we use the adjunction between bialgebras and coalgebras via the free algebra functor. This result allows us to do classical homotopical algebra in various categories such as associative bialgebras, Lie bialgebras or Poisson bialgebras in chain complexes.
Classifying spaces of algebras over a prop, to appear in Algebraic & Geometric Topology (Version arXiv).
Abstract. We prove that a weak equivalence between cofibrant props induces a weak equivalence between the associated classifying spaces of algebras. This statement generalizes to the prop setting a homotopy invariance result which is well known in the case of algebras over operads. The absence of model category structure on algebras over a prop leads us to introduce new methods to overcome this difficulty. We also explain how our result can be extended to algebras over colored props in any symmetric monoidal model category tensored over chain complexes.
Keywords : props, classifying spaces, bialgebras category, homotopical algebra, homotopy invariance.
Simplicial localization of homotopy algebras over a prop, 2013 (Version arXiv).
Abstract.We prove that a weak equivalence between two cofibrant (colored) props in chain complexes induces a Dwyer-Kan equivalence between the simplicial localizations of the associated categories of algebras. This homotopy invariance under base change implies that the homotopy category of homotopy algebras over a prop P does not depend on the choice of a cofibrant resolution of P, and gives thus a coherence to the notion of algebra up to homotopy in this setting. The result is established more generally for algebras in combinatorial monoidal dg categories.
I worked on my thesis Théorie de l'homotopie des algèbres sur un prop in the Topology and Geometry team of the Paul Painlevé laboratory. My advisor was Benoit Fresse.
I defended my thesis the 13 september 2013.
An electronic version is available here:
Théorie de l'homotopie des algèbres sur un PROP
Key words: PROP, bialgebra, homotopical algebra, classifying space.
Groupes de travail |
I organized a reading seminar "Deformation theory and derived algebraic stacks" at Paul Painlevé laboratory in fall 2013. A detailed schedule is available here: [Programme]