Sinan Yalin
UFR de Mathématiques
Cité Scientifique - Bâtiment M2
Laboratoire Paul Painlevé
F-59655 Villeneuve d'Ascq Cedex
France
Email: sinan.yalin[at]math.univ-lille1.fr
I am working on a thesis entitled Homotopy theory of algebras over a prop under the supervision of Benoit Fresse, in the Geometry and Topology team.
Operads, props and their variants
Moduli spaces of algebraic structures and deformation theory
Homotopical algebra
Quantization and quantum groups
Field theories and string topology
Knot theory
Classifying spaces and moduli spaces of algebras over a prop, 40 pages, submitted, 2012 (Version arXiv).
Abstract. The purpose of this article is two-fold. First we show that a weak equivalence between cofibrant props induces a weak equivalence between the associated classifying spaces of bialgebras. 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. Then we provide a generalization of a theorem of Charles Rezk in the setting of algebras over a (colored) prop: we introduce the notion of moduli space of algebra structures over a prop, and prove that under certain conditions such a moduli space is the homotopy ber of a map between classifying spaces.
Keywords : props, classifying spaces, moduli spaces, bialgebras category, homotopical algebra, homotopy invariance.
The homotopy theory of bialgebras over pairs of operads, 27 pages, submitted, 2013 (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.
Preliminary version available here