Models for classifying spaces and derived deformation theory

Using the theory of extensions of L-infinity algebras, we construct rational homotopy models for classifying spaces of fibrations, giving answers in terms of classical homological functors, namely the Chevalley-Eilenberg and Harrison cohomology. We also investigate the algebraic structure of the Chevalley-Eilenberg complexes of L-infinity algebras and show that they possess, along with the Gerstenhaber bracket, an L-infinity structure that is homotopy abelian.Comment: 23 pages. This version contains minor technical corrections and a new section with a list of open problems. To appear in Proceedings of the LM

Convolution algebras and the deformation theory of infinity-morphisms

Given a coalgebra C over a cooperad, and an algebra A over an operad, it is often possible to define a natural homotopy Lie algebra structure on hom(C,A), the space of linear maps between them, called the convolution algebra of C and A. In the present article, we use convolution algebras to define the deformation complex for infinity-morphisms of algebras over operads and coalgebras over cooperads. We also complete the study of the compatibility between convolution algebras and infinity-morphisms of algebras and coalgebras. We prove that the convolution algebra bifunctor can be extended to a bifunctor that accepts infinity-morphisms in both slots and which is well defined up to homotopy, and we generalize and take a new point of view on some other already known results. This paper concludes a series of works by the two authors dealing with the investigation of convolution algebras.Comment: 17 pages, 1 figure; (v2): Expanded some proofs, corrected typos, updated references. Final versio