http://hdl.handle.net/2445/7502 Sun, 19 May 2013 10:34:42 GMT 2013-05-19T10:34:42Z http://hdl.handle.net/2445/34755 Title: Sur le théorème locale des cycles invariants Authors: Guillén Santos, Francisco; Navarro, Vicenç (Navarro Aznar) Abstract: Si on considère une famille de variétés projectives complexes non singulières, c"est un fait aujourd'hui bien connu que les possibles variétés singulières vers lesquelles peut dégénerer cette famille doivent vérifier certaines contraintes, parmi lesquelles une importante relation entre la cohomologie de la fibre singulière, la cohomologie de la fibre générique et la monodromie de la famille, qui est precise par le théorème local des cycles invariants prouvé par Clemens, Deligne et Steenbrink ([1], [4], [13]) : tous les cocycles de la fibre générique qui sont invariants par la monodromie autour d¡une fibre singulière proviennent par spécialisation de la cohomologie de cette fibre singulière. Tue, 31 Jul 1990 00:00:00 GMT http://hdl.handle.net/2445/34755 1990-07-31T00:00:00Z http://hdl.handle.net/2445/34655 Title: A Cartan-Eilenberg approach to Homotopical Algebra Authors: Guillén Santos, Francisco; Navarro, Vicenç (Navarro Aznar); Pascual, Pere (Pascual i Gainza); Roig Martí, Agustí Abstract: In this paper we propose an approach to homotopical algebra where the basic ingredient is a category with two classes of distinguished morphisms: strong and weak equivalences. These data determine the cofibrant objects by an extension property analogous to the classical lifting property of projective modules. We define a Cartan-Eilenberg category as a category with strong and weak equivalences such that there is an equivalence of categories between its localisation with respect to weak equivalences and the relative localisation of the subcategory of cofibrant objects with respect to strong equivalences. This equivalence of categories allows us to extend the classical theory of derived additive functors to this non additive setting. The main examples include Quillen model categories and categories of functors defined on a category endowed with a cotriple (comonad) and taking values on a category of complexes of an abelian category. In the latter case there are examples in which the class of strong equivalences is not determined by a homotopy relation. Among other applications of our theory, we establish a very general acyclic models theorem. Wed, 13 May 2009 00:00:00 GMT http://hdl.handle.net/2445/34655 2009-05-13T00:00:00Z http://hdl.handle.net/2445/34541 Title: The differentiable chain functor is not homotopy equivalent to the continuous chain functor Authors: Guillén Santos, Francisco; Navarro, Vicenç (Navarro Aznar); Pascual, Pere (Pascual i Gainza); Roig Martí, Agustí Abstract: Let $S_*$ and $S_*^\{infty}$ be the functors of continuous and differentiable singular chains on the category of differentiable manifolds. We prove that the natural transformation $i: S_*^\infty \rightarrow S_*$, which induces homology equivalences over each manifold, is not a natural homotopy equivalence. Thu, 01 Jan 2009 00:00:00 GMT http://hdl.handle.net/2445/34541 2009-01-01T00:00:00Z http://hdl.handle.net/2445/34464 Title: Monoidal functors, acyclic models and chain operads Authors: Guillén Santos, Francisco; Navarro, Vicenç (Navarro Aznar); Pascual, Pere (Pascual i Gainza); Roig Martí, Agustí Abstract: We prove that for a topological operad $P$ the operad of oriented cubical singular chains, $C^{\ord}_\ast(P)$, and the operad of simplicial singular chains, $S_\ast(P)$, are weakly equivalent. As a consequence, $C^{\ord}_\ast(P\nsemi\mathbb{Q})$ is formal if and only if $S_\ast(P\nsemi\mathbb{Q})$ is formal, thus linking together some formality results which are spread out in the literature. The proof is based on an acyclic models theorem for monoidal functors. We give different variants of the acyclic models theorem and apply the contravariant case to study the cohomology theories for simplicial sets defined by $R$-simplicial differential graded algebras. Tue, 01 Apr 2008 00:00:00 GMT http://hdl.handle.net/2445/34464 2008-04-01T00:00:00Z