Function spaces and classifying spaces of algebras over a prop

The goal of this paper is to prove that the classifying spaces of categories of algebras governed by a prop can be determined by using function spaces on the category of props. We first consider a function space of props to define the moduli space of algebra structures over this prop on an object of the base category. Then we mainly prove that this moduli space is the homotopy fiber of a forgetful map of classifying spaces, generalizing to the prop setting a theorem of Rezk. The crux of our proof lies in the construction of certain universal diagrams in categories of algebras over a prop. We introduce a general method to carry out such constructions in a functorial way.Comment: 28 pages, modifications mainly in section 2 (more details in some proofs and additional explanations), typo corrections. Final version, to appear in Algebr. Geom. Topo

Moduli stacks of algebraic structures and deformation theory

We connect the homotopy type of simplicial moduli spaces of algebraic structures to the cohomology of their deformation complexes. Then we prove that under several assumptions, mapping spaces of algebras over a monad in an appropriate diagram category form affine stacks in the sense of Toen-Vezzosi's homotopical algebraic geometry. This includes simplicial moduli spaces of algebraic structures over a given object (for instance a cochain complex). When these algebraic structures are parametrised by properads, the tangent complexes give the known cohomology theory for such structures and there is an associated obstruction theory for infinitesimal, higher order and formal deformations. The methods are general enough to be adapted for more general kinds of algebraic structures.Comment: several corrections, especially in sections 6 and 7. Final version, to appear in the J. Noncommut. Geo

The homotopy theory of bialgebras over pairs of operads

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.Comment: 27 pages, final version, to appear in the Journal of Pure and Applied Algebr

Classifying spaces of algebras over a prop

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.Comment: Final version, to appear in Algebraic \& Geometric Topolog

Homotopy theory of algebras over a PROP

Le but de cette thèse est de mettre en place une théorie d’homotopie générale pour les catégories de bigèbres différentielles graduées. Une première partie est consacrée au cas des catégories de bigèbres définies par un couple d’opérades en distribution. Les bigèbres classiques, les bigèbres de Lie, les bigèbres de Poisson fournissent des exemples de telles structures de bigèbres. Le résultat principal de cette partie montre que la catégorie des bigèbres associée a un couple d’opérades en distribution hérite d’une structure de catégorie de modèles. La notion de PROP donne un cadre pour étudier des structures de bigèbres générales, impliquant des opérations à plusieurs entrées et plusieurs sorties comme générateurs de la structure, par opposition aux opérades en distribution qui ne permettent de coder que des opérations à une seule entrée ou à une seule sortie seulement. Les PROPs forment une catégorie, dans laquelle on peut définir une notion d’objet cofibrant avec de bonnes propriétés homotopiques.La seconde partie de la thèse est consacrée à la théorie homotopique des bigèbres sur un PROP. Le résultat principal de la thèse est que les catégories de bigèbres associées à des PROPs cofibrants faiblement équivalents ont des catégories homotopiques équivalentes. En fait, on prouve un théorème plus précis qui donne une équivalence au niveau des localisations simpliciales des catégories. Notre théorème entraine que la catégorie des bigèbres associée à une résolution cofibrante d’un PROP donné P définit une notion de bigèbre à homotopie près sur P indépendante du choix de la résolution, et permet de donner un sens à des problèmes de réalisation homotopiques dans ce cadre.The purpose of this thesis is to set up a general homotopy theory for categories of differential graded bialgebras. A first part is devoted to the case of bialgebras defined by a pair of operads in distribution. Classical bialgebras, Lie bialgebras and Poisson bialgebras provide examples of such bialgebra structures. The main result of this part asserts that the category of bialgebras associated to a pair of operads in distribution inherits a model category structure. The notion of a PROP provides a setting for the study of general bialgebras structures, involving operations with multiple inputs and multiple outputs as generators of the structure, in contrast to operads in distribution which only encode operations with either one single input or one single output. PROPs form a category, in which one can define a notion of cofibrant object with good homotopical properties. The second part of the thesis is devoted to the homotopy theory of bialgebras over a PROP. The main result of the thesis asserts that the categories of bialgebras associated to weakly equivalent cofibrant props have equivalent homotopy categories. We actually prove a more precise theorem asserting that this equivalence holds at the level of a simplicial localization of the categories. Our theorem implies that the category of bialgebras associated to a cofibrant resolution of a given PROP P defines a notion of bialgebra up to homotopy over P independent of the choice of the resolution, and enables us to give a sense to homotopical realization problems in this setting

SIMPLICIAL LOCALIZATION OF HOMOTOPY ALGEBRAS OVER A PROP

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

Maurer-Cartan spaces of filtered L-infinity algebras

We study several homotopical and geometric properties of Maurer- Cartan spaces for L-infinity algebras which are not nilpotent, but only filtered in a suitable way. Such algebras play a key role especially in the deformation theory of algebraic structures. In particular, we prove that the Maurer-Cartan simplicial set preserves fibrations and quasi-isomorphisms. Then we present an algebraic geometry viewpoint on Maurer-Cartan moduli sets, and we compute the tangent complex of the associated algebraic stack

Realization spaces of algebraic structures on chains

Given an algebraic structure on the homology of a chain complex, we define its realization space as a Kan complex whose vertices are the structures up to homotopy realizing this structure at the homology level. Our algebraic structures are parametrised by props and thus include various kinds of bialgebras. We give a general formula to compute subsets of equivalences classes of realizations as quotients of automorphism groups, and determine the higher homotopy groups via the cohomology of deformation complexes. As a motivating example, we compute subsets of equivalences classes of realizations of Poincar\'e duality for several examples of manifolds