000135617 001__ 135617
000135617 005__ 20190316234528.0
000135617 0247_ $$2doi$$a10.5075/epfl-thesis-4381
000135617 02470 $$2urn$$aurn:nbn:ch:bel-epfl-thesis4381-8
000135617 02471 $$2nebis$$a5752142
000135617 037__ $$aTHESIS
000135617 041__ $$afre
000135617 088__ $$a4381
000135617 245__ $$aCoalgèbres d'Alexander-Whitney$$bun modèle algébrique pour les espaces topologiques
000135617 269__ $$a2009
000135617 260__ $$aLausanne$$bEPFL$$c2009
000135617 300__ $$a148
000135617 336__ $$aTheses
000135617 520__ $$aThe goal of this work is to study Alexander-Whitney coalgebras (first defined in [HPST06]) from a topological point of view. An Alexander-Whitney coalgebra is a coassociative chain coalgebra over Z with an extra algebraic structure : the comultiplication must respect the coalgebra structure up to an infinite sequence of homotopies (this sequence is part of the data of the Alexander-Whitney coalgebra structure). Alexander-Whitney coalgebras are interesting for topologists because the normalized chain complex C(K) of a simplicial set K is endowed with an Alexander-Whitney coalgebra structure. This theorem is proved for the first time here (generalising a result proven in [HPST06]). This theorem gives the hope that the Alexander-Whitney coalgebra structure of C(K) contains interesting information that can be used to solve topological problems. This hope is strengthened by the success already obtained in the work of several topologists. Among others, [HPST06], [HL07], [Boy08], and [HR] use the Alexander-Whitney coalgebra structure of the normalized chains of a simplicial set in an essential way to solve topological problems. This thesis begins with some background material. In particular, the definition of a DCSH morphism between two coassociative chain coalgebras is recalled in complete detail. For example, signs are determined with great precision. Next we devote a chapter to the definition of Alexander-Whitney coalgebras and to their importance in topology. In the following chapter we begin the conceptual study of Alexander-Whitney coalgebras. A global study of these objects had not yet been carried out even if the Alexander-Whitney coalgebra structure has been studied and used in order to answer some specific questions. With the aim of studying Alexander-Whitney coalgebras in a nice setting, we develop an operadic description of these coalgebras in the following chapter. More precisely, we show that there is an explicit operad AW such that the coalgebras over this operad are exactly the Alexander-Whitney coalgebras. Furthermore, AW is shown to be a Hopf operad, so that the category formed by the Alexander-Whitney coalgebras is actually a monoidal category. These results are proven in a reasonably general framework. In fact, we associate an operad to each bimodule (over the associative operad) of a certain type, such that we get AW if this bimodule is well chosen. In particular, these results enable us to study Alexander-Whitney coalgebras from the standpoint of operads. This strategy is recognised to be successful in various mathematical situations, and especially in algebraic topology. Moreover, we develop a minimal model notion in the setting of right module over a chosen operad (which has to satisfy some reasonable conditions), with the aim of applying this result to the special case of the Alexander-Whitney coalgebras. This is possible because coalgebras over some fixed operad P can be seen as right modules over P. And the category of right modules over P has some nice features which do not appear to hold in the category of P-coalgebras. The inspiration for this part of our work comes from the notion of minimal model developed in the framework of rational homotopy theory. The two following facts show that it is reasonable to try to adapt some ideas of rational homotopy theory to the category of Alexander-Whitney coalgebras. A. There is a theorem that says that studying topological spaces up to rational equivalences is, essentially, equivalent to studying cocommutative chain coalgebras over the field of rational numbers. This is false if the ring of integers replaces the field of rational numbers, but Alexander-Whitney coalgebras are "almost" cocommutative in the sense which is explained in this thesis. B. It could be that the Alexander-Whitney coalgebra structure of the normalized chains of a simplicial set is weak enough to allow explicit computations. At least, it is clear that the Alexander-Whitney coalgebra structure on the normalized chains is far from being an E∞-structure (such a structure determines the homotopy type of the considered simplicial set, at least under some conditions). The chapter about minimal models in the framework of right modules over an operad includes an existence theorem and a discussion of the unicity of this model. In the second part of this chapter, we construct an explicit path-object in the model category of right modules over an operad. This path-object is then used to investigate the topologically relevant information that could stem from the minimal model in the case of the operad AW. Finally, we present and examine some interesting open questions about Alexander-Whitney coalgebras. These questions give a nice outlook on future research in this area.
000135617 6531_ $$aAlexander-Whitney coalgebra
000135617 6531_ $$aoperad
000135617 6531_ $$aAlexander-Whitney operad
000135617 6531_ $$aright module over an operad
000135617 6531_ $$aminimal model
000135617 6531_ $$aDCSH category
000135617 6531_ $$acoalgèbre d'Alexander-Whitney
000135617 6531_ $$aopérade
000135617 6531_ $$aopérade d'Alexander-Whitney
000135617 6531_ $$amodule à droite sur une opérade
000135617 6531_ $$amodèle minimal
000135617 6531_ $$acatégorie DCSH
000135617 700__ $$aNaïto, Théophile
000135617 720_2 $$0240499$$aHess-Bellwald, Kathryn$$edir.$$g105396
000135617 8564_ $$s1173220$$uhttps://infoscience.epfl.ch/record/135617/files/EPFL_TH4381.pdf$$yTexte intégral / Full text$$zTexte intégral / Full text
000135617 909C0 $$0252139$$pUPHESS$$xU10968
000135617 909CO $$ooai:infoscience.tind.io:135617$$pthesis$$pthesis-bn2018$$pDOI$$pSV$$qDOI2$$qGLOBAL_SET
000135617 919__ $$aGR-HE
000135617 918__ $$aSB$$cIGAT$$dEDMA
000135617 920__ $$b2009
000135617 973__ $$aEPFL$$sPUBLISHED
000135617 970__ $$a4381/THESES
000135617 980__ $$aTHESIS