Topological space

Summary

In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate. A topological space is the most general type of a mathematical space that allows for the definition of limits, continuity, and connectedness. Common types of topological spaces include Euclidean spaces, metric spaces and manifolds. Although very general, the concept of topological spaces is fundamental, and used in virtually every branch of modern mathematics. The study of topological spaces in

Official source

https://en.wikipedia.org/wiki/Topological_space

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Catégories simpliciales enrichies et K-théorie de Waldhausen

Ilias Amrani

This thesis, which presents a new approach to the algebraic K-theory, is divided into two parts. The first one is devoted to the category of small simplicial categories. First, we construct a new model structure on sCat = [Δop,Cat] which is called the diagonal model structure, in reference to the diagonal model structure of Moerdijk on bisimplicial sets sSet2. Then we show that the new structure is proper and cellular. Note that this new model structure is not tensored and cotensored over the category of simplicial sets sSet in a manner consistent with the model structure. To remedy this, we use another model structure on sSet2 defined in the article of Cegarra and Remedios [3], which is equivalent to the Moerdijk structure. So we build a second new model structure on [Δop,Cat], which is cofibrantly generated, left proper, cellular and (co)tensored on sSet in a compatible way. Based on the work of [13], we construct the stable category of spectra (not symmetric) SpN(sCat*, Σ). It garantees the existence of Ω-spectra, which allows us to define thenotion of "weak Waldhausen category". The calculation of the simplicial enrichment map of the model category SpN(sCat*, Σ), leads to our new definition of algebraic K-theory of weak Waldhausen categories . The second part of this thesis is an attempt to generalize the previous results for enriched categories. First we begin by recalling the theory of ∞-categories and ∞-groupoids, based on the work of Joyal [14] and Lurie [18]. Then we make comparisons of ∞-categories with the category of simplicial sets equipped with the usual model structure. Our first result is the construction of a model structure on Top – Cat , the category of small categories enriched over the category of topological spaces Top, based on the work of Bergner [1] . The category Top – Cat is Quillen equivalent to sSet – Cat. Note that all objects in Top – Cat are fibrant ; this remark will play an important role in this theory. Our second result is the construction of a new model structure on the category of small simplicial categories enriched over Top, denoted by Top – sCat = [Δop,Top – Cat]. We show that this structure is proper and cellular. The fact that Top – sCat is not (co)tensored over sSet poses a barrier to defining the category of spectra SpN(sCat*, Σ).

EPFL2010

Coalgèbres d'Alexander-Whitney

Théophile Naïto

The 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.

EPFL2009

Recent proposals of large and infinite extra dimensions triggered a strong research activity in theories in which our universe is considered as a sub-manifold of some higher-dimensional space-time, a so-called 3-brane. In this context, it is generally assumed that some mechanism is at work which binds Standard Model particles to the 3-brane, an effect often referred to as the localization of matter on the brane. Gravity, however, is allowed to propagate in general also in the extra dimensions. As demonstrated by Randall and Sundrum in 1999, it is also possible to localize gravity itself on a 3-brane. In the setup they proposed. the 3-brane is realized as a singular domain wall separating two patches of 3-dimensional anti-de-Sitter (AdS5) space-time. The potential between two test masses on the brane at distances larger than the AdS5-radius was shown to be the usual 4-dimensional Newtonian 1/r potential with strongly suppressed corrections. The model of Randall and Sundrum, usually referred to as the Randall-Sundrum II setup, constitutes the center of interest for this thesis. The main goal of this work is to find possible generalizations to higher dimensions of the simple setup considered by Randall and Sundrum. One of the motivations for such a generalization is that a realistic theory should possibly be able to explain the chiral nature of 4-dimensional fermions on the brane. One way to explain chiral fermions from higher dimensions is to consider 3-braves identified with the cores of topological defects located in a higher-dimensional transverse space. Naturally a richer topological structure of the field configuration in transverse space provides the possibility of a more realistic spectrum of chiral fermions localized on the 3-brane. After two introductory chapters on extra dimensions and non-factorizable geometries which are relevant for the Randall-Sundrum II model, we briefly discuss basics of topological defects in the following third chapter. In the rest of the third chapter we consider various solutions to higher-dimensional Einstein equations coupled to a series of physically different sources and discuss their properties of localization of gravity. Due to their asymptotic nature, these solutions are only valid far from the cores of the defects in transverse space. Therefore, it seems reasonable to complement the consideration by presenting a particular numerical example of a solution to the Einstein equations coupled to a set of scalar and gauge fields: this solution describes a 3-brave realized as a 't Hooft-Polyakov monopole residing in the 3-dimensional transverse space of a 7-dimensional space-time. The last chapter of this work is dedicated to the study of a modification of the original Randall-Sundrum II model of another kind. The motivation is given by the geodesic incompleteness of the latter scenario with respect to time-like and light-like geodesics. We will describe a model which resembles the Randall-Sundrurn II model with respect to its properties of gravity localization but with the advantage that the underlying space-time manifold is geodesically complete. Parts of the calculations related to the properties of gravity at low energies in this model are rather technical in nature and we therefore preferred to assemble them in several appendices. We finally add some concluding remarks and discuss possible further developments.

EPFL2004