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Concept# Simplicial set

Summary

In mathematics, a simplicial set is an object composed of simplices in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and . Formally, a simplicial set may be defined as a contravariant functor from the to the . Simplicial sets were introduced in 1950 by Samuel Eilenberg and Joseph A. Zilber.
Every simplicial set gives rise to a "nice" topological space, known as its geometric realization. This realization consists of geometric simplices, glued together according to the rules of the simplicial set. Indeed, one may view a simplicial set as a purely combinatorial construction designed to capture the essence of a "well-behaved" topological space for the purposes of homotopy theory. Specifically, the category of simplicial sets carries a natural , and the corresponding is equivalent to the familiar homotopy category of topological spaces.
Simplicial sets are used to define , a basic notion of . A construction analogous

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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).

Kan spectra provide a combinatorial model for the stable homotopy category. They were introduced by Dan Kan in 1963 under the name semisimplicial spectra. A Kan spectrum is similar to a pointed simplicial set, but it has simplices in negative degrees as well and all its simplices have infinitely many faces and degeneracies. In the first part of this thesis, we define a functor from the category of Gamma-spaces to the category of Kan spectra without passing through any other category of spectra. We show that the resulting Kan spectrum of a Gamma-space A agrees with the usual object associated to A in the stable homotopy category by comparing it to Bousfield-Friedlander's spectrum construction. In particular, applying our construction to the Gamma-space associated to a symmetric monoidal category provides a combinatorial model of its algebraic K-theory spectrum. For the Gamma-space associated to an abelian group, this yields via the stable Dold-Kan correspondence the unbounded chain complex with the abelian group concentrated in degree zero. The second part of this work concerns group spectra and twisting structures. Group spectra are the group objects in the category of Kan spectra. They provide an algebraic, combinatorial model for the stable homotopy category. We transfer Brown's model structure from the category of Kan spectra to a Quillen equivalent model structure on the category of group spectra. We then construct the analogues of Kan's loop group functor and its right adjoint Wbar together with corresponding classifying bundles, so that the category of Kan spectra becomes a twisted homotopical category in the sense of Farjoun and Hess.

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.