Groupe d'homotopie

Résumé

En mathématiques, et plus particulièrement en topologie algébrique, les groupes d'homotopie sont des invariants qui généralisent la notion de groupe fondamental aux dimensions supérieures. Définition Il y a plusieurs définitions équivalentes possibles. ;Première définition Soit X un espace topologique et x_0 un point de X. Soit \mathcal{B}^i la boule unité de dimension i de l'espace euclidien \mathbb{R}^i. Son bord \partial \mathcal{B}^i = \mathcal{S}^{i-1} est la sphère unité de dimension i-1. Le i-ième groupe d'homotopie supérieur \pi_i(X,x_0) est l'ensemble des classes d'homotopie relative à \mathcal{S}^{i-1} d'applications continues f : \mathcal{B}^i\to X telle que : f(\mathcal{S}^{i-1}) = {x_0}. Un élément de \pi_i(X,x_0) est donc représenté par une fonction continue de la i-boule vers X, qui envoie la (i-1)-sphère vers

Source officielle

https://fr.wikipedia.org/wiki/Groupe_d'homotopie

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There is a classical "duality" between homotopy and homology groups in that homotopy groups are compatible with homotopy pullbacks (every homotopy pullback gives rise to a long exact sequence in homotopy), while homology groups are compatible with homotopy pushouts (every homotopy pushout gives rise to a long exact sequence in homology). This last statement is sometimes referred to as the Mayer-Vietoris or excision axiom. The classical Blakers-Massey theorem (or homotopy excision theorem) asks to what extent the excision property for homotopy pushouts remains true if we replace homology groups by homotopy groups and gives a range in which the excision property holds. It does so by estimating the connectivity of a certain comparison map, which is a rather crude measure, as it is just a single number. Since connectivity is a special case of a cellular inequality, the hope is that there is a stronger statement hidden behind the connectivity result in terms of such inequalities. This process of generalising the homotopy excision theorem has been initiated by Chachólski in the 90s, where he proved a more general version for homotopy pushout squares. The caveat was that one had to suspend the comparison map in question first and the goal of our project -- which we obtained -- was to lose this suspension and then move on to cubical diagrams, rather than squares. To do so, there are a few basic ingredients that are necessary. We first talk about our abstract approach to derived functors, then construct left Bousfield localisations of combinatorial model categories and finally, generalise the foundational concepts in the theory of closed classes to non-connected spaces.

EPFL2016

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The theory of persistence, which arises from topological data analysis, has been intensively studied in the one-parameter case both theoretically and in its applications. However, its extension to the multi-parameter case raises numerous difficulties, where it has been proven that no barcode-like decomposition exists. To tackle this problem, algebraic invariants have been proposed to summarize multi-parameter persistence modules, adapting classical ideas from commutative algebra and algebraic geometry to this context. Nevertheless, the crucial question of their stability has raised little attention so far, and many of the proposed invariants do not satisfy a naive form of stability. In this paper, we equip the homotopy and the derived category of multi-parameter persistence modules with an appropriate interleaving distance. We prove that resolution functors are always isometric with respect to this distance. As an application, this explains why the graded-Betti numbers of a persistence module do not satisfy a naive form of stability. This opens the door to performing homological algebra operations while keeping track of stability. We believe this approach can lead to the definition of new stable invariants for multi-parameter persistence, and to new computable lower bounds for the interleaving distance (which has been recently shown to be NP-hard to compute in [2]).

SPRINGER INTERNATIONAL PUBLISHING AG2021

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Towards an (∞,2)-category of homotopy coherent monads in an ∞-cosmos

Dimitri Zaganidis

This thesis is part of a program initiated by Riehl and Verity to study the category theory of (infinity,1)-categories in a model-independent way. They showed that most models of (infinity,1)-categories form an infinity-cosmos K, which is essentially a category enriched in quasi-categories with some additional structure reminiscent of a category of fibrant objects. Riehl and Verity showed that it is possible to formulate the category theory of (infinity,1)-categories directly with infinity-cosmos axioms. This should also help organize the category theory of (infinity,1)-categories with structure. Given a category K enriched in quasi-categories, we build via a nerve construction a stratified simplicial set N_Mnd(K) whose objects are homotopy coherent monads in K. If two infinity-cosmoi are weakly equivalent, their respective stratified simplicial sets of homotopy coherent monads are also equivalent. We also provide an (infinity,2)-category Adj_r(K) whose objects are homotopy coherent adjunctions in K, that we use to classify the 1-simplices of N_Mnd(K) up to homotopy.

EPFL2017

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