Extended etale Homotopy groups from profinite Galois categories

Click to access Extended_etale.pdf

Cet article s’appuie sur SGA1

https://arxiv.org/abs/math/0206203v1

et part d’un schéma X (scheme) notion explicitée dans EGA1

Click to access PMIHES_1960__4__5_0.pdf

Mais je veux juste souligner le 1.2 page 5 sur 9, qui met en place l’∞-catégorie des ∞-catégories notée Cat∞ et dans ce blog (∞,1)Cat, la sous ∞-catégorie Top∞ des ∞-topoi et la sous ∞-catégorie Spc des espaces qui est un ∞-topos archétypique analogue à Set pour les 1-catégories :

https://anthroposophiephilosophieetscience.wordpress.com/2018/01/11/scienceinternelle-l∞-topos-s-spaces-joue-dans-le-domaine-des-∞-categories-le-role-du-1-topos-set-dans-le-domaine-des-categories/

https://anthroposophiephilosophieetscience.wordpress.com/2017/03/01/highertopostheory-11-lanalogue-du-1-topos-set-pour-la-theorie-des-∞-categories-l-∞-categorie-spaces/

Spc est engendrée par les ∞-groupoides, idées développées par Grothendieck dans « A la poursuite des champs »:

https://ncatlab.org/nlab/show/Pursuing+Stacks

« Grothendieck considered among other things, the notion of n and ∞ homotopy types and how to model them, homology and cohomology theories defined on categories of models and schematisation of homotopy types. This last is an attempt to define homotopy theory relative to a base ring, say, such that over ℤ ordinary homotopy theory is recovered.

The following summary is due to David Roberts (from this discussion on MO):

Pursuing Stacks is composed of (if memory serves correctly) three themes. The first was homotopy types as higher (non-strict) groupoids. This part was first considered in Grothendieck’s letters to Larry Breen from 1975, and is mostly contained in the letter to Daniel Quillen which makes up the first part of PS (about 12 pages or so). Georges Maltsiniotis has extracted Grothendieck’s proposed definition for a weak ∞-groupoid, and there is by Dimitri Ara (Ara 10) towards showing that this definition satisfies the homotopy hypothesis

Rappelons que (∞,1)Cat

https://ncatlab.org/nlab/show/(infinity,1)Cat

est considérée dans ce blog comme l’Idée de l’Un, ou de Dieu :

https://anthroposophiephilosophieetscience.wordpress.com/2017/04/16/scienceinternelle-19-recherches-sur-lidee-de-dieu-qui-est-dieu-∞-categorie-des-∞-categories/

la plus importante donc.

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