Applied Categorical Structures

, Volume 21, Issue 5, pp 469–522 | Cite as

The Fundamental Pro-groupoid of an Affine 2-scheme

Article

Abstract

A natural question in the theory of Tannakian categories is: What if you don’t remember Forget? Working over an arbitrary commutative ring R, we prove that an answer to this question is given by the functor represented by the étale fundamental groupoid π1(spec(R)), i.e. the separable absolute Galois group of R when it is a field. This gives a new definition for étale π1(spec(R)) in terms of the category of R-modules rather than the category of étale covers. More generally, we introduce a new notion of “commutative 2-ring” that includes both Grothendieck topoi and symmetric monoidal categories of modules, and define a notion of π1 for the corresponding “affine 2-schemes.” These results help to simplify and clarify some of the peculiarities of the étale fundamental group. For example, étale fundamental groups are not “true” groups but only profinite groups, and one cannot hope to recover more: the “Tannakian” functor represented by the étale fundamental group of a scheme preserves finite products but not all products.

Keywords

Higher category theory Presentable categories Fundamental groupoids Galois theory Categorification Affine 2-schemes Tannakian reconstruction 

Mathematics Subject Classifications (2010)

18D05 18D10 18D35 14F35 14H30 

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© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.University of CaliforniaBerkeleyUSA

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