The fundamental pro-groupoid of an affine 2-scheme

TL;DR

This paper introduces a new perspective on the étale fundamental groupoid of affine 2-schemes, connecting it to categories of modules over commutative 2-rings, and clarifies its properties and limitations.

Contribution

It defines a novel notion of fundamental groupoid for affine 2-schemes using commutative 2-rings, expanding the Tannakian framework beyond traditional settings.

Findings

The étale fundamental groupoid can be characterized via categories of modules.

Introduces the concept of commutative 2-rings encompassing Grothendieck topoi.

Clarifies that étale fundamental groups are profinite and not true groups.

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 \'etale fundamental groupoid $\pi_1(\spec(R))$, i.e.\ the separable absolute Galois group of $R$ when it is a field. This gives a new definition for \'etale $\pi_1(\spec(R))$ in terms of the category of $R$-modules rather than the category of \'etale 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 $\pi_1$ for the corresponding "affine 2-schemes." These results help to simplify and clarify some of the peculiarities of the \'etale fundamental group. For example, \'etale fundamental groups are not "true" groups but only profinite groups, and one cannot hope to recover more: the "Tannakian" functor represented by the \'etale fundamental group of a scheme preserves finite products but not all products.