On the profinite fundamental group of a connected Grothendieck topos

TL;DR

The paper defines an intrinsic notion of the fundamental group for connected Grothendieck toposes by focusing on finite objects and Galois theory within topos theory. It introduces a robust finiteness framework (locally finite and decomposition-finite objects) that yields an embedded pretopos of finite objects and an atomic, locally coherent extension for sums. For connected, finitely generated toposes, it constructs canonical Galois points whose profinite automorphism groups realize the topos as the classifying topos of a profinite group, providing functoriality and a topos-theoretic Chevalley analogue. This leads to a profinite fundamental group hat{π}_1(E,x) that generalizes the classical fundamental group to a broad topos-theoretic setting and specializes to familiar objects like absolute Galois groups in étale toposes, while preserving essential exact sequences in the topos framework.

Abstract

We show that finite (i.e. locally finite and decomposition-finite) objects of a connected Grothendieck topos span a Boolean pretopos with an essentially unique Galois point. The automorphism group of this point carries a profinite topology whose classifying topos is equivalent to the given Grothendieck topos if the latter is finitely generated. This leads to an intrinsic definition of the fundamental group of any connected Grothendieck topos.

Theorems & Definitions (89)

• Lemma 1.1
• proof
• Lemma 1.2
• proof
• Lemma 1.3
• proof
• Lemma 1.4
• proof
• Proposition 1.5
• proof