On the profinite fundamental group of a connected Grothendieck topos

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.