nLab
finite étale morphism of anabelioids

Contents

Introduction

Observing that an anabelioid is in particular a topos, a finite étale morphism 𝒳𝒴\mathcal{X} \rightarrow \mathcal{Y} of anabelioids is the same as an étale geometric morphism: it is a ‘local isomorphism’, in some sense.

Definition

Recall that a morphism of anabelioids? 𝒳𝒴\mathcal{X} \rightarrow \mathcal{Y} is simply an exact functor 𝒴𝒳\mathcal{Y} \rightarrow \mathcal{X}, that is to say, a functor 𝒴𝒳\mathcal{Y} \rightarrow \mathcal{X} preserving both finite limits and finite colimits. The following definition follows Mochizuki2004.

Definition

A morphism of anabelioids? f:𝒳𝒴f: \mathcal{X} \rightarrow \mathcal{Y} is finite étale if there is an object SS of 𝒴\mathcal{Y} such that f=i Sif = i_S \circ i for some isomorphism of anabelioids i:𝒳𝒴 /Si: \mathcal{X} \rightarrow \mathcal{Y}_{/ S}, where 𝒴 /S\mathcal{Y}_{/ S} denotes the overcategory of objects of 𝒴\mathcal{Y} over SS, and where i S:𝒴 /S𝒴i_{S}: \mathcal{Y}_{/ S} \rightarrow \mathcal{Y} is the canonical functor.

References

Last revised on April 17, 2020 at 19:38:49. See the history of this page for a list of all contributions to it.