# Contents

## Introduction

Observing that an anabelioid is in particular a topos, a finite étale morphism $\mathcal{X} \rightarrow \mathcal{Y}$ 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? $\mathcal{X} \rightarrow \mathcal{Y}$ is simply an exact functor $\mathcal{Y} \rightarrow \mathcal{X}$, that is to say, a functor $\mathcal{Y} \rightarrow \mathcal{X}$ preserving both finite limits and finite colimits. The following definition follows Mochizuki2004.

###### Definition

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