nLab finite étale morphism of anabelioids

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

• The geometry of anabelioids, Shinichi Mochizuki, 2004, Publ. Res. Inst. Math. Sci., 40, No. 3, 819-881. paper Zentralblatt review