The \'etendue of a combinatorial space and its dimension

TL;DR

This paper introduces the concept of 'étendue' for simplicial sets, linking their geometric dimension to the equivalence of their associated étendues, with implications for presheaf toposes over well-founded sites.

Contribution

It defines the étendue of a simplicial set and establishes a criterion connecting étendue equivalence to geometric dimension, extending to presheaf toposes.

Findings

Étendue captures geometric information of simplicial sets.

Equivalence of étendues corresponds to equal dimensions for non-singular objects.

Results extend to presheaf toposes over well-founded sites.

Abstract

To each simplicial set $X$ we naturally assign an \'etendue ${\'E X}$ whose internal logic captures information about the geometry of $X$. In particular, we show that, for 'non-singular' objects $X$ and $Y$, the \'etendues ${\'E X}$ and ${\'E Y}$ are equivalent if, and only if, $X$ and $Y$ have the same dimension. Many of the results apply to presheaf toposes over 'well-founded' sites.