The \'etendue of a combinatorial space and its dimension

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.