The étendue of a combinatorial space and its dimension

Abstract

To each simplicial set X we naturally assign an étendue 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 étendues 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.