The smallest $n$-pure subtopos and dimension theory
Jens Hemelaer
TL;DR
This work develops a comprehensive dimension theory for Grothendieck toposes via n-pure geometric morphisms, introducing smallest n-pure subtopoi $ opos_{ le n+1}$ and a dimension $ ext{dim}( opos)$ measured against the content $ opos_{< in}$. It blends topos-theoretic localization, descent, and homotopy-theoretic methods (simplicial toposes, injective model structure, EM cohomology) to analyze dimension and boundary phenomena, including base-to-topos independence and connections to Galois theory and fundamental groups. The paper computes dimensions in key contexts: Boolean toposes (zero-dimensional), toposes of manifolds (dimension matches manifold dimension with boundary handling), rational lines, and monoid-related toposes; it then proves that petit étale toposes of excellent regular schemes in characteristic $0$ have dimension $2d$ for Krull dimension $d$, with $ ext{Spec}(b Z)$ giving dimension $2$. The results offer a robust framework to compare topological intuition with topos-theoretic dimension and provide calculational tools across spaces, monoids, and schemes using localization, descent, and EM cohomology. This has potential implications for understanding dimension in mixed characteristic settings and guidance for further topos-theoretic approaches to arithmetic geometry.
Abstract
We introduce the notion of $n$-pure geometric morphism between Grothendieck toposes, over a Grothendieck base topos $\mathcal{T}$. This is a higher-dimensional analogue of the concepts of dense and pure geometric morphism. We extend the construction of the smallest dense subtopos and smallest pure subtopos by constructing a smallest $n$-pure subtopos, for each natural number $n$. Based on this, we then propose a concept of dimension for a Grothendieck topos, in this way also arriving naturally at a distinction between toposes with boundary and toposes without boundary. We show that the zero-dimensional toposes without boundary are precisely the Boolean toposes, and that the topos associated to an $n$-manifold is again $n$-dimensional (with boundary if the manifold has a boundary). Some other toposes for which we calculate the dimension are the topos associated to the rational line and the toposes associated to a right Ore monoid or free monoid. Finally, we move to algebraic geometry: for a scheme $X$ of characteristic $0$ and Krull dimension $d$, we prove that the dimension of the associated petit étale topos is $2d$, assuming that $X$ is excellent and regular, or that $X$ is variety. As a first example in mixed characteristic, we show that the petit étale topos associated to $\mathrm{Spec}(\mathbb{Z})$ is two-dimensional.