The smallest $n$-pure subtopos and dimension theory

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 \'etale 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 \'etale topos associated to $\mathrm{Spec}(\mathbb{Z})$ is two-dimensional.