2025/10/11 by Hemelaer, Jens
#18F10 #Algebraic Geometry (math.AG) #Category Theory (math.CT) #FOS: Mathematics #General Topology (math.GN)
paper · doi:10.48550/arxiv.2510.10349
We introduce the notion of n-pure geometric morphism between Grothendieck toposes, over a Grothendieck base topos 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 Spec(ℤ) is two-dimensional.