Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.10349 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911205609701376 |
|---|---|
| author | Hemelaer, Jens |
| author_facet | Hemelaer, Jens |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_10349 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The smallest $n$-pure subtopos and dimension theory Hemelaer, Jens Category Theory Algebraic Geometry General Topology 18F10 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. |
| title | The smallest $n$-pure subtopos and dimension theory |
| topic | Category Theory Algebraic Geometry General Topology 18F10 |
| url | https://arxiv.org/abs/2510.10349 |