Timeline for Topological regularity for toposes
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 3, 2017 at 23:24 | comment | added | Simon Henry | For the relation between finite object and the hausdorff condition it is more subtle and a little conjectural. It manifest itself for hyperconnected toposes. I have a first result on the subject in arxiv.org/abs/1505.04987 | |
| Jan 3, 2017 at 23:20 | comment | added | Simon Henry | For subterminal objects $U $ and $V$ having internally "There exists $F $ kuratowski finite such that $ U \subset F \subset V $ " is equivalent to " U is empty or V is inhabited " depending on whether the $n$ appearing in the definition of finiteness is zero or non zero. But "U is empty or V is inhabited " is "$\neg U \cup V$" saying that it is true is a possible definition of $U \triangleleft V $. | |
| Jan 3, 2017 at 20:52 | comment | added | Mike Shulman | @SimonHenry That's interesting! It's not obvious to me why that should be; can you explain (maybe in chat)? | |
| Jan 3, 2017 at 13:49 | comment | added | Simon Henry | A remark that might be useful but that I haven't been able to use: the relation $U \triangleleft V$ that appears in the definition of regularity is equivalent to "internally, there exists a finite object $F$ such that $U \subset F \subset V$ ". Moreover, finite objects in toposes are closely related to the Hausdorff conditions. | |
| Dec 31, 2016 at 21:23 | comment | added | Mike Shulman | Yes, I thought of those, but none of them is really topos-theoretic. One good criterion for correctness would be that it can be categorified, the way proper categorifies to tidy and open categorifies to locally connected. | |
| Dec 30, 2016 at 21:41 | comment | added | Simon Henry | I would say that there is several notion that can be introduced that answer the question, but no real criterion to say that one is better than others. You can ask for the localic reflection to be regular. Or ask that every object of the topos can be covered by an object $X$ such that $\mathcal{T}_{/X}$ has a regular localic reflection. Or ask stronger conditions that will also implies that the topos is separated (proper diagonal). | |
| Dec 30, 2016 at 17:30 | answer | added | Christopher Townsend | timeline score: 1 | |
| Dec 30, 2016 at 11:44 | history | asked | Mike Shulman | CC BY-SA 3.0 |