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Morgan Rogers (he/him) (Jan 20 2021 at 11:45):Hi all! I uploaded my rather long preprint last week. It's on a couple of big classes of Grothendieck toposes, those with enough supercompact objects or compact objects, respectively, where here I mean "compact object" in the weaker sense that every covering family on such an object has a finite subcover.
The supercompactly generated toposes include all presheaf toposes, regular toposes, atomic toposes, and interestingly for me, all toposes of topological monoid actions (although that last part is only hinted at in the paper). The compactly generated toposes include all of these, plus all coherent toposes.
The highlights are that I find an intrinsic characterisation of morphisms whose inverse images preserve supercompact/compact objects, I give canonical sites for these toposes (which interestingly are characterised purely in terms of their colimits rather than their limits), as well as more general sites, and I use the whole lot to recover some familiar 2-categorical dualities for posets and semilattices.
Morgan Rogers (he/him) (Jul 06 2021 at 11:48):In this paper, I have generalize the well-established "proper" geometric morphisms to stronger notions called "polite" and "precise" morphisms, which respectively correspond (roughly) to gms whose direct image functor preserves inhabited and all unions of subobjects, where proper gms have direct image functor preserving directed unions.
My reviewer suggested the name "superproper" for 'precise'. In light of the results of the paper, this is a valid name, but it begs the question of what the intermediate 'polite' morphisms should be called to match this convention. Can anyone come up with something more compelling than "quasisuperproper"?
Matteo Capucci (he/him) (Jul 06 2021 at 12:58):Maybe you could go with
precise -> hyperproper
polite -> superproper
proper -> proper
Matteo Capucci (he/him) (Jul 06 2021 at 12:58):Or something like 'very proper' and 'extremely proper'
Matteo Capucci (he/him) (Jul 06 2021 at 12:58):I like 'polite' and 'precise' better by the way :laughing:
Morgan Rogers (he/him) (Jul 06 2021 at 13:02):Matteo Capucci (he/him) said:
Maybe you could go with
precise -> hyperproper
polite -> superproper
proper -> proper
The trouble is that it turns out that "proper" is to "compact" as "precise" is to "supercompact", so it would make the most sense to put superproper at the top, but finding a prefix that conveys a notion 'slightly weaker than super' is a challenge
Zhen Lin Low (Jul 06 2021 at 13:21):I think I can see where you're trying to go with proper → polite, but considering PTJ's use of "tidy" and the French meaning of "propre" perhaps "clean" is the direction you should consider...
Morgan Rogers (he/him) (Jul 06 2021 at 13:44):proper → clean → fragrant? :yum:
I considered something similar, but proper → tidy is a strengthening in an orthogonal direction to proper → polite/precise.
That said, I suppose something can be clean without being tidy and vice versa!
Morgan Rogers (he/him) (Jul 06 2021 at 13:46):proper → polished → pristine?
Morgan Rogers (he/him) (Jul 06 2021 at 13:49):Darn I like that more than my original names, but it would be rude to my reviewer to say "thanks for the suggestion, but I came up with something even better". I'll suggest it in the email I send with my new draft.
Fawzi Hreiki (Jul 06 2021 at 13:54):British?
Fawzi Hreiki (Jul 06 2021 at 13:54):Posh?
Jens Hemelaer (Jul 06 2021 at 13:55):I think the name "superproper" makes a lot of sense, given that it corresponds to supercompact objects in the same way as proper corresponds to compact objects. As a non-native speaker, it's also not obvious what the different gradations are between e.g. proper/polished/pristine.
Maybe "supracompact" makes sense for something that is a bit weaker than supercompact.
Morgan Rogers (he/him) (Jul 06 2021 at 14:03):That does convey the right meaning, but since I have to refer to these things side by side, I'm hesitant to make the names quite so similar. I'll add that to the possibilities.
Morgan Rogers (he/him) (Jul 06 2021 at 14:06):It's one of those cases where the definitions are very similar (involving "all" unions vs "all inhabited" unions), but one can show that the subtlety is in the details. Any G' topos has at most one precise/pristine/superproper point, whereas for a localic topos, all points are polite/polished/supraproper.
Morgan Rogers (he/him) (Jul 06 2021 at 14:06):Oof supraproper is hard to say
Chad Nester (Jul 06 2021 at 14:50):I mean ... "properer"?
Morgan Rogers (he/him) (Jul 06 2021 at 14:55):good'n'proper?
John Baez (Jul 06 2021 at 18:40):Propest? :upside_down: