You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.
Morgan Rogers (he/him) (Jan 20 2021 at 11:32):I've been considering some 2-categorical aspects of 1-topos theory, and it seems to me that the literature is rather limited on this subject. It's not clear what the (pseudo)-monomorphisms are, for example. There's Section B of the Elephant which deals with some issues of weighted limits and colimits, but that similarly leaves some basic questions unanswered.
I have shown, for example, that geometric inclusions are full and faithful in the category of toposes, and that localic geometric morphisms are faithful. Is this a known fact? Is the converse known? I'm leaning towards the conjecture that the surjection-inclusion and hyperconnected-localic factorisation systems correspond to the (eso,ff) and (eso+full,faithful) factorisation systems, but there's some more work to do to verify that. If it's not the case, that could be quite exciting.
It is never stated explicitly in the Elephant that geometric surjections are stable under pullback, so even if the above is true, it's still unclear whether is a (locally) regular 2-category.
@Simon Henry, @Mike Shulman, since I'm referring to your answer and blog respectively, do you have any further insights?
Morgan Rogers (he/him) (Jan 20 2021 at 14:13):Oh I worked out why surjections aren't stable under pullback, at least, so indeed there's no chance of local regularity if my preceding conjecture holds.
Morgan Rogers (he/him) (Jan 20 2021 at 14:18):I have also worked out that surjections are co-faithful, but unless I find a way to show that the ff morphisms in are precisely the inclusions, it will be impossible to verify that surjections are precisely the eso morphisms according to the pullback definition.
Mike Shulman (Jan 20 2021 at 15:56):I don't remember whether I've seen it written down explicitly that geometric inclusions are ff and localic morphisms are faithful, but the first one is pretty obvious and I remember observing the latter myself. But I don't think that these factorization systems coincide with (eso,ff) and (eso+full,faithful); it's better to think of the 2-category as analogous to the 1-category of topological spaces (or, better, the 1-category of locales), in which not every monomorphism is is a subspace inclusion.
Mike Shulman (Jan 20 2021 at 15:57):If you haven't read it already, you might be interested in section 5.4 of Lectures on n-Categories and Cohomology.
Morgan Rogers (he/him) (Jan 20 2021 at 16:27):Thanks for the response Mike! Those notes (especially the subsequent Section 5.5) have reinforced the intuition I was developing through your blog that the range of notions of monicness expands naturally with the dimension one is working in.
Mike Shulman said:
it's better to think of the 2-category as analogous to the 1-category of topological spaces (or, better, the 1-category of locales), in which not every monomorphism is is a subspace inclusion.
I know this fact, and the stackexchange link I mentioned earlier contains a specific counterexample to it. The problem is that saying "not every is a " doesn't answer the question of precisely which are ! I'll keep chipping away at this.
Morgan Rogers (he/him) (Jan 21 2021 at 16:15):fyi @Mike Shulman, the pullback squares before Definition 20 in Section 5.5 of those notes seem dodgy! They should have in the top left corner, I think.
Mike Shulman (Jan 21 2021 at 16:29):Yeah, the top and bottom rows should be exchanged.