Category Theory
Zulip Server
Archive

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.

Stream: learning: questions

Topic: toposes and quasitoposes

Jon Sterling (Feb 21 2022 at 19:44):

Mike Shulman said:

The best thing to do, in my opinion, is embed the quasitopos in a topos, such as [[subsequential spaces]] sitting inside [[Johnstone's topological topos]]. Then the standard internal logic of the topos (using arbitrary monos) has unique choice and a subobject classifier, plus there is generally a lex modality allowing you to recover the original quasitopos and its strong monos as the separated objects -- it's just that the classifier of arbitrary subobjects is not separated.

Is the recipe to take sheaves on the quasitopos with the coherent topology? I suppose this must not be what you have in mind...

Mike Shulman (Feb 21 2022 at 19:50):

If it's a Grothendieck quasitopos, meaning the category of separated sheaves on a bisite $(C,J,K)$, then it embeds as the $K$-separated objects in the topos of $(C,J)$-sheaves.

Mike Shulman (Feb 21 2022 at 19:54):

You can certainly take sheaves for the coherent topology on any (small) quasitopos, but then the resulting topos will be much larger than the quasitopos, and it's unclear whether the whole topos will admit a modality whose separated objects are relevant. But in general, of course, as you well know, such embeddings are a reasonable way to use topos logic to reason about non-toposes, it's just that in the Grothendieck-quasitopos case one can do better.

Jon Sterling (Feb 21 2022 at 19:56):

Gotcha. I was just now reading on the nlab about coarse objects in a quasitopos, and I think I am starting to get it.

Let $E$ be a quasitopos; the full subcategory $\mathsf{Cs}(E)\subseteq E$ spanned by coarse objects is a topos. When $E$ is viewed as the category of separated objects for some site, then $\mathsf{Cs}(E)$ is the category of sheaves for that site.

Jon Sterling (Feb 21 2022 at 19:59):

Jonathan Sterling said:

Gotcha. I was just now reading on the nlab about coarse objects in a quasitopos, and I think I am starting to get it.

Let $E$ be a quasitopos; the full subcategory $\mathsf{Cs}(E)\subseteq E$ spanned by coarse objects is a topos. When $E$ is viewed as the category of separated objects for some site, then $\mathsf{Cs}(E)$ is the category of sheaves for that site.

So just to keep thinking out loud, it sounds like the quasitopos is sandwiched between two topoi --- one a lex localization of another.

Mike Shulman (Feb 21 2022 at 20:00):

Jonathan Sterling said:

it sounds like the quasitopos is sandwiched between two topoi --- one a lex localization of another.

Yes, exactly! At least, for a Grothendieck quasitopos. For an elementary one, it's not always clear whether there is an "outer" topos containing it. For instance, it's unclear how to embed the quasitopos of [[convergence spaces]] into a topos.

Jon Sterling (Feb 21 2022 at 20:01):

So now I have one last question in this line of thinking --- all this makes good sense for grothendieck quasitopoi, but I have a non-grothendieck quasitopos for which the same relationship occurs: in particular, the category of assemblies is sandwiched between SET and Eff, where SET is the double-negation-sheaves subtopos of Eff.

Jon Sterling (Feb 21 2022 at 20:02):

So are there some rules of thumb for when I should be able to do something like this, or do we just have a bunch of examples and counterexamples?

Mike Shulman (Feb 21 2022 at 20:02):

That's a good question! I wish I had an equally good answer. (-:

Jon Sterling (Feb 21 2022 at 20:03):

That's fair :)

Jon Sterling (Feb 21 2022 at 20:05):

Lately I have been having some reasons to bother suffering with quasitopoi... Previously I had managed to avoid working explicitly with them.

One property of an embedding of a quasitopos into a topos that does not seem to be satisfied for the nice embeddings that we are discussing (separated objects for a lex modality, etc.) is that it should preserve dependent products. In contrast, I suppose that the yoneda version that I proposed would preserve dependent products --- at the cost of some other trade-offs.

Mike Shulman (Feb 21 2022 at 20:05):

Really? I thought the separated objects for any lex modality were always closed under dependent products.

Mike Shulman (Feb 21 2022 at 20:06):

(I wonder if the last few messages in this thread should be moved to a new thread about quasitoposes.)

Jon Sterling (Feb 21 2022 at 20:06):

Well, maybe I am confused about something --- but I think if the inclusion of a reflective subcategory preserves dependent products, then the reflection is lex. (But yes, moving to a new thread sounds good!)

Mike Shulman (Feb 21 2022 at 20:08):

Lemma 1.26 in RSS says that any reflective subuniverse is closed under dependent products, where the base doesn't even have to be in the subuniverse.

Jon Sterling (Feb 21 2022 at 20:09):

Huh... So why can't I use this + adjoint play to show that the left adjoint preserves pullbacks?

Jon Sterling (Feb 21 2022 at 20:21):

Huh, I just did some back of the envelope computation and I no longer see why I thought this was untrue. Well, perhaps I spoke too soon in terms of needing to suffer through quasitopoi after all! So thanks for that! You saved me a lot of trouble.

Jon Sterling (Feb 21 2022 at 21:19):

So just to record what was confusing me in case anyone else is having the same problem --- there is Day's result that a reflective subcategory is an exponential ideal if and only if its reflector preserves products. For some reason I thought I envisioned a "dependent" version of this that says "a reflective subcategory is closed under dependent products [with arbitrary base] if and only if its reflector preserves pullbacks". I think that is my mistake --- the proof for products and exponentials doesn't seem to generalize to pullbacks and dependent products. @Mike Shulman would you concur?

Mike Shulman (Feb 21 2022 at 21:42):

Yes, I agree; it's because dependent products are not "adjoint" to pullbacks in the same way that exponentials are adjoint to products. There is a "dependently typed" version of Day's result, but what it says is that a reflective subcategory is a "slice-wise exponential ideal" if and only if the reflector preserves all pullbacks over an object of the subcategory. This condition is called having "stable units" and is due to Cassidy, Hébert, and Kelly; I wrote something about it here.

Jon Sterling (Feb 22 2022 at 07:12):

Great, thanks! This saves me a ton of trouble and resolves something that was gnawing at me...