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Stream: learning: reading & references

Topic: Sheaves on a Site

Benjamin Merlin Bumpus (he/him) (Apr 02 2024 at 15:26):

Hi everyone!
I'm quite comfortable with the definition of sheaves on a category $C$ equipped with a coverage $K$ . I know that, when thinking of sheaves on a topological space $X$ , these can be identified with Etále spaces. Now my question is: is there a nice analogue for this construction for sites: given a sheaf $F \colon C^{\mathsf{op}} \to Set$ on a site $(C, K)$ , can I always write this as a morphism of sites $(E, K_E) \to (C, K)$ ? Something like this must exist, but I don't know what to look up, any help would be greatly appreciated :)

Morgan Rogers (he/him) (Apr 02 2024 at 15:59):

Riccard Zanfa's PhD thesis deals with this at a very high level. For a more elementary take, you should know that for a presheaf $X$ on $\mathcal{C}$ , $\mathrm{PSh}(\mathcal{C})/X$ is equivalent to the category of presheaves over the category of elements of $X$ , and by pulling back you can deduce that one choice of site for $\mathrm{Sh}(\mathcal{C},K)/X$ is the category of sheaves for a topology on the category of elements of $X$ derived from $K$ . This is made explicit in Caramello's work somewhere (possibly in her "Denseness conditions..." paper)

Benjamin Merlin Bumpus (he/him) (Apr 02 2024 at 16:00):

My naïve idea for one direction is to just integrate everything in sight:

modulo suitable size considerations, a coverage $K$ on a category $C$ is just a special kind of presheaf $K \colon C^{op} \to Set$
integrating the sheaf $F$ , we should get the following situation $\int F \xrightarrow{\pi_F} C^{op} \xrightarrow{K} Set$
integrating the composite, you get a thing that looks like $\int (K \circ \pi_F) \to \int F$ which my intuition tells me ought to be unaravelable into a presheaf $K' \colon (\int F)^{op} \to Set$ which I'm hoping is actually defining a coverage on $\int F$ ....

Is this at all in the right direction?

Benjamin Merlin Bumpus (he/him) (Apr 02 2024 at 16:00):

Morgan Rogers (he/him) said:

Riccard Zanfa's PhD thesis deals with this at a very high level. For a more elementary take, you should know that for a presheaf $X$ on $\mathcal{C}$ , $\mathrm{PSh}(\mathcal{C})/X$ is equivalent to the category of presheaves over the category of elements of $X$ , and by pulling back you can deduce that one choice of site for $\mathrm{Sh}(\mathcal{C},K)/X$ is the category of sheaves for a topology on the category of elements of $X$ derived from $K$ . This is made explicit in Caramello's work somewhere (possibly in her "Denseness conditions..." paper)

oh, thanks Morgan! I'll check this out :heart:

Morgan Rogers (he/him) (Apr 02 2024 at 16:01):

It sounds like you would have gotten to an idea that worked ;)

Benjamin Merlin Bumpus (he/him) (Apr 02 2024 at 16:02):

Morgan Rogers (he/him) said:

It sounds like you would have gotten to an idea that worked ;)

ah damn, I spoiled my own fun :rolling_on_the_floor_laughing:

Benjamin Merlin Bumpus (he/him) (Apr 02 2024 at 16:03):

The thesis you linked looks amazing though, thanks so much! I might get in touch with @Riccardo Zanfa and @Olivia Caramello about this stuff :heart:

Morgan Rogers (he/him) (Apr 02 2024 at 16:24):

Riccardo left academia a few years ago but I can help you get in touch with him if you struggle :)

Zoltan A. Kocsis (Apr 02 2024 at 17:41):

Benjamin Merlin Bumpus (he/him) said:

identified with _Etále_ spaces

Since we're in the learning stream: the word is étalé, three syllables, meaning something like "laid out flat".

Let me know if you plan a deep-delve into the resources Morgan pointed out. My PhD thesis has a section giving a nonstandard-analytic treatment of sheaves on topological spaces, relying heavily on étalé spaces. This prompted questions about whether similar concepts could extend to sheaves on a site. I couldn't pursue this line of research at the time, and one of my stumbling blocks was my lack of clarity on how to do étalé spaces without ordinary topology. It might be time to revisit this question.

Spencer Breiner (Apr 02 2024 at 17:51):

Let's not be prescriptive. I prefer the 2-syllable pronunciation "étale", which is quite common in English (but not French). See the (surprisingly heated) discussion here.

Reid Barton (Apr 02 2024 at 17:55):

I think it's okay to accept "étale" (and even "etale") while rejecting "etále", especially considering that "á" is not used in French at all as far as I know.

Eric M Downes (Apr 02 2024 at 18:12):

Interesting historical note -- evidently, Grothendieck, at least when speaking to a primarily english audience, seems to have pronounced it flatly without multiple syllables "etale". Or at least he did so in the audio recordings Lawvere et al. rescued; Colin McLarty (don't recall timecode sadly) ... not looking to argue, just a quirky historical tidbit.

Benjamin Merlin Bumpus (he/him) (Apr 02 2024 at 18:44):

Zoltan A. Kocsis said:

Benjamin Merlin Bumpus (he/him) said:

identified with _Etále_ spaces

Since we're in the learning stream: the word is étalé, three syllables, meaning something like "laid out flat".

Let me know if you plan a deep-delve into the resources Morgan pointed out. My PhD thesis has a section giving a nonstandard-analytic treatment of sheaves on topological spaces, relying heavily on étalé spaces. This prompted questions about whether similar concepts could extend to sheaves on a site. I couldn't pursue this line of research at the time, and one of my stumbling blocks was my lack of clarity on how to do étalé spaces without ordinary topology. It might be time to revisit this question.

Ciao bellissimo! Yes, I plan to, I have an application in mind you might like! (also thanks for pointing out the correct spelling: can you tell that I was writing in Portuguese moments before typing my question? :rolling_on_the_floor_laughing: )

Rémy Tuyéras (Apr 03 2024 at 11:32):

Just a note on the naming thing: I believe the adjective "étale" is the word that we want to use when talking about topologies (as in "mer étale", a completly flat sea). The adjective étalé is a different word with a similar meaning:
Étale = slack
Étalé = spread

Zoltan A. Kocsis (Apr 03 2024 at 13:36):

Correct, the étalé of "étalé space" and the étale of "étale cohomology" are unrelated.

John Baez (Apr 03 2024 at 18:09):

I decided to act like an American and just write "etale": in this case, the math is hard enough without worrying about accents.

Matteo Capucci (he/him) (Apr 04 2024 at 07:04):

I always wondered why these terms weren't translated to English...? Slack and spread seem fine to me

Mike Shulman (Apr 04 2024 at 13:54):

In the case of étalé space there's a perfectly good English term "local homeomorphism".

Damiano Mazza (Apr 04 2024 at 14:35):

Yes if $X\to B$ is a vector bundle or fibration of some sort, $X$ is usually called the "total space". I wonder why this terminology is not applied to local homeomorphisms... I mean, it still makes sense to me (when you apply the Grothendieck construction to a (pre)sheaf $F$ , people call $\int\!F$ the "total category" if I'm not mistaken).

John Baez (Apr 04 2024 at 17:40):

Well, etale spaces don't look like your typical "total space" - they look like a stack of filo, so we can do very special things with them. So it makes some sense to give them a special name, even though they are total spaces.

Mike Shulman (Apr 04 2024 at 19:02):

But "local homeomorphism" is precise and clear.

Mike Shulman (Apr 04 2024 at 19:03):

Of course it sounds like it is talking about the map rather than the total space, but you can't talk about the map without talking about its domain, and really when you say "étalé space" you're referring to the space together with the map, since a space can't be "étalé" unless you specify what it's étalé over and how. So I don't really see this as an objection.

Matteo Capucci (he/him) (Apr 05 2024 at 06:59):

Mike Shulman said:

In the case of étalé space there's a perfectly good English term "local homeomorphism".

Ah, right! I actually like that, but it's so long I end up using étale again :big_smile: