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Stream: learning: questions

Topic: Grothendieck topologies

Naso (Apr 30 2021 at 01:00):

Do Grothendieck topologies on a category form a lattice?

Dmitri Pavlov (Apr 30 2021 at 01:07):

Yes: https://mathoverflow.net/questions/39212/infima-and-suprema-of-grothendieck-topologies/39218#39218

Dmitri Pavlov (Apr 30 2021 at 01:08):

They even form a locale, as I explain on MathOverflow.

Morgan Rogers (he/him) (Apr 30 2021 at 07:58):

Hmm be careful there! The subtoposes form a frame (which you can think of as a locale if you like); since the ordering on Grothendieck topologies is the reverse of the ordering on subtoposes, they form a co-frame.

John Baez (Apr 30 2021 at 13:15):

It's a random sort of question, but: what's a sheaf on the locale of Grothendieck topologies on a category like? Is this concept interesting at all?

(Here we'd better use the suitably reversed ordering to get a frame and thus a locale, per Morgan's warning.)

Mike Shulman (Apr 30 2021 at 13:22):

As a special case, the locale of (reversed) sublocales of a given locale is called the "dissolution" and studied somewhere in the Elephant. It's kind of like a constructive approximation to a Booleanization. So sheaves on the locale of reversed subtoposes of a given topos is probably a sort of "dissolution of a topos".

Fawzi Hreiki (Apr 30 2021 at 13:32):

For a general (not necessarily Grothendieck) topos, does the poset of subtoposes still form a co-frame or just a co-Heyting algebra?

Mike Shulman (Apr 30 2021 at 14:39):

Just a co-Heyting algebra.

Fawzi Hreiki (Apr 30 2021 at 14:46):

Right that makes sense, although morally there should be a way to say that this co-Heyting algebra is internally complete (from the POV of the topos itself).

Jens Hemelaer (Apr 30 2021 at 15:41):

John Baez said:

It's a random sort of question, but: what's a sheaf on the locale of Grothendieck topologies on a category like? Is this concept interesting at all?

This is a bit of an advertisement, but sheaves on the dissolution locale appear in a paper by Karin Cvetko-Vah, Lieven Le Bruyn and me.

The result is that if $Y$ is a locale (with associated frame $\mathcal{O}(Y)$), then sheaves (with global support) on the dissolution locale $Y_d$ correspond precisely to the (left-handed) noncommutative frames that have $\mathcal{O}(Y)$ as commutative quotient.

If you start with a topological space $X$, then to create noncommutative frames you can take the new topology on $X$ generated by both the open and the closed sets (this defines a new space $X_f$). If you then take a sheaf $\mathcal{F}$ with global support on $X_f$, then the associated noncommutative frame has elements $(U,s)$, with $U \subseteq X$ open and $s \in \mathcal{F}(U)$. The noncommutative meet and join are defined as:
$(U,s) \wedge (V,t) = (U \cap V, s|_{U \cap V})\qquad$ (restrict)
$(U,s) \vee (V,t) = (U \cup V, s|_{U-V} \cup t)\quad$ (overwrite).
But not every noncommutative frame with commutative quotient $\mathcal{O}(X)$ is of this form. To get all noncommutative frames, you need to look at the dissolution locale of $X$ (which is a bit of a constructive version of $X_f$, in the spirit of @Mike Shulman 's remark).

Mike Shulman (Apr 30 2021 at 15:53):

Fawzi Hreiki said:

Right that makes sense, although morally there should be a way to say that this co-Heyting algebra is internally complete (from the POV of the topos itself).

Of course it is, since any elementary topos is a Grothendieck topos over itself. (-:

Benjamin Merlin Bumpus (he/him) (Jun 23 2022 at 09:42):

Hi everyone! :wave:

I've been starting to learn a little about sheaves and Grothendieck topologies and I have a naïve question that you might be able to help me out with.

Just so we're on the same page (and since this is the only definition I know at the moment) let's recall the definition of a Grothendieck topology from Mac Lane and Moerdijk's book:
A Grothendieck topology on a category $\mathbf{C}$ is a function $J$ assigning to each object $c \in \mathbf{C}$ a collection $Jc$ of sieves on $\mathbf{C}$ such that:
(i) the maximal sieve $t_x = \{f | \mathbf{cod} f = c\} \in J c$
(ii) [Stability] given any arrow $h : d \to c$, if $s \in Jc$, then $h^*(s) \in Jd$
(iii) [Transitivity] if $s \in J c$ and $r$ is a sieve on $c$ such that, $\forall h : d \to c \text{ in } s$ we have $h^* (r) \in j d$, then $r \in J c$.

So here's my question:

Why do we want the first condition? I think I see why we want it if the goal is to recover the topological spaces example that Mac Lane and Moerdijk use to motivate this whole thing. However, it seems to me like the important parts are really (ii) and (iii). Does not having (i) break the appropriate equalizer diagram that features in the definition of a sheaf?

Anyway, let me know if the question makes sense. Also any general insights about this are appreciated; it's often really nice to get a glimpse of how other people "see" these concepts.

Zhen Lin Low (Jun 23 2022 at 11:40):

It is difficult to answer questions like these. You could certainly consider the definition where axiom (i) is omitted. It would simply be a different concept. What would you consider to be a satisfactory answer?

Patrick Nicodemus (Jun 23 2022 at 12:15):

The maximal sieve is generated as a sieve (i.e., under precomposition) by the identity morphism on $c$. Each sieve in a Grothendieck topology adds a gluing criterion to the definition of a sheaf. Here, that criterion would be something like: if $\{f\}$ is a singleton subset of $P(c)$, regarded as a trivial "matching family", then it admits a unique gluing that restricts to $f$ along the identity map, namely $f$ itself.

To me this seems like a tautology which should be trivially satisfied by every presheaf so this suggests that we are justified in assuming that all topologies contain these maximal sieves without reducing the generality of the theory.

Zhen Lin Low (Jun 23 2022 at 13:45):

In fact it is true that every presheaf automatically satisfies the sheaf condition for the maximal sieve – this is the Yoneda lemma in disguise! – so indeed axiom (i) is redundant if we think of Grothendieck topologies only in terms of the sheaf condition they imply.

Benjamin Merlin Bumpus (he/him) (Jun 23 2022 at 14:35):

Thanks for the answers.

I'm quite confused by the fact that you're saying that $\{f\} \subseteq P c$ is a matching family, since, when comparing to the book (p.g. 121 Eqn 1), I'm expecting a matching family for a sieve $(s \in Jc)$ w.r.t. a presheaf $P: \mathbf{C}^{op} \to \mathbf{Sets}$ to consist of pairs of the form $\bigl( (f: d \to c) \in s, \; x_f \in P(d) \bigr)$ (which, b.t.w. looks very Grothendieck-construction-like to me...)

Is there another use fo the term "matching family", or am I terribly misunderstanding something? :rolling_on_the_floor_laughing:

Mike Shulman (Jun 23 2022 at 16:19):

There's a nice explanation of the various conditions on a Grothendieck topology in Sketches of an Elephant. The conclusion is that the only really important axiom is that covering families can be "pulled back" in some very weak sense. A collection of "covering families" (not necessarily sieves) with this property is there called a [[coverage]]. If you omit this condition, you get truly different notions of "sheaf". But the category of sheaves for any coverage on a small category is a Grothendieck topos, and any Grothendieck topos can be presented as the category of sheaves for some coverage on some small category.

The other conditions in the usual definition of Grothendieck topology, including the fact that the covering families are sieves, and also the inclusion of the maximal sieve and the transitivity condition, are "saturation" conditions that "might as well be there". Specifically, any coverage can be saturated to one that satisfies these conditions and has the same sieves. The main reason for including them is that there is then a one-to-one correspondence between Grothendieck topologies on a small category and geometric subtoposes of its presheaf category, which is not the case for arbitrary coverages: two different coverages on the same category can give rise to the same sheaf topos.

Benjamin Merlin Bumpus (he/him) (Jun 23 2022 at 18:59):

@Mike Shulman oh this is really interesting!
I'm currently have been playing with an application of Grothendieck topologies to combinatorics and the one axiom that was causing me trouble was the first, so it's reassuring that it's in some sense not necessary.
I'll definitely have a look at Sketches of an Elephant since this sounds really interesting!

A follow-up question on nomenclature: would one still call it a Grothendieck topology when omitting the first (saturation) axiom or would that be very confusing for those who are used to this sort of thing? (would it be weird to call it an unsaturated Grothendieck topology?)

Mike Shulman (Jun 23 2022 at 19:02):

It's generally better to stick with established terminologies; that's why Johnstone introduced a different word "coverage". (Of course, Johnstone also eschewed the word "topology" entirely, talking instead about "Grothendieck coverages".) It might be better to add adjectives to "coverage" than add non-adjectives to "Grothendieck topology".

Benjamin Merlin Bumpus (he/him) (Jun 23 2022 at 20:04):

Yeah, that makes complete sense; thanks a lot!

Matteo Capucci (he/him) (Jun 24 2022 at 09:29):

Benjamin Merlin Bumpus said:

I'm currently have been playing with an application of Grothendieck topologies to combinatorics and the one axiom that was causing me trouble was the first, so it's reassuring that it's in some sense not necessary.

I found myself in the same pickle in different situations. In my experience the first axiom is nasty when you want to define a topology where covers are given by surjective coverings comprised of nice objects. Then the first axiom breaks because if you can't trivially cover nicely a non-nice object. Just to make a stupid example: suppose you want to get a topology of covers of open sets by connected open sets. Then given a disconnected open set, its trivial cover is not connected so you're out of luck.
Usually when this happens it's because your concept of 'niceness' is non-local (like connectedness). So you can still go around the problem by defining a coverage instead. But then you find out that once you saturate it to a Grothendieck topology, your nice covers have been watered down (e.g. doing it with the above, you'd get covers of open sets by open sets which are the union of connected sets, hence only 'locally connected').
This smells like sheafificaiton, and I suspect it is, in some sense: you'd want saturation to be the sheafification of the presheaf $U \mapsto$ "coverings of $U$" with respect to the topology given by saturating the given coverage. It's quite circular, so I'm not sure it's correct.

Benjamin Merlin Bumpus (he/him) (Jun 24 2022 at 09:38):

Ciao @Matteo Capucci (he/him), grazie per la risposta!

"This smells like sheafification.."

I'm pretty new to this, but I agree and I'm excited to think about it more!

I'll keep you updated on whether my combinatorial/algorithmic application works out :tada:

Mike Shulman (Jun 24 2022 at 15:58):

@Matteo Capucci (he/him) In my understanding, that's exactly the situation that coverages are designed for. Why do you say you end up with unions of connected sets? I think you can have a coverage where the covering families are all required to be literally connected. (In case this is the confusion, note that coverages are even weaker than "Grothendieck pretopologies" -- their covering families are not required to be closed under actual pullback, only "up to factorization".)

Zhen Lin Low (Jun 24 2022 at 16:05):

Incidentally, the business of defining a general notion of "locally P" for various properties P is a lot trickier than you might think. I believe you can't get a satisfactory definition using only a Grothendieck topology. It may be possible using a coverage but I think it is better to have a Grothendieck topology + a notion of neighbourhood.

Patrick Nicodemus (Jun 24 2022 at 16:47):

Benjamin Merlin Bumpus said:

Thanks for the answers.

I'm quite confused by the fact that you're saying that $\{f\} \subseteq P c$ is a matching family, since, when comparing to the book (p.g. 121 Eqn 1), I'm expecting a matching family for a sieve $(s \in Jc)$ w.r.t. a presheaf $P: \mathbf{C}^{op} \to \mathbf{Sets}$ to consist of pairs of the form $\bigl( (f: d \to c) \in s, \; x_f \in P(d) \bigr)$ (which, b.t.w. looks very Grothendieck-construction-like to me...)

Is there another use fo the term "matching family", or am I terribly misunderstanding something? :rolling_on_the_floor_laughing:

Benjamin I meant the singleton consisting of the pair $(f, \operatorname{id}_c)$.

Matteo Capucci (he/him) (Jun 24 2022 at 17:10):

Matteo Capucci (he/him) In my understanding, that's exactly the situation that coverages are designed for. Why do you say you end up with unions of connected sets? I think you can have a coverage where the covering families are all required to be literally connected. (In case this is the confusion, note that coverages are even weaker than "Grothendieck pretopologies" -- their covering families are not required to be closed under actual pullback, only "up to factorization".)

@Mike Shulman yeah exactly, I was trying to say that coverages work but when you saturate them to proper topologies your notion of 'nice' has to become 'locally nice' in some sense

Matteo Capucci (he/him) (Jun 24 2022 at 17:16):

It's been a while since I regularly fiddled with this stuff though, so maybe I'm missing something. If I have a topological space and I look at the coverage given by open covers whose members are all connected, it seems to me then when I saturate to get a Grothendieck topology I end up including open covers with just locally connected members. No?

Mike Shulman (Jun 24 2022 at 17:17):

Yes. It sounded to me like you were saying that this happens already with coverages. If you meant to say instead that coverages solve the problem, then yes, I agree.

Matteo Capucci (he/him) (Jun 24 2022 at 17:18):

Incidentally, the business of defining a general notion of "locally P" for various properties P is a lot trickier than you might think. I believe you can't get a satisfactory definition using only a Grothendieck topology. It may be possible using a coverage but I think it is better to have a Grothendieck topology + a notion of neighbourhood.

I guess it depends on what one informally means by 'locally' but as far as I remember in a presheaf topos a property P is local iff it forms a sheaf, hence if you can reduce checking P on U to checking P on the members of some covering sieve of U

Zhen Lin Low (Jun 24 2022 at 23:41):

That notion is rather weak. Try fitting "locally compact" into that framework.

Benjamin Merlin Bumpus (he/him) (Nov 21 2022 at 22:38):

Hey everyone, thanks again for all the answers!

After a long time thinking about other things, I've come back to the idea of Grothendieck topologies and I wrote a blog post about it (you can find it here). I'm still very new to all of this, but, after some of @John Baez 's encouragement at the MRC in Buffalo, I've decided to share my notes in blog-format as I go along.

If any of you have feedback and/or insights, I'd love to hear about it! :smile:

Morgan Rogers (he/him) (Nov 22 2022 at 09:30):

If there are any sheaf-theorists out there who can tell me why it’s a bad idea to promote K to a functor, then leave a comment, I’d be very happy to hear about it!

Your post-script handles it: we can generate sieves from covering families, and these behave functorially, but sieves are almost always far too large to compute with directly, so being able to phrase the sheaf condition in terms of small covering families is more convenient.

Benjamin Merlin Bumpus (he/him) (Nov 22 2022 at 13:54):

Morgan Rogers (he/him) said:

If there are any sheaf-theorists out there who can tell me why it’s a bad idea to promote K to a functor, then leave a comment, I’d be very happy to hear about it!

Your post-script handles it: we can generate sieves from covering families, and these behave functorially, but sieves are almost always far too large to compute with directly, so being able to phrase the sheaf condition in terms of small covering families is more convenient.

Oh! That's a great answer, thanks so much!