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Stream: deprecated: topos theory

Topic: filterquotients and stalks of stacks

sarahzrf (Oct 20 2020 at 16:02):

so i was squinting at the filterquotient construction a while back and i worked out that it seemed an awful lot like what it is is, essentially, taking the stalk of a stack, at very least when your topos is localic—does anybody know of any sources that discuss this, or maybe at least sources that discuss taking stalks of stacks?

sarahzrf (Oct 20 2020 at 16:06):

to elaborate: suppose your topos is, say, Sh(X) for X a topological space or locale. then we can consider the stack Sh : O(X)^op → Cat which sends each open to the topos of sheaves on that subspace, w/ restriction as the action on morphisms. if we correspond O(X) to the subterminals in Sh(X), this stack should correspond to what we get if we restrict Sh(X)'s self-indexing to have just the subterminals as its domain, U ↦ Sh(X)/U. then...

sarahzrf (Oct 20 2020 at 16:07):

a filter of subterminals is gonna be a filter of opens, which you can use to take a stalk—you send the filter through your stack and then take a colimit

sarahzrf (Oct 20 2020 at 16:08):

...which is exactly what the filterquotient does—you have a filter of subterminals, you form a diagram out of the slices over them, connected by the base change functors, you take a colimit

sarahzrf (Oct 20 2020 at 16:08):

...right?

Morgan Rogers (he/him) (Oct 20 2020 at 18:01):

@Riccardo Zanfa might be able to answer that, if he's around; I know he's been working on some stacks recently.

Peter Arndt (Oct 21 2020 at 09:51):

Yes, that sounds perfectly right.

David Michael Roberts (Oct 21 2020 at 21:59):

I like this idea!

sarahzrf (Oct 21 2020 at 22:32):

ty :)

David Michael Roberts (Oct 21 2020 at 22:40):

It links in with the result (I'm working from rough memory here) that every topos is the total topos of a fibred topos, where the fibres are well-pointed. Might be due to Awodey? I'll have to think about/hunt where to find it, if you are interested in seeing it.

David Michael Roberts (Oct 21 2020 at 22:45):

Well, it was easier to find than I thought, so for your amusement, here it is https://doi.org/10.1016/S0022-4049(98)00076-0 Steve Awodey, Sheaf representation for topoi JPAA (2000)

David Michael Roberts (Oct 21 2020 at 22:46):

PS: A "hyperlocal topos" is most of what Mike Shulman calls a "constructively well-pointed topos". Only nontriviality of the topos, and 1 being a strong generator is missing

sarahzrf (Oct 21 2020 at 23:04):

...what's a fibred topos?

sarahzrf (Oct 21 2020 at 23:05):

is it maybe some kind of grothendieck fibration of topoi w/ nice properties like each fiber being a topos?

David Michael Roberts (Oct 22 2020 at 02:55):

It's a Groth. fibration whose fibres are toposes (or equiv. and indexed category landing inside $\mathbf{Topos}$ instead of $\mathbf{Cat}$ ). Awodey works over the site given by the starting topos, and proves that a) he's got a stack and b) it's strictifiable to a sheaf. I don't really like step b), tbh.

sarahzrf (Oct 22 2020 at 03:45):

in the groth fibration version youd also want to impose the condition that the pullbacks are geometric morphisms, then?

sarahzrf (Oct 22 2020 at 03:45):

...which component? inverse image?

sarahzrf (Oct 22 2020 at 03:46):

thatd be my guess, but if youre saying it factors thru something forgetful U : Topos → Cat, well, thatd have to take direct image components on morphisms, right?

David Michael Roberts (Oct 22 2020 at 05:12):

I think you get to choose. The case that Awodey is interested in is the codomain fibration, where the pullbacks (i.e. the re-indexing functors between fibres) are logical. But the version in SGA4 (and Verdier's thesis has an essentially equivalent version) uses geometric morphisms, so basically the right-adjoint part of the geometric morphism.

David Michael Roberts (Oct 22 2020 at 05:21):

Here's Quillen writing about them: http://www.claymath.org/library/Quillen/Working_papers/quillen%201971/1971-5.pdf#page=60 (other recent sources that come up with a Google search don't seem to want to define "morphism of topoi", taking it to be obvious)

Jens Hemelaer (Oct 22 2020 at 19:20):

David Michael Roberts said:

Well, it was easier to find than I thought, so for your amusement, here it is https://doi.org/10.1016/S0022-4049(98)00076-0 Steve Awodey, Sheaf representation for topoi JPAA (2000)

Interesting paper! With the terminology there, a 'hyperlocal topos' seems to be the same thing as what in the Elephant is called a 'local topos', do you know if this is the case?

sarahzrf (Oct 22 2020 at 19:37):

btw, is there some nice theory of what kinds of properties show up when you take stalks of sheaves? (...i guess maybe this reduces to asking about filtered colimits, but it does feel a bit more specific??)

sarahzrf (Oct 22 2020 at 19:37):

like is there a nice general categorical explanation of the notion of a local ring and why stalks tend to be local

sarahzrf (Oct 22 2020 at 19:39):

...also, since the topic has drifted a bit, i'd like to restate my original question:

sarahzrf (Oct 22 2020 at 19:40):

does anybody know of any sources that discuss filterquotients as being an example of taking stalks of stacks, or at least sources that discuss taking stalks of stacks at all?

Reid Barton (Oct 22 2020 at 19:45):

What is the filterquotient construction? Is it the same as an ultraproduct?

Reid Barton (Oct 22 2020 at 19:46):

I know there's a way to describe ultraproducts in terms of pushing a sheaf on $X$ forward to $\beta X$ and then taking the stalk at the ultrafilter.

Reid Barton (Oct 22 2020 at 19:52):

sarahzrf said:

like is there a nice general categorical explanation of the notion of a local ring and why stalks tend to be local

This is probably a separate topic, but my understanding is that it's not so much that stalks tend to be local (for example, the stalks of the constant sheaf on a non-local ring aren't local) but that we want to turn a ring $R$ into a local ring, and we can do that in a universal way, at the cost of also turning it from an ordinary ring, that is, a ring object in the category of sheaves over a point, into a ring object in some other topos.

Reid Barton (Oct 22 2020 at 19:54):

I don't know why it's specifically "local ring" that appears here other than, of course, the more similar a ring is to a field, the better, and maybe local ring is the best you can do for some reason.

Reid Barton (Oct 22 2020 at 19:55):

At least subject to the condition that the topos be a locale, perhaps.

Jens Hemelaer (Oct 22 2020 at 20:29):

sarahzrf said:

like is there a nice general categorical explanation of the notion of a local ring and why stalks tend to be local

One possible explanation is as follows:

Let $\mathcal{C}$ be the opposite of the category of finitely presented commutative rings. Let $J$ be the Zariski topology. Then $\mathcal{E} = \mathbf{Sh}(\mathcal{C},J)$ is the classifying topos of local rings. In particular, a geometric morphism $\mathbf{Sh}(X) \to \mathcal{E}$ is the same thing as a sheaf of rings on $X$ such that each stalk is a local ring.

Reid Barton (Oct 22 2020 at 20:35):

Right, I was remembering something similar but a bit less fancy and more explicit (a ring $S$ is local if for any $R \in \mathcal{C}$ , any map $f : R \to S$ and any co-covering family $(R \to R[1/a_i])_{i \in I}$ , the map $f$ factors through at least one of the $R \to R[1/a_i]$ ).

Reid Barton (Oct 22 2020 at 20:36):

Is it possible to give an a priori justification of the Zariski topology on $\mathcal{C}$ without first knowing about algebraic geometry?

Jens Hemelaer (Oct 22 2020 at 20:42):

sarahzrf said:

does anybody know of any sources that discuss filterquotients as being an example of taking stalks of stacks, or at least sources that discuss taking stalks of stacks at all?

I did not hear about filterquotients before you started this thread. One source that discusses stalks of stacks is Lurie's Higher Topos Theory. He talks about infinity-sheaves, but it seems that 2-sheaves/stacks are a special case. Lurie defines what a geometric morphism is between infinity-toposes, and then a point of an infinity-topos $\mathcal{E}$ is a geometric morphism $\mathbf{Sets} \to \mathcal{E}$ . Like for 1-toposes, the definition is that the inverse image functor preserves colimits and finite limits, but then in the right higher categorical sense. My guess would be that the filterquotient is indeed the same as taking a stalk according to Lurie's definition, but I wouldn't know how to prove this.

Reid Barton (Oct 22 2020 at 20:42):

I mean, it's already not such a complicated thing. But it's not clear to me how to explain why $R \to R[1/a]$ should correspond to an open subset in a way that's not circular.

Reid Barton (Oct 22 2020 at 21:32):

The nlab page https://ncatlab.org/nlab/show/ultrapower is relevant to the original question.

sarahzrf (Oct 24 2020 at 16:03):

Jens Hemelaer said:

sarahzrf said:

does anybody know of any sources that discuss filterquotients as being an example of taking stalks of stacks, or at least sources that discuss taking stalks of stacks at all?

I did not hear about filterquotients before you started this thread. One source that discusses stalks of stacks is Lurie's Higher Topos Theory. He talks about infinity-sheaves, but it seems that 2-sheaves/stacks are a special case.

I was under the impression that Lurie works tend to use ∞ to mean (∞, 1)? or is that only true of some of them?

sarahzrf (Oct 24 2020 at 16:03):

definitely sounds apropos tho!

sarahzrf (Oct 24 2020 at 16:04):

Reid Barton said:

I know there's a way to describe ultraproducts in terms of pushing a sheaf on $X$ forward to $\beta X$ and then taking the stalk at the ultrafilter.

Reid Barton said:

The nlab page https://ncatlab.org/nlab/show/ultrapower is relevant to the original question.

yeah, i'm pretty familiar with these constructions—that's what prompted me to look at filterquotients this way

sarahzrf (Oct 24 2020 at 16:06):

Reid Barton (Oct 24 2020 at 16:12):

sarahzrf said:

Jens Hemelaer said:

I did not hear about filterquotients before you started this thread. One source that discusses stalks of stacks is Lurie's Higher Topos Theory. He talks about infinity-sheaves, but it seems that 2-sheaves/stacks are a special case.

I was under the impression that Lurie works tend to use ∞ to mean (∞, 1)? or is that only true of some of them?

I think the claim is that the definition of a stack of categories only uses the invertible 2-morphisms in Cat, and so it's a special case of "sheaf valued in an (∞, 1)-category", namely the (2, 1)-category Cat.

sarahzrf (Oct 24 2020 at 16:15):

eesh, im too hungry to think about that easily rn, let me see

Reid Barton (Oct 24 2020 at 16:18):

You might want to consider the claim
$(*)$ A diagram (indexed by an ordinary category) is a limit diagram in the 2-category Cat if and only if it is a limit diagram in the (2,1)-category Cat' obtained by discarding the noninvertible 2-morphisms
and then convince yourself that $(*)$ implies the claim above, and separately that $(*)$ is in fact true

sarahzrf (Oct 24 2020 at 16:19):

hmm, hold on, doesnt the descent condition involve like... lax limits, if you phrase it in terms of universal objects

sarahzrf (Oct 24 2020 at 16:20):

i dont rly have actual experience working with stacks at all tbh, the only definition i know for them that i could plausibly expand out to any level of detail is shaped like "local objects in [C^op, Cat] with respect to inclusions of covering sieves, but make it 2"

Reid Barton (Oct 24 2020 at 16:41):

To be honest, I would generally take "sheaf valued in the (2, 1)-category Cat" as the definition of a stack of categories in the first place.

Reid Barton (Oct 24 2020 at 16:42):

But I think it unwinds to a definition of descent that involves only isomorphisms in the descent data, while still allowing there to be noninvertible morphisms in the categories themselves.

Reid Barton (Oct 24 2020 at 16:44):

For the simplest example let's say we have a cover of $X$ by just two open subsets $U$ and $V$ , and a stack of categories $\mathcal{C}$ on $X$ . Then there are four relevant categories $\mathcal{C}(X)$ , $\mathcal{C}(U)$ , $\mathcal{C}(V)$ , $\mathcal{C}(U \cap V)$ and restriction functors between them that form a square.

Reid Barton (Oct 24 2020 at 16:46):

The sheaf condition says this square is a (pseudo)pullback, so giving an object $A$ of $\mathcal{C}(X)$ amounts to giving objects $A_U \in \mathcal{C}(U)$ , $A_V \in \mathcal{C}(V)$ , $A_{U \cap V} \in \mathcal{C}(U \cap V)$ together with isomorphisms $A_U|_{U \cap V} \xrightarrow{\sim} A_{U \cap V}$ and $A_V|_{U \cap V} \xrightarrow{\sim} A_{U \cap V}$ .

Reid Barton (Oct 24 2020 at 16:47):

So being a stack here means satisfying a descent condition involving isomorphisms.

Reid Barton (Oct 24 2020 at 16:47):

If there were more than two open subsets, then we would have some equations relating the isomorphisms which I'm too lazy to write out.

Reid Barton (Oct 24 2020 at 16:49):

I guess you could also consider "lax stacks" or "oplax stacks" where you replace the (pseudo)limits by lax limits or oplax limits, and correspondingly the isomorphisms in the descent data by arbitrary maps, but I don't know of any examples of those.

Reid Barton (Oct 24 2020 at 16:50):

In the traditional stacks valued in groupoids, the maps in the descent data automatically have to be isomorphisms of course, because then there are no noninvertible morphisms in $\mathcal{C}(U \cap V)$ in the first place.

sarahzrf (Oct 24 2020 at 16:58):

oh hmmmm yeah im not sure where i got the lax limit idea from

Reid Barton (Oct 24 2020 at 17:02):

sarahzrf said:

i dont rly have actual experience working with stacks at all tbh, the only definition i know for them that i could plausibly expand out to any level of detail is shaped like "local objects in [C^op, Cat] with respect to inclusions of covering sieves, but make it 2"

I think with this description it is indeed less obvious why you can do without the noninvertible 2-morphisms. Since the codomain of the covering sieve is some representable $X$ , and then the category of maps from $X$ to $\mathcal{C}$ is $\mathcal{C}(X)$ by Yoneda, but the groupoid of maps is just the maximal groupoid of $\mathcal{C}(X)$ , and this isn't enough to determine $\mathcal{C}(X)$ .

Reid Barton (Oct 24 2020 at 17:07):

You really need $\mathcal{C}$ to be local in the Cat-enriched sense, i.e., $\mathbf{Hom}(X, C) \to \mathbf{Hom}(\mathcal{U}, C)$ is an equivalence of categories (where $\mathcal{U}$ represents the covering sieve), and not just an equivalence of underlying groupoids.

Reid Barton (Oct 24 2020 at 17:08):

So, for this approach you could either work in a genuine 2-category, or fudge things using the same idea used to prove $(*)$ .

Reid Barton (Oct 24 2020 at 17:18):

Which is to say (spoiler!) that we can describe the condition " $\mathbf{Hom}(B, C) \to \mathbf{Hom}(A, C)$ is an equivalence of categories" using only the (2, 1)-category structure of Cat (or of Cat-valued presheaves) as " $\mathrm{Map}(B \times I, C) \to \mathrm{Map}(A \times I, C)$ is an equivalence of groupoids for every small category $I$ " (here $\mathrm{Map}$ denotes the Hom-groupoids in the (2, 1)-category).

Reid Barton (Oct 24 2020 at 17:44):

(Or every $I$ that's a string of $n \ge 0$ arrows, or in the case of 2-categories, every $I$ that's a string of $n \in \{0, 1\}$ arrows.)

David Michael Roberts (Oct 25 2020 at 02:09):

Reid Barton said:

To be honest, I would generally take "sheaf valued in the (2, 1)-category Cat" as the definition of a stack of categories in the first place.

And how would you define the 1-arrows and 2-arrows for these?

Reid Barton (Oct 25 2020 at 13:15):

Never really thought about it before, but I would assume that at least the (2, 1)-category structure is inherited from the ambient (2, 1)-category.

David Michael Roberts (Oct 25 2020 at 23:24):

And the full (2,2)-category structure comes from the (2,2)-category structure on Cat. In all honesty, I don't understand the need/desire to use (2,1)-categories here.

Reid Barton (Oct 26 2020 at 00:07):

And likewise, I don't see the need/desire to use (2,2)-categories.

Reid Barton (Oct 26 2020 at 00:11):

There's nothing intrinsically 2-categorical (such as a lax limit) happening here, so I prefer to stay on more familiar ground

David Michael Roberts (Oct 26 2020 at 02:23):

I'm not sure what you mean by "here", in that case. I'm just talking about actually using the 2-arrows that arise naturally in examples (not just in abstract category theory). The (2,2)-category structure doesn't stop the use of invertible stuff; the (2,1)-category structure stops the use of non-invertible stuff. I prefer having all the relevant structure available. But I can see I'm not going to win any hearts and minds :-)

Reid Barton (Oct 26 2020 at 04:05):

I'm just saying that the notion of "a sheaf of Xs" (on an ordinary site) makes sense as soon as you have an $(\infty,1)$ -category of Xs. Moreover, if you plug in X = category then you get out what people call a "stack of categories".
Of course, if the category of Xs has some extra structure like being a k-linear category or having noninvertible 2-morphisms or having a notion of graded morphisms or whatever then the category of sheaves of Xs will probably inherit that extra structure, and this extra structure will probably be useful; but it's not required for understanding what a sheaf of Xs is.

Reid Barton (Oct 26 2020 at 04:11):

As opposed to, say, if I had a V-enriched category and I want to talk about "V-enriched presheaves of Xs" on it, then I would need a V-enrichment on X even to make sense of the objects.
That doesn't happen for stacks of categories; they're a (2,1)-categorical concept.

David Michael Roberts (Oct 26 2020 at 06:39):

Oh, I totally grant that considering just the definition of a single stack uses only the invertible 2-arrows. But I'm Australian :-) I want the whole 2-category of them.

Morgan Rogers (he/him) (Oct 26 2020 at 09:45):

Ahhh hearing people talk about studying individual stacks out of the context of the whole 2-category of them makes me cringe, I'm totally on board @David Michael Roberts

Reid Barton (Oct 26 2020 at 13:50):

This isn't about individual stacks though. The "sheaf of Xs" gives you the whole (2,1)-category of stacks, which is the basic object.
Then, if you want, you can introduce the extra structure of the noninvertible 2-morphisms, which comes from the noninvertible 2-morphisms of Cat.
The situation is just like sheaves of R-modules (R a fixed commutative ring). The Hom sets in R-Mod have the extra structure of being R-modules themselves, and therefore the Hom sets between sheaves of R-modules also have an R-module structure. But I'm sure you'll agree that it's perfectly reasonable to talk about "the category of sheaves of R-modules", without (or prior to) introducing this extra structure as an R-Mod-enriched category.
That's all that's going on with stacks of categories; there's no major difference between R-Mod-enrichment and Cat-enrichment.

David Michael Roberts (Oct 26 2020 at 21:40):

which is the basic object.

I disagree. There's no reason to say why that is the basic object. Whether you say Cat (i.e. the sheaves of categories on a single point) is groupoid enriched or category enriched is a bigger distinction than thinking of RMod as enriched over itself or just Set. The base change functors are quite different, I think. And, don't forget, the basic reason for introducing categories was to introduce natural transformations, not just isomorphisms. It's literally the genesis of the subject.

How to you feel about the (2,2)-category of fibred (or indexed) categories? Is it justifiable from your point of view to take the underlying (2,1)-category as basic?

Jens Hemelaer (Oct 27 2020 at 09:42):

The reference for filterquotients that I was looking at (Mac Lane and Moerdijk, Sheaves in Geometry and Logic, V.9) completely avoids any discussion about natural transformations and defines the filterquotient as a filtered colimit in the 1-category of categories (see Equation 12). Maybe @sarahzrf was using a different reference.

I think it's unfortunate that the original question is still unresolved. It would be great to find a definition of stalk of a stack in print somewhere, and then have a proof that the filterquotient is a special instance of this.

David Michael Roberts (Oct 27 2020 at 11:12):

@Jens Hemelaer yes, sorry for getting the thread off-track.

Reid Barton (Oct 27 2020 at 12:50):

Sorry @David Michael Roberts, there must have been a major miscommunication from the start, if you got the impression that I'm trying to deny the nature of Cat as a (2,2)-category.
All I'm saying is that the property of a presheaf of categories being a sheaf (aka "stack") is something happening entirely within the underlying (2,1)-category. Consequently, if one understands the theory of sheaves valued in a (2,1)-category (or $(\infty,1)$ -category), it just applies to this situation and so, for example, there is no need to re-prove the existence of sheafification ("stackification"). At most, one would need to prove that the sheafification has the right (2,2)-universal property and not just the right (2,1)-universal property. But this is basically automatic, precisely because the concept of "sheaf" is oblivious to the noninvertible 2-morphisms, and because of properties of the 2-category Cat like $(*)$ above.

2-category theory does have a bunch of genuinely new concepts, so if the purpose of category theory is to make trivial things trivially trivial, likewise it makes sense to keep track of which things in 2-category theory are really just (2,1)-category theory.

Reid Barton (Oct 27 2020 at 12:51):

Jens Hemelaer said:

The reference for filterquotients that I was looking at (Mac Lane and Moerdijk, Sheaves in Geometry and Logic, V.9) completely avoids any discussion about natural transformations and defines the filterquotient as a filtered colimit in the 1-category of categories (see Equation 12). Maybe sarahzrf was using a different reference.

I think it's unfortunate that the original question is still unresolved. It would be great to find a definition of stalk of a stack in print somewhere, and then have a proof that the filterquotient is a special instance of this.

Filtered colimits in the 1-category Cat are also colimits in the (2,1)- or 2-category Cat, so I don't think there will be any difficulty here.

David Michael Roberts (Oct 27 2020 at 12:56):

@Reid Barton Ah, well. I'm just being overly grumpy, maybe. It grinds my gears (in a mild way) when people restrict attention to the (2,1)-case as if that's all that exists. Thanks for clearing up your rationale.

Jens Hemelaer (Oct 27 2020 at 14:44):

Reid Barton said:

Filtered colimits in the 1-category Cat are also colimits in the (2,1)- or 2-category Cat, so I don't think there will be any difficulty here.

Do you know whether filtered 2-colimits commute with finite 2-limits (for a suitable definition of filtered 2-colimit)? This seems to be one of the things that would need to be satisfied in order to get an analogy with stalks.

Reid Barton (Oct 27 2020 at 14:51):

The (2,1)-category Cat is locally finitely presentable, so filtered colimits commute with finite limits there; and so in the 2-category Cat, at least filtered conical 2-colimits commute with finite conical 2-limits.

Reid Barton (Oct 27 2020 at 15:03):

At some point I came across a fully 2- version of this statement but I can't seem to find it now. One would have to decide how to define both filtered weights and finite weights.

Reid Barton (Oct 27 2020 at 15:06):

Ah, I think it was in http://cms.dm.uba.ar/academico/carreras/licenciatura/tesis/2016/Nicolas_Canevali.pdf

sarahzrf (Oct 27 2020 at 18:48):

Jens Hemelaer said:

The reference for filterquotients that I was looking at (Mac Lane and Moerdijk, Sheaves in Geometry and Logic, V.9) completely avoids any discussion about natural transformations and defines the filterquotient as a filtered colimit in the 1-category of categories (see Equation 12). Maybe sarahzrf was using a different reference.

I think it's unfortunate that the original question is still unresolved. It would be great to find a definition of stalk of a stack in print somewhere, and then have a proof that the filterquotient is a special instance of this.

sorry no, that's the only reference i knew of when posting—i just thought that might be the shadow of a weak higher object

sarahzrf (Oct 27 2020 at 18:48):

although it might be of interest to note that i have since run across a definition of filterquotients in the elephant which manages to sidestep the issue even further!

sarahzrf (Oct 27 2020 at 18:57):

actually i believe this is how mac lane & moerdijk strictify the construction too

sarahzrf (Oct 27 2020 at 18:57):

so i guess maybe this is just spelling out the colimit explicitly?

sarahzrf (Oct 27 2020 at 18:57):

let me crack open sheaves in geometry in logic

sarahzrf (Oct 27 2020 at 18:59):

image.png

sarahzrf (Oct 27 2020 at 18:59):

image.png

sarahzrf (Oct 27 2020 at 19:01):

image.png

sarahzrf (Oct 27 2020 at 19:01):

okay yeah

sarahzrf (Oct 27 2020 at 19:02):

i havent actually read this section in any detail :sweat_smile: my extremely limited knowledge of filterquotients comes from nlab and an mse thread

Jens Hemelaer (Oct 27 2020 at 20:06):

Reid Barton said:

Ah, I think it was in http://cms.dm.uba.ar/academico/carreras/licenciatura/tesis/2016/Nicolas_Canevali.pdf

Thank you very much, that's a wonderful reference.

Jens Hemelaer (Oct 27 2020 at 20:08):

sarahzrf said:

although it might be of interest to note that i have since run across a definition of filterquotients in the elephant which manages to sidestep the issue even further!

I didn't realize they were described in the Elephant as well... and apparently filterquotients of Grothendieck toposes are not necessarily again Grothendieck toposes, that's something I missed.

sarahzrf (Oct 27 2020 at 20:59):

im not convinced there's anything that's not described in the elephant

David Michael Roberts (Oct 28 2020 at 01:41):

Jens Hemelaer said:

sarahzrf said:

although it might be of interest to note that i have since run across a definition of filterquotients in the elephant which manages to sidestep the issue even further!

I didn't realize they were described in the Elephant as well... and apparently filterquotients of Grothendieck toposes are not necessarily again Grothendieck toposes, that's something I missed.

Absolutely. You need them to not be Grothendieck toposes in order to get a new model of ETCS satisfying different conditions to the one you started from, which is the point of forcing. Otherwise the fact Set is the terminal Grothendieck topos gets you. And I imagine there are other applications, too :-)