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Stream: community: general

Topic: Grothendieck on topos theory

John Baez (Oct 06 2020 at 21:55):

Have you ever wondered how Grothendieck would start a class on topos theory? Here's how:

Grothendieck.mp3

I got this from a little bird who put the tape on my window sill.

John Baez (Oct 06 2020 at 21:56):

Please don't ask me any questions about it!

Fawzi Hreiki (Oct 06 2020 at 22:06):

Cool!

Fawzi Hreiki (Oct 06 2020 at 22:07):

I'm surprised his english is so good actually, although maybe I need not be

John Baez (Oct 06 2020 at 22:12):

Yes. It's interesting that he has a German accent - maybe from his mother? I don't know where or when he learned English.

Fawzi Hreiki (Oct 06 2020 at 22:33):

I remember reading (on wikipedia I think) that he insisted on spelling his name 'Alexander' as the Germans do instead of 'Alexandre' as the French do. So perhaps the accent was also a semi-conscious effort on his part.

Nikolaj Kuntner (Oct 07 2020 at 07:33):

There's a video of a Connes lecture on youtube that plays that bit. There's also a few hours of him in French. And here's a sort of 20 page bio that goes through some of the stages of his life. A book bio exists too.

Matteo Capucci (he/him) (Oct 07 2020 at 10:19):

John Baez said:

Have you ever wondered how Grothendieck would start a class on topos theory? Here's how:

Grothendieck.mp3

I got this from a little bird who put the tape on my window sill.

Wow I'm excited like a groupie finding out a new album of their favourite band

David Michael Roberts (Oct 07 2020 at 22:14):

I know John said no questions, but I presume this recording was made in Buffalo? See discussion at https://www.youtube.com/watch?v=5AR55ZsHmKI, where one of the slides mention 17 hours of tape recordings (!!!!)

Fawzi Hreiki (Oct 07 2020 at 22:16):

I doubt (most of) those will ever see the light of day unfortunately

Fawzi Hreiki (Oct 07 2020 at 22:23):

There are - in principle - also meant to be some category theory related recordings here http://archmathsci.org/ but they have never been uploaded

Dan Doel (Oct 07 2020 at 23:26):

Heh, I didn't know it was actually Grothendieck who said that the topos is not really the category of sheaves.

David Michael Roberts (Oct 07 2020 at 23:49):

Looking at model-indepedence there, thinking the category is not the actual object!

Dan Doel (Oct 08 2020 at 00:09):

I think it's not just that. The topos is the space, and the category is the opens. At least, that's what I've heard.

John Baez (Oct 08 2020 at 00:19):

What's the difference between "the category" and "the topos"? Isn't a topos a category?

Reid Barton (Oct 08 2020 at 00:24):

Isn't an affine scheme just a ring?

John Baez (Oct 08 2020 at 00:26):

Yes, but the category of affine schemes is the opposite of the category of rings.

John Baez (Oct 08 2020 at 00:27):

Joyal talks about a 2-category of "logoi" that's opposite to the 2-category of "topoi". Is that what you're getting at?

John Baez (Oct 08 2020 at 00:27):

(Or maybe he says logoses and toposes - let's not get into that!)

John Baez (Oct 08 2020 at 00:29):

I'm just trying to understand what's meant by "the topos is not the category of sheaves".

Reid Barton (Oct 08 2020 at 00:29):

I can't say whether this is what Grothendieck had in mind, of course

Dan Doel (Oct 08 2020 at 00:29):

I thought the point was that the category is like the frame of opens, and "the topos" is the space/locale.

Dan Doel (Oct 08 2020 at 00:29):

The frame is not the space.

John Baez (Oct 08 2020 at 00:30):

Okay, it sounds like an issue of opposite categories then: the category of frames is the opposite of the category of locales.

Reid Barton (Oct 08 2020 at 00:30):

Here is another thing you could say. Let's take for granted there is some 2-category(!) of topoi. Then there is some particular topos S such that the category corresponding (if you will) to a topos X is the Hom-category Hom_Topoi(X, S).

Reid Barton (Oct 08 2020 at 00:33):

Then as it turns out, the whole topos X is determined by the category Hom_Topoi(X, S) and so we could just identify it with this category.

Reid Barton (Oct 08 2020 at 00:33):

But if you believed this was the real story, then you might say "the category is not the actual object".

Dan Doel (Oct 08 2020 at 00:34):

I'm not sure Grothendieck was just thinking of it as reversed arrows, though (judging from the lecture, and also that isn't what I thought the point was). The point is that the space is much 'smaller' than the frame, and the way you think of the maps might be very different depending on the direction, or how you're thinking of the objects.

Dan Doel (Oct 08 2020 at 00:34):

One example in the lecture is that the whole category of sets is just the (category of sheaves for the) one point topos.

Reid Barton (Oct 08 2020 at 00:38):

Right, and more generally the (generalized) points of the topos are a totally different kind of thing from objects of the (corresponding) category, which are like functions on the topos--or more accurately, functors to the classifying topos of objects

Reid Barton (Oct 08 2020 at 00:41):

So the category could be seen as dual to some geometric object we don't know how to describe directly. Since the category contains all the data anyways, we can just call it the topos.

Reid Barton (Oct 08 2020 at 00:41):

I guess another analogy could be quantum groups.

John Baez (Oct 08 2020 at 00:44):

Okay, thanks. I think it all boils down to the duality between geometry and (mostly commutative but sometimes noncommutative) algebra, where we sometimes define a category of rather mystical geometrical objects to be the opposite of a very nice category of algebraic gadgets.

John Baez (Oct 08 2020 at 00:45):

Joyal and Anel really run with this idea applied to topos theory in their paper Topo-logie.

Reid Barton (Oct 08 2020 at 00:46):

This also "explains" why a "geometric morphism" is a right adjoint between topoi, when it's the left adjoint which preserves the algebraic structure (colimits and finite limits).

John Baez (Oct 08 2020 at 00:47):

Here's the abstract of Topo-logie:

We claim that Grothendieck topos theory is best understood from a dual algebraic point of view. We are using the term logos for the notion of topos dualized, i.e. for the category of sheaves on a topos. The category of topoi is here defined to be the opposite of that of logoi. A logos is a structure akin to commutative rings and we detail many analogies between the topos–logos duality and the duality between affine schemes and commutative rings.

John Baez (Oct 08 2020 at 00:48):

I should read this whole paper - I haven't yet. :cry:

Dan Doel (Oct 08 2020 at 00:48):

Yeah, I just didn't realize it went all the way back to Grothendieck. :smile:

John Baez (Oct 08 2020 at 00:55):

He knew what he was doing....

Reid Barton (Oct 08 2020 at 00:55):

I guess it's not surprising that the inventor of schemes would have thought about topoi this way. On the other hand, I can also imagine an alternate timeline where the notion of (Grothendieck) topos (as a category) became established before the connection to geometry.

Fawzi Hreiki (Oct 08 2020 at 00:58):

I remember seeing somewhere Lawvere writing or saying that topology is in fact at base more algebraic than geometric, with the geometry arising via duality

Fawzi Hreiki (Oct 08 2020 at 00:59):

Which of course we all know now because of frames and locales and so on

Fawzi Hreiki (Oct 08 2020 at 01:00):

The lecture by Joyal based on the topo-logie paper explains this quite well

Dan Doel (Oct 08 2020 at 01:01):

Yeah, I guess I had just assumed that the frame/locale thing came first, but it sounds like that was actually backed out of toposes.

Fawzi Hreiki (Oct 08 2020 at 01:03):

Yeah as far as I know, toposes actually did come first

Dan Doel (Oct 08 2020 at 01:06):

Or I guess more specifically, I thought the 'the topos is not the category of sheaves' came after recognizing it as a larger example of frames/locales. But it was frames/locales that came from dialing back that topos insight.

Dan Doel (Oct 08 2020 at 01:06):

I think I did know that toposes were invented first.

Fawzi Hreiki (Oct 08 2020 at 01:31):

Ah ok. Makes sense

David Michael Roberts (Oct 08 2020 at 03:25):

To my mind it's more like an oo-category vs a model category presenting it. Or, another way, using Joyal–Tierney, the localic groupoid underlying the (Grothendieck) topos is the geometric object. Though this is really just one way to present the corresponding stack over locales.

David Michael Roberts (Oct 08 2020 at 03:39):

cf https://ncatlab.org/nlab/show/classifying+topos+of+a+localic+groupoid#properties

Moerdijk then finishes the deal, showing the equivalence of 2-categories of Grothendieck toposes, and of localic groupoids, some kind of generalised morphisms (some kind of bibundles between a kind of 'completion') and some reasonably obvious 2-arrows (equivariant maps between bispaces).

John Baez (Oct 08 2020 at 03:55):

Yes, that's very nice - it turns the "mystical" geometric entity that a topos is the algebraic counterpart of into something concrete.

John Baez (Oct 08 2020 at 03:56):

It shows that topoi are essentially just topological groupoids in disguise - well, localic groupoids, but that's just a refinement of the same idea.

John Baez (Oct 08 2020 at 03:57):

The real trick is that the morphisms aren't the obvious maps but something subtler!

Fawzi Hreiki (Oct 08 2020 at 09:57):

This intuition/fact that a topos 'has' an underlying space only applies to Grothendieck toposes though right?

Fawzi Hreiki (Oct 08 2020 at 10:01):

Although I guess you can define a Grothendieck topos over any elementary topos $E$ so you can just regard $E$ as the sheaf topos for the 1 point space (in $E$ itself), but this just shifts the perspective from $\textbf{Set}$ to $E$.

Matteo Capucci (he/him) (Oct 08 2020 at 11:15):

My guess was that he meant topoi are not categories in the sense 'geometric morphisms' are not just 'functors'. Since Grothendieck was so morphism-first, it wouldn't surprise me that he thought of categories as things that transform like categories, and topoi don't.

Dan Doel (Oct 08 2020 at 14:52):

I didn't mention every quote in the video, but there were a bunch about the sheaves being something that arises from or describe an underlying geometric object that he wanted to actually refer to as "the topos." He just didn't have any more direct way to describe it than the category of sheaves.

Fawzi Hreiki (Oct 08 2020 at 15:19):

If you can motivate the notion of a geometric morphism without using sheaves, then you can say that a Grothendieck topos is a subtopos of a presheaf topos and then the site (and the sheaves on that site) are just a 'basis'. But I dont know if thats what he meant.

Matteo Capucci (he/him) (Oct 08 2020 at 15:25):

Dan Doel said:

I didn't mention every quote in the video, but there were a bunch about the sheaves being something that arises from or describe an underlying geometric object that he wanted to actually refer to as "the topos." He just didn't have any more direct way to describe it than the category of sheaves.

Uh I see... I should probably listen to the whole thing then, this looks intriguing. It might be just a philosophical take then.

Dan Doel (Oct 08 2020 at 15:27):

I'd definitely recommend watching the video.

It was definitely more of a "how you should think about toposes" thing. How to think to get the best use out of the concepts, whereas the actual definitions are technical details.

Dan Doel (Oct 08 2020 at 15:28):

Of course, the best way to think might depend on what your application is.

Dan Doel (Oct 08 2020 at 15:29):

A lot of applications are interested in toposes being similar to the category of sets, and I don't think that's because they're thinking of sets as the 1-point space exactly.

Dan Doel (Oct 08 2020 at 15:33):

But then, I guess that's the point of also talking about the 'logos'.

Mike Shulman (Oct 08 2020 at 16:45):

David Michael Roberts said:

Moerdijk then finishes the deal, showing the equivalence of 2-categories of Grothendieck toposes, and of localic groupoids, some kind of generalised morphisms (some kind of bibundles between a kind of 'completion') and some reasonably obvious 2-arrows (equivariant maps between bispaces).

Can't the morphisms also be described more intuitively as an appropriate kind of anafunctor between localic groupoids?

John Baez (Oct 08 2020 at 16:48):

What's the difference between "some kind of bibundle" of localic groupoids and "an appropriate kind of anafunctor"? They seem like different terms for the same thing. I guess the devil lies in the details... and also David's introduction of a kind of "completion". (What's this completion business? Can we avoid it?)

Mike Shulman (Oct 08 2020 at 16:51):

Depending on what "some kind of bibundle" means, there might be no difference at all. But I think calling it an anafunctor makes it clear that we are really talking about the natural 2-category of localic groupoids (for internal categories in a category without AC, we have to use anafunctors instead of functors to get good category-theoretic behavior) rather than imposing some weird alternative notion of morphism on them.

Jens Hemelaer (Oct 08 2020 at 19:47):

As far as I know, localic groupoids and maps between them are extremely complicated.

Take for example the classifying topos of abelian groups $\mathcal{E}$. By definition, the points of this topos are the abelian groups. The theorem by Butz and Moerdijk then says that there is a topological groupoid $\mathcal{G}$ such that $\mathcal{E}$ is the topos of sheaves on $\mathcal{G}$.

Jens Hemelaer (Oct 08 2020 at 19:48):

But the elements of $\mathcal{G}$ are not the same thing as the points of $\mathcal{E}$, because the former is a set and the latter is a proper class (all abelian groups).

Jens Hemelaer (Oct 08 2020 at 19:56):

So for the 'right' notion of maps between topological groupoids, you have a proper class of maps $\ast \to \mathcal{G}$.

Mike Shulman (Oct 08 2020 at 20:04):

Of course. In a category not satisfying choice, there can certainly be a proper class of anafunctors between two small categories.

Jens Hemelaer (Oct 08 2020 at 21:34):

Yes, I guess all I wanted to say is that I don't have geometric intuition for what anafunctors of topological groupoids look like, so in most cases I don't see how to gain geometric intuition by looking at the topological groupoid rather than at the topos itself.

David Michael Roberts (Oct 08 2020 at 22:55):

@Mike Shulman not the way Moerdijk does it (this is in the second paper). He forms a localic category out of the localic groupoid and shows they have equivalent categories of sheaves. This localic category seems to be a presentation of the stack of sheaves on the original localic groupoid. What one does is embed locales into toposes by taking sheaves on them and then does a comma object construction using the localic cover given by the objects of the groupoid.
The morphisms $X \to Y$ ($X,Y$ localic groupoids) are then bibundles for the associated localic categories (there is an asymmetry, one of the anchor maps needs to be open, and there's a flatness condition).

So perhaps a better theorem would be to say that every Grothendieck topos comes from a localic category, and we can dispense with the funky morphisms, just taking bibundles as Moerdijk gives them. Personally I like this approach better.

David Michael Roberts (Oct 08 2020 at 22:58):

@John Baez really think we can't avoid it, since toposes are not groupoids! We have to account, somehow, for the non-invertible natural transformations of geometric morphisms, which Moerdijk does.

David Michael Roberts (Oct 08 2020 at 23:09):

I agree though, that if one could show that Moerdijk's"complete flat bispaces" were the same as anafunctors between the corresponding localic categories, this would the best possible situation.

Mike Shulman (Oct 09 2020 at 01:07):

I don't remember whether I've read Moerdijk's paper, but I think I got my impression from Pronk's "Etendues and stacks as bicategories of fractions" where she says

The category of toposes and isomorphism classes of geometric morphisms can be viewed as a category of frarctions... of a specific category of groupoids with respect to the class of weak equivalences (see (Moerdijk, 1988b)).... We want to understand this equivalence also on the level of 2-cells. One approach which is totally independent of the category of fractions theory is presented in (Moerdijk, 1990). A similar result is obtained in (Bunge, 1990).

The two citations to Moerdijk are to the two "classifying topos of a continuous groupoid" papers. The Bunge one is "An application of descent to a classification theorem for toposes". This suggests that at the level of homotopy 1-categories, what I said is what Moerdijk does in the first paper. Pronk's paper generalizes this to deal with 2-cells in the case of $T_1$-etendues, in which case all the 2-cells are invertible.

Mike Shulman (Oct 09 2020 at 01:10):

I suspect it should be possible to get the noninvertible 2-cells by doing a similar localization that takes account of the fact that the category of locales is not just a 1-category but a (1,2)-category, so that localic groupoids are not just a (2,1)-category (like internal groupoids in an arbitrary 1-category) but are enriched over double categories that are thin in one direction and invertible in the other. I feel like I might even have seen someone recently doing this, but I can't remember it off the top of my head.

David Michael Roberts (Oct 09 2020 at 02:48):

This suggests that at the level of homotopy 1-categories, what I said is what Moerdijk does in the first paper.

I concur. I had some vague thoughts, since last posting, that it might be easier with anafunctors directly rather than trying to deal with bibundles for categories that aren't groupoids. The anafunctor treatment requires very little structure to work compared to the bibundle approach for internal groupoids. It's clear that generalising the latter to internal categories is nontrivial, one can compare the matter under discussion, and also Moerdijk's "Classifying spaces and classifying topoi", and the torsors in Street's "Categorical and combinatorial aspects of descent theory"

Leopold Schlicht (Sep 01 2021 at 16:08):

John Baez said:

Have you ever wondered how Grothendieck would start a class on topos theory? Here's how:

Grothendieck.mp3

I got this from a little bird who put the tape on my window sill.

For those of you who have ever wondered how Grothendieck would proceed in his course on topos theory: you can find all audio files at https://agrothendieck.github.io/, clicking on "1973".

John Baez (Sep 01 2021 at 16:14):

Oh, good! I have all of them, but I never made them available, mainly because I wasn't sure it was allowed.

Graham Manuell (Sep 02 2021 at 20:42):

This is somewhat unrelated, but after the initial post it appears that there was some discussion saying that toposes predated frames/locales. For what it's worth, this is not true. Frames were first defined (under different names) and studied in the mid to late 1950s (though there is some related work even before this), but toposes were only defined and studied in the early 1960s. It is of course still surprising that these dates are so close together (and even more so because it is also not clear if Grothendieck was aware of this earlier work at the time).

John Baez (Sep 02 2021 at 20:56):

Interesting! Who studied frames back then, and under what names?

Graham Manuell (Sep 02 2021 at 21:01):

I know of Charles Ehresmann, Seymour Papert and Dona Strauss/Papert, but there were probably others too. One name used was "local lattice".

Mateo Carmona (Sep 03 2021 at 19:24):

Leopold Schlicht said:

John Baez said:

Have you ever wondered how Grothendieck would start a class on topos theory? Here's how:

Grothendieck.mp3

I got this from a little bird who put the tape on my window sill.

For those of you who have ever wondered how Grothendieck would proceed in his course on topos theory: you can find all audio files at https://agrothendieck.github.io/, clicking on "1973".

Thanks for sharing!

Mateo Carmona (Sep 03 2021 at 19:25):

John Baez said:

Interesting! Who studied frames back then, and under what names?

Here is a very interesting reading http://www.neverendingbooks.org/did-nobeling-discover-toposes-2
did Nöbeling invent topos theory as some say Krull invented scheme theory? No, of course not, they both lacked the crucial ingredient of sheaf theory.

David Michael Roberts (Sep 04 2021 at 03:26):

For those wondering why Krull is relevant, he more or less got the Zariski topology on an affine scheme, IIRC, but no structure sheaf