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Stream: deprecated: topos theory

Topic: petit and gros

sarahzrf (Apr 02 2020 at 17:48):

i'm looking at this page https://ncatlab.org/nlab/show/big+and+little+toposes
and it seems kinda confusingly worded, but let me see if i follow...

sarahzrf (Apr 02 2020 at 17:49):

the "big topos" pov seems straightforward enough, i think

sarahzrf (Apr 02 2020 at 17:51):

as for "small topos"—is this correct: the idea is that we can regard a topos not so much as itself being a generalized space, but as encoding the data of a generalized space, by thinking about a hypothetical base space which it is the topos of sheaves over?

Reid Barton (Apr 02 2020 at 18:00):

Topos theorists like to identify a topological space X (maybe with some extra adjectives) with its category of sheaves Shv(X)--that's why they consider morphisms to go in (unpopular opinion) the "wrong" direction.

sarahzrf (Apr 02 2020 at 18:01):

hmm.

Reid Barton (Apr 02 2020 at 18:01):

Maybe "identify" is not the perfect word, but the idea is that a topos is a generalization of a space.

sarahzrf (Apr 02 2020 at 18:02):

this seems uncomfortably close in concept space to sheaves themselves being generalized spaces :thinking:

sarahzrf (Apr 02 2020 at 18:02):

and/or $X \mapsto \operatorname{Sh}(X)$ feels uncomfortably close in concept space to the yoneda embedding

sarahzrf (Apr 02 2020 at 18:02):

what am i smelling?

sarahzrf (Apr 02 2020 at 18:04):

oh, and slice categories come in here somewhere too >:/

Reid Barton (Apr 02 2020 at 18:05):

I like to think of $X \mapsto \mathrm{Sh}(X)$ as a souped up version of sending $X$ to the ring of continuous functions on $X$. Now they are Set-valued functions, and they can be evaluated on points of $X$ (producing stalks), but there's some extra "stuff" (no longer a property, like continuity) saying how the stalks relate.

Reid Barton (Apr 02 2020 at 18:06):

Beware that I have not tested this analogy in any serious way, I just like the way it sounds.

sarahzrf (Apr 02 2020 at 18:06):

i think i have heard that you can redefine things like sheaves as functors between ∞-groupoids or something

sarahzrf (Apr 02 2020 at 18:07):

like the 0-cells of your ∞-groupoid are the points, you send them to stalks; 1-cells are paths, you send them to isos between stalks; 2-cells are homotopies of paths, you send them to coherences or something? idk

Dan Doel (Apr 02 2020 at 18:08):

I've been told that some of this is somehow related to locales vs. frames, in ways that I don't understand enough to say. Like, you should distinguish between a topos and a category of sheaves like you distinguish between a locale and its frame of opens (and they are related by $^{op}$).

sarahzrf (Apr 02 2020 at 18:08):

wait, srsly?

sarahzrf (Apr 02 2020 at 18:08):

that's cute

Dan Doel (Apr 02 2020 at 18:08):

Maybe someone else can explain that to you. :)

sarahzrf (Apr 02 2020 at 18:09):

no i think i get it hahaha

Dan Doel (Apr 02 2020 at 18:09):

(And re-explain it to me.)

Reid Barton (Apr 02 2020 at 18:09):

The trouble is that because left adjoints, well, have right adjoints, it's so tempting to just call the right adjoint the "morphism"

sarahzrf (Apr 02 2020 at 18:09):

it sounds to me like they're saying that "topos" has been defined the wrong way, with the arrows going the wrong way

sarahzrf (Apr 02 2020 at 18:10):

and that we should think of the objects we currently refer to as "topoi" as formal duals of the things that really deserve to be named after the greak word for "place"

Dan Doel (Apr 02 2020 at 18:10):

Yeah, probably.

Reid Barton (Apr 02 2020 at 18:10):

I think Joyal calls this a "logos".

sarahzrf (Apr 02 2020 at 18:11):

(not to mention that a frame is a (0, 1)-grothendieck topos, so what's a (1, 1)-locale...?)

Dan Doel (Apr 02 2020 at 18:12):

Well, probably the locale should be the topos, and the (0,1)-grothendieck topos should be called something else?

Dan Doel (Apr 02 2020 at 18:12):

The logos?

sarahzrf (Apr 02 2020 at 18:12):

no i mean at level (0, 1) we have frames and locales

sarahzrf (Apr 02 2020 at 18:13):

the corresponding terms at level (1, 1) are "grothendieck topos" and... nothing, we don't talk about the formal duals of those

Dan Doel (Apr 02 2020 at 18:13):

Oh, yeah, sorry. The (1,1)-frame is the logos.

Reid Barton (Apr 02 2020 at 18:13):

I think the table is

frame   locale
logos   topos
ring    affine scheme

Reid Barton (Apr 02 2020 at 18:14):

However in the last line we have the handy notation Spec for the antiequivalence between the two columns, while in the second line (and the first I guess) people don't write anything.

sarahzrf (Apr 02 2020 at 18:14):

hmm, im going out on a real limb by saying "we don't talk about the formal duals of those"

sarahzrf (Apr 02 2020 at 18:14):

as if i know what topos theorists talk about

sarahzrf (Apr 02 2020 at 18:14):

using "we" as if im part of that group

sarahzrf (Apr 02 2020 at 18:14):

lmao

Reid Barton (Apr 02 2020 at 18:19):

So where in algebraic geometry we have for an affine scheme
$X = \operatorname{Spec} \Gamma(X, \mathcal{O}_X)$
we should write in topos theory
$X = \operatorname{Spec} \mathrm{Sh}(X)$
where "Spec" just denotes passing to the opposite category; and now it's okay to identify the topos $X$ on the left with the (sober) topological space $X$ on the right.

Reid Barton (Apr 02 2020 at 18:19):

But instead people just regard Sh(X) as living in the opposite category, and don't write Spec.

Reid Barton (Apr 02 2020 at 18:28):

https://ncatlab.org/nlab/show/logos

sarahzrf (Apr 02 2020 at 20:33):

u know i think this topic has made a couple things click for me 🤯

sarahzrf (Apr 02 2020 at 20:35):

like....... i swear to god i think i see that a sheaf is a categorification of an open subset @__@

sarahzrf (Apr 02 2020 at 20:37):

i think it should be more standard to recommend poring over the correspondences between (0, 0) and (0, 1) and (1, 1), it's helped me a ton

sarahzrf (Apr 02 2020 at 20:39):

nPOV for n > (1, 1) gets too much focus!!! more representation for nPOV with n < (1, 1) nowww

Matteo Capucci (he/him) (Apr 02 2020 at 20:51):

Reid Barton said:

I think Joyal calls this a "logos".

I knew logos as Heyting categories, but this usage makes much much more sense!!

Gershom (Apr 02 2020 at 21:09):

to the original question, the way i think of as "petit" and "gros" is say you have a space and want to think of the category of sheaves over that space, then its still sort of "stuff regarding that space" and you have a petit topos. Now, suppose you have a "nice category of spaces" and show that it itself is a topos, then that is a "gros" topos. There is a lot of rather mystifying to me stuff about how one can relate the two notions, and give this intuition some formal weight or talk about how you might have a "gros" topos where the objects themselves can be thought of as "petit" toposes (which I would think could be fleshed-out in stacky/2-categorical approaches), but that's the general sense at least.

sarahzrf (Apr 02 2020 at 21:13):

its too late ive already become convinced that i was right with my initial analysis :upside_down:

sarahzrf (Apr 02 2020 at 21:14):

(but i guess it's not really very different from what you said anyway)

Morgan Rogers (he/him) (Apr 03 2020 at 09:15):

Reid Barton said:

Topos theorists like to identify a topological space X (maybe with some extra adjectives) with its category of sheaves Shv(X)--that's why they consider morphisms to go in (unpopular opinion) the "wrong" direction.

They are called geometric morphisms...

Morgan Rogers (he/him) (Apr 03 2020 at 09:21):

sarahzrf said:

using "we" as if im part of that group

Come join us :heart_kiss:

Morgan Rogers (he/him) (Apr 03 2020 at 09:28):

Reid Barton said:

Topos theorists like to identify a topological space X (maybe with some extra adjectives) with its category of sheaves Shv(X)--that's why they consider morphisms to go in (unpopular opinion) the "wrong" direction.

It would be more accurate to say that we identify locales with their toposes of sheaves. Which we can do because the 2-category of locales (internal to Set) is equivalent to the 2-category of localic toposes over Set :innocent: We can recover a locale from a localic topos from its subterminal objects.

Morgan Rogers (he/him) (Apr 03 2020 at 09:38):

Gershom said:

to the original question, the way i think of as "petit" and "gros" is say you have a space and want to think of the category of sheaves over that space, then its still sort of "stuff regarding that space" and you have a petit topos. Now, suppose you have a "nice category of spaces" and show that it itself is a topos, then that is a "gros" topos. There is a lot of rather mystifying to me stuff about how one can relate the two notions, and give this intuition some formal weight or talk about how you might have a "gros" topos where the objects themselves can be thought of as "petit" toposes (which I would think could be fleshed-out in stacky/2-categorical approaches), but that's the general sense at least.

In the same vein, to get the petit topos corresponding to a space (object) in a gros topos, one simply takes the slice over that object.
For intuition about this, suppose I have a nice category C of spaces which isn't too big (eg has a set of objects). Then I have a natural Grothendieck topology J on it that identifies covering collections of morphisms, so (C,J) is a site for the Grothendieck topos Sh(C,J), which is the gros topos for those spaces. Alternatively, I can try to build a topos by other means to act as this gros topos. Lawvere, Johnstone and company have put a lot of effort into finding such toposes, or at least identifying the properties they should have in order to be able to bring geometric techniques to bear on them.
Taking the slice at a representable object X, I get the petit topos for that space which in particular contains all of the spaces which lay over X in my original category. But I can think of any object of Sh(C,J) as a space built (via suitable colimits) from those in my original category, and can take slices over them to get the corresponding petit toposes which embody the properties of these "imaginary spaces".

Reid Barton (Apr 03 2020 at 14:02):

Yes, it's quite appreciated that there is this reminder "geometric" that the morphism goes the wrong way :upside_down:

Reid Barton (Apr 03 2020 at 14:03):

and the real issue is not with the direction of the morphisms between topoi, but with the identification "a topos is a category with blah blah properties".

Reid Barton (Apr 03 2020 at 14:04):

But "a topos is not a category" is probably an even more unpopular opinion.

Morgan Rogers (he/him) (Apr 03 2020 at 14:10):

Care to explain what you mean?

Morgan Rogers (he/him) (Apr 03 2020 at 14:12):

Because I would really like there to be a clearer distinction between "unpopular opinion" and "nonsense" :rolling_on_the_floor_laughing:

Reid Barton (Apr 03 2020 at 14:27):

A logos/category of sheaves/Grothendieck topos (as a category) is like a ring. (Small) colimits play the role of addition, finite limits play the role of multiplication. A morphism is a functor which preserves small colimits and finite limits, namely the left adjoint part of a geometric morphism.

Reid Barton (Apr 03 2020 at 14:27):

It should be apparent a priori that "right adjoint whose left adjoint preserves finite limits" isn't a great definition.

Reid Barton (Apr 03 2020 at 14:28):

Nobody would say a morphism in the category of rings from A to B is a ring homomorphism from B to A. But because left adjoints happen to have right adjoints which go in the opposite direction, it is possible to do this in the context of topoi.

Reid Barton (Apr 03 2020 at 14:39):

In general there are many places in category theory where it's clear that, when considering categories which are built out of Set or otherwise resemble Set, it's better to regard an adjunction between such categories as a morphism in the direction of the left adjoint. The most basic example being presheaves on a category A as the free cocompletion of A.

Dan Doel (Apr 03 2020 at 14:39):

Also it's the equivalent of saying, "a locale is not a completely distributive lattice." Or, "a set is not a complete atomic boolean algebra."

Dan Doel (Apr 03 2020 at 14:43):

But for the reasons Reid mentioned, people associated the name with the wrong thing historically.

Dan Doel (Apr 03 2020 at 14:48):

@Reid Barton Are things not backwards for the gros topos case? I.E. should one think of 'a generalized category of sets' as being a logos?

Dan Doel (Apr 03 2020 at 14:50):

Or do the morphisms of toposes not really matter as much in that scenario?

Morgan Rogers (he/him) (Apr 03 2020 at 15:08):

Reid Barton said:

It should be apparent a priori that "right adjoint whose left adjoint preserves finite limits" isn't a great definition.

But a geometric morphism isn't the right adjoint functor, it's the whole adjunction! We just choose the direction to match the direction of the things that actually generate geometric morphisms (originally continuous maps, but also functors between small categories for presheaf toposes).

Morgan Rogers (he/him) (Apr 03 2020 at 15:13):

Dan Doel said:

Also it's the equivalent of saying, "a locale is not a completely distributive lattice." Or, "a set is not a complete atomic boolean algebra."

Okay, we often give the objects of a dual category different names for concision/clarity, but saying that an object in the dual category is fundamentally other is a stretch, isn't it?

Morgan Rogers (he/him) (Apr 03 2020 at 15:16):

I'm happy to concede that the "logos" naming convention is a good idea, but that won't stop me from talking about objects of a topos in the same way that I talk about open sets of a topological space.

Cody Roux (Apr 03 2020 at 15:27):

I think there's a lot of value to objectively false but "morally true" statements

Dan Doel (Apr 03 2020 at 15:28):

I don't know. do you actually think of a set as being the same thing as a CABA? That's saying that a set is the same thing as its lattice of subsets. That doesn't seem to be the way people think about them at all.

Morgan Rogers (he/him) (Apr 03 2020 at 15:29):

I certainly take for granted the presence of subsets and their behaviour, and make arguments about sets that employ those, so I suppose I implicitly do, yes!

Dan Doel (Apr 03 2020 at 15:31):

So, you don't think a set is a collection of points.

Morgan Rogers (he/him) (Apr 03 2020 at 15:32):

I smell a false dichotomy here. You're saying I'm only allowed to think of a set either in terms of its elements or in terms of its subsets? Clearly it's both!

Morgan Rogers (he/him) (Apr 03 2020 at 15:34):

I choose my morphisms of sets to go in the direction described by mappings of points because there's less data to specify, though, and that direction was simply inherited by topology and then topos theory

Morgan Rogers (he/him) (Apr 03 2020 at 15:38):

Of course, for toposes with enough points, the same reasoning about the direction also works. Otherwise one has to look at points valued in a topos with enough models of a theory classified by your "domain" topos, but since many mathematicians care more about internal models of things than the toposes themselves (I can justify this if someone wants to pick me up on it), I think this is a good argument for the choice of direction too.

sarahzrf (Apr 04 2020 at 04:24):

Morgan Rogers said:

I'm happy to concede that the "logos" naming convention is a good idea, but that won't stop me from talking about objects of a topos in the same way that I talk about open sets of a topological space.

okay, but how about "elements of a frame"?

sarahzrf (Apr 04 2020 at 04:24):

would you say "elements of a locale" to mean the same thing?

sarahzrf (Apr 04 2020 at 04:25):

suppose i said "cantor space forms a locale. let's pick an element of it..."—what would you assume i mean?

sarahzrf (Apr 04 2020 at 04:26):

if you say "well, i would say a point of cantor space" then you are dodging the question :-)

sarahzrf (Apr 04 2020 at 04:27):

here's a better argument, btw:

sarahzrf (Apr 04 2020 at 04:27):

what is the product of two locales? what is the product of two frames?

Morgan Rogers (he/him) (Apr 04 2020 at 10:02):

As I said above, we use different names for the objects in the dual of a category for concision/clarity about which way around the morphisms go; in your last example the universal property of the product is obviously dependent on which way the morphisms go and the names tell me which it is. That typically extends to the data too: an element of a frame is an open of a locale; a completely prime filter of a frame is a point of a locale etc. But using different names doesn't impose a concrete separation between a category and its opposite. If I perform a construction in a set or in a locale, I choose one of the two terminological perspectives for consistency, but I can always express that argument from the other perspective if someone forces me to.

Morgan Rogers (he/him) (Apr 04 2020 at 10:06):

sarahzrf said:

suppose i said "cantor space forms a locale. let's pick an element of it..."—what would you assume i mean?

I would assume you're talking about a point, because Cantor space is a pretty classical space from before people thought so hard about open sets as frames. It's easy to take for granted how modern the locale/frame perspective is! But I would recommend that you refer to an element as a point if you insist on calling the space a locale :stuck_out_tongue_wink: