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Stream: learning: questions

Topic: Motivating Sheaves with Locality

John Onstead (Nov 04 2024 at 12:21):

The first thing we learn about motivating sheaves is that they are helpful in some ill-defined class of local-global problems. But now I want to make this more specific and figure out the precise class of local-global problems that sheaves solve. To start off, in a previous discussion I learned that the local behavior of continuous or smooth sections (including Taylor series for the latter) for a bundle $E \rightarrow B$ can be extracted from the germs of those sections at a point. In turn, the germs themselves come from the sheaf of sections $\mathsf{Sh}_E$, where we can take a "stalk" construction at a point to get the set/space of all germs of functions at that point $\mathsf{Sh}_E(p)$.

But here's my problem: the stalk construction that gives us germs is a direct limit construction that can be done for ANY functor $F: \mathsf{Open}(B)^{op} \rightarrow \mathsf{Set}$ (or $\mathsf{Ring}$, etc.), not just sheaves. In fact, so long as the target category has all colimits you can do this construction. So it seems, at least on the surface, that the local behavior of a section of $E$ can be obtained without reference to the fact that the functor $\mathsf{Sh}_E$ is a sheaf. Is this truly the case? And if so, is there something special about stalks of sheaves compared to stalks of non-sheaves that highlights their importance?

Chris Grossack (they/them) (Nov 04 2024 at 15:17):

I think you're right to observe that this works for presheaves, and thus that this isn't one of the problems that sheaf theory solves.

Indeed, in taking a section of a presheaf (over $U$, say) and passing to its germ at $x$, we've made the data more local. We took something defined on $U$ and restricted it to get something defined only at a point $x$. In general, we can pass from "more global" to "more local" easily, by restriction (as we basically did here).

Sheaves let you solve the opposite problem. If you have local data in multiple places, when can you "glue" that local data together in order to get something globally defined. There's lots of examples, but here's one relating explicitly to stalks:

Theorem: If $\mathcal{F}$ is a sheaf and $\varphi$ is a geometric formula about $\mathcal{F}$ (and its elements) then $\varphi$ is true at the stalk $x$ if and only if $\varphi$ is true in some neighborhood of $x$!

So we see sheaf-ness lets geometric truth "spread out". By checking the property at a single point, we learn for free that the property holds in some open neighborhood! So we've moved from "more local" (truth at a point) to "more global" (truth in a neighborhood). This is great, because moving from local to global is generally quite hard!

Chris Grossack (they/them) (Nov 04 2024 at 15:23):

For example, if $\mathcal{F}$ is a sheaf of rings (for instance, the sheaf of continuous $\mathbb{R}$-valued functions) then "$f$ is invertible" is a geometric question, since we can write it as $\exists g . fg = 1$.

The theorem tells us that if "$f$ is invertible" holds at a point $x$, then it must hold in a neighborhood of $x$, but this compiles down to the usual statement that if $f(x) \neq 0$ then actually $f$ is nonzero in a neighborhood of $x$.

Mike Shulman (Nov 04 2024 at 15:57):

Another manifestation of the same idea is that a sheaf is determined by its stalks in a certain sense, while a presheaf is not. Specifically, for a topological space $B$, the category of sheaves on $B$ is equivalent to the full subcategory of $\mathrm{Top}/B$ spanned by the local homeomorphisms, and in the local homeomorphism $E\to B$ corresponding to a sheaf $X$, the fiber over $b\in B$ is precisely the stalk at $b$. So if you remember all the stalks at all points, together with an induced topology on them, you can recover the original sheaf (as the sheaf of sections of this local homemorphism).

This is not true for an arbitrary presheaf. In fact, it is true for a presheaf precisely when that presheaf is a sheaf! See for instance this post by @Tom Leinster .

John Baez (Nov 04 2024 at 16:36):

John Onstead said:

But here's my problem: the stalk construction that gives us germs is a direct limit construction that can be done for ANY functor $F: \mathsf{Open}(B)^{op} \rightarrow \mathsf{Set}$ (or $\mathsf{Ring}$, etc.), not just sheaves.

Right - in our course on topos theory here on Zulip, @David Egolf showed that:

• from any presheaf on a space $B$, i.e. any functor

$F: \mathsf{Open}(B)^{op} \rightarrow \mathsf{Set}$,

you can form a bundle $p: \Lambda(F) \to B$ whose fiber at $b \in B$ is the set of all germs of $F$ at $B$.

• $p: \Lambda(F) \to B$ is an etale space, i.e. a local homeomorphism.

• Conversely, any bundle $p: E \to B$ gives a presheaf on $B$ called its presheaf of sections, $\Gamma(E)$... but this is actually a sheaf!

• The functor $\Lambda$ sending presheaves to bundles is left adjoint to the functor $\Gamma$ sending bundles to presheaves.

• The composite functor $\Gamma \circ \Lambda$ is called sheafification because it sends any presheaf to a sheaf.

Indeed sheaves on $B$ are a reflective subcategory of presheaves on $B$ and $\Gamma \circ \Lambda$ is the reflector.

John Onstead (Nov 04 2024 at 19:37):

Chris Grossack (they/them) said:

I think you're right to observe that this works for presheaves, and thus that this isn't one of the problems that sheaf theory solves.

Indeed, in taking a section of a presheaf (over $U$, say) and passing to its germ at $x$, we've made the data more local. We took something defined on $U$ and restricted it to get something defined only at a point $x$. In general, we can pass from "more global" to "more local" easily, by restriction (as we basically did here).

Sheaves let you solve the opposite problem. If you have local data in multiple places, when can you "glue" that local data together in order to get something globally defined. There's lots of examples, but here's one relating explicitly to stalks

Thanks, this is really helpful! Let me see if I understand. So while it's easy to go from global to local, sheaves help when going the other direction- from local to global. And I really liked the example of geometric formulas- it gives a really good example of a class of local to global problems that sheaves are well suited to study. It seems that whenever we have some functions or sections in Top (or related categories of spaces), sheaves are what allow for the extension of local properties (at least, when those properties are definable via a geometric formula) of these functions or sections to global properties, such as in the case of invertibility (or maybe also the value, derivatives, etc. when applicable). I hope I understood correctly!

John Onstead (Nov 04 2024 at 19:45):

Mike Shulman said:

This is not true for an arbitrary presheaf. In fact, it is true for a presheaf precisely when that presheaf is a sheaf! See for instance this post by Tom Leinster .

Wow, that blew my mind! I've always seen sheaves mentioned from the algebraic geometry perspective so I was not expecting them to come from a general categorical construction like "given $A \to B$ with $B$ having small colimits, take $\mathrm{Psh}(A)$, find the corresponding adjunction to a functor $\mathrm{Psh}(A) \to B$, then restrict the adjunction to an equivalence"!

John Onstead (Nov 04 2024 at 19:48):

John Baez said:

Right - in our course on topos theory here on Zulip, David Egolf showed that:

• from any presheaf on a space $B$, i.e. any functor

$F: \mathsf{Open}(B)^{op} \rightarrow \mathsf{Set}$,

you can form a bundle $p: \Lambda(F) \to B$ whose fiber at $b \in B$ is the set of all germs of $F$ at $B$.

• $p: \Lambda(F) \to B$ is an etale space, i.e. a local homeomorphism.

• Conversely, any bundle $p: E \to B$ gives a presheaf on $B$ called its presheaf of sections, $\Gamma(E)$... but this is actually a sheaf!

• The functor $\Lambda$ sending presheaves to bundles is left adjoint to the functor $\Gamma$ sending bundles to presheaves.

• The composite functor $\Gamma \circ \Lambda$ is called sheafification because it sends any presheaf to a sheaf.

Indeed sheaves on $B$ are a reflective subcategory of presheaves on $B$ and $\Gamma \circ \Lambda$ is the reflector.

Ah, that helps put everything together nicely! I didn't know etale spaces corresponded to local homeomorphisms, that's good to know as well!

John Onstead (Nov 04 2024 at 19:53):

There's a lot of potential ways to follow this up... I'll think more about that and come back in a bit!

John Baez (Nov 04 2024 at 21:08):

Sure. 'Etale space' is just a weird name for 'local homeomorphism'. More precisely it's a name for the domain of a local homeomorphism. But you can't just look at a space and tell if it's an etale space: it's the map that decides that!

Mike Shulman (Nov 04 2024 at 21:20):

Because of that, I wouldn't say that an etale space is a name for the domain, but rather a name for the space equipped with the map.

John Baez (Nov 04 2024 at 21:35):

Yes, better. I realized there are other examples of such linguistic trickery: 'total space' and 'base space'.

John Baez (Nov 04 2024 at 21:36):

Never ask "Is this space base?"

John Onstead (Nov 04 2024 at 21:47):

Here's my next question... in order to learn more about the full range of problems sheaves can be applied to, I'll need to go beyond topology. (We'll return to topology later since I still have many questions to ask about that.) As mentioned above, sheaves on topological spaces can be understood in terms of Leinster's adjunction restriction construction, and this in turn gives rise to the notions of sheafification and the duality between etale spaces/local homeomorphisms and sheaves.

What I want to know is: how general is this situation? For instance, the most general setting for discussion of sheaves is that of sheaves over a site (a category equipped with a notion of coverage or Grothendieck topology). Can I always guarantee that an analogue of sheafifcation and the etale space-sheaf correspondence holds in this most general of cases? In particular, given some (small) site $C$, is there always some functor $C \to D$ (into some $D$ with of course small colimits) such that the corresponding $\mathrm{Psh}(C) \to D$ is an adjunction that restricts to an equivalence of form $\mathrm{Sh}(C) \to E$?

John Baez (Nov 04 2024 at 21:54):

I don't know the answer to that. Maybe someone here does.

I'd prefer to say this: the category of sheaves on any site is always a [[reflective subcategory]] of the category of presheaves on that site, so there's a reflector called 'sheafification' that turns presheaves into sheaves.

In fact there's a really slick way to talk about sheaf categories that sidesteps the stuff about Grothendieck topologies and just describes them as reflective subcategories of presheaf categories with a nice extra property!

Mike Shulman (Nov 04 2024 at 21:59):

John Onstead said:

given some (small) site $C$, is there always some functor $C \to D$ (into some $D$ with of course small colimits) such that the corresponding $\mathrm{Psh}(C) \to D$ is an adjunction that restricts to an equivalence of form $\mathrm{Sh}(C) \to E$?

Yes: take $D = \mathrm{Sh}(C)$. (-:O

Mike Shulman (Nov 04 2024 at 22:00):

But no, for a general site there isn't a really satisfying analogue of "etale space" that acts this way. For posetal sites you can replace Top by the category of [[locales]]. But for an arbitrary site you have to basically use the (2-)category of toposes, and then it kind of feels like begging the question.

John Onstead (Nov 04 2024 at 22:45):

John Baez said:

I'd prefer to say this: the category of sheaves on any site is always a [[reflective subcategory]] of the category of presheaves on that site, so there's a reflector called 'sheafification' that turns presheaves into sheaves.

That makes sense, I guess I was just curious about how to generalize the observation made in your blog that sheafification can be thought of as taking the etale space followed by taking the sheaf of the etale space. In that case it seems that the usual direct notion of sheafification between the sheaf and presheaf categories (such as you mention here) could "factor through" (in a particular way) an intermediate notion, that of bundles over a topological space.

John Onstead (Nov 04 2024 at 22:49):

Mike Shulman said:

For posetal sites you can replace Top by the category of [[locales]]. But for an arbitrary site you have to basically use the (2-)category of toposes, and then it kind of feels like begging the question.

I'm really curious about how this would work! If your category can be made into a site, you can do sheaf theory "on it", but only for some special categories like Top can you do sheaf theory "inside" it. That is, you can define some notion of "decomposition" of an object in this special kind of category (like the decomposition of a topological space into its open sets via the generation of its category of opens) such that sheaf and sheaf-like constructions make sense and are useful in some way. Is what you are saying imply that the category of locales is also another category like Top with this special property? If so, what exactly is this special property (of being able to do sheaf theory "inside of" the category in a way analogous to Top) and what classes of categories possess it?

John Baez (Nov 04 2024 at 23:31):

John Onstead said:

That makes sense, I guess I was just curious about how to generalize the observation made in your blog that sheafification can be thought of as taking the etale space followed by taking the sheaf of the etale space.

Yeah, I don't know how to do that, and Mike seems to be saying you can't.

Mike Shulman (Nov 04 2024 at 23:44):

For locales, it works basically like for topological spaces: the only real difference is that the colimits in Loc aren't just topologies on an underlying set of germs, but may not even have very many "points" at all.

Mike Shulman (Nov 04 2024 at 23:45):

A local homeomorphism of locales is defined in a fairly straightforward way generalizing a local homeomorphism of spaces, and you get again that sheaves on a locale $X$ are equivalent to local homeomorphisms with codomain $X$. Moreover, if a locale "is" a topological space, then so is the domain of any local homeomorphism over it, so this generalizes the topological space version directly.

Mike Shulman (Nov 04 2024 at 23:47):

For toposes it feels more tautological, because a local homeomorphism of toposes $\mathcal{Y} \to \mathcal{X}$ is essentially defined to be one that identifies its domain $\mathcal{Y}$ with a slice topos $\mathcal{X}/F$ of its codomain. So "of course" the category of such local homemorphisms, for fixed $\mathcal{X}$, is equivalent to $\mathcal{X}$ itself qua category, and hence to the category of sheaves on any site presentation of $\mathcal{X}$.

Mike Shulman (Nov 04 2024 at 23:48):

I wouldn't be surprised if someone had tried to define and study "categories in which you can do this" more abstractly, but I don't off the top of my head know of such.

John Baez (Nov 05 2024 at 00:35):

Mike Shulman said:

For toposes it feels more tautological, because a local homeomorphism of toposes $\mathcal{X} \to \mathcal{X}$ is essentially defined to be one that identifies its domain $\mathcal{Y}$ with a slice topos $\mathcal{X}/F$ of its codomain.

You meant $\mathcal{Y} \to \mathcal{X}$.

John Onstead (Nov 05 2024 at 00:42):

Mike Shulman said:

I wouldn't be surprised if someone had tried to define and study "categories in which you can do this" more abstractly, but I don't off the top of my head know of such.

There's certainly an enormous amount to unpack here, and it's something I think is worth doing. I'll have to think about this a bit more!
But before we move on to this kind of topic I want to review and make sure we don't lose sight of what got this conversation started: the claim that sheaves are the presheaves that are, in some sense, fully defined by their stalks. So perhaps I should have tried to define what a "stalk" is for a presheaf for a general category (site). (After all the etale spaces/local homeomorphisms/etc were just the objects we had to be able to "string together" all the stalks of a sheaf into one single object.) This seems tricky because, in a sense, the direct limit definition of a stalk relies on the posetal nature of the category of open sets, but maybe it can be done? But if it can't, then it seems clear that the local to global nature of sheaves as discussed above isn't directly dependent on being able to reconstruct a sheaf from its stalks. Instead, the local to global nature of a sheaf entails being able to do more than that, and it just so happens that in the special case of topological spaces (and similar categories) this local to global nature can be specialized and used to conclude that we can reconstruct the sheaf from its stalks. This is certainly relevant to my main quest here, which is to find the most general class of local-global problems sheaves help solve: this would clarify that extending stalks to sheaves is just one subset of this class, but not the entirety of the class itself.

John Onstead (Nov 05 2024 at 00:48):

John Baez said:

Sure. 'Etale space' is just a weird name for 'local homeomorphism'. More precisely it's a name for the domain of a local homeomorphism. But you can't just look at a space and tell if it's an etale space: it's the map that decides that!

While I think more about the above, I also want to consider a surprising realization I just had based on this quote. A topological space is considered a manifold if it has a local homeomorphism to $\mathbb{R}^n$, right? And an etale space is the domain/total space of a local homeomorphism (perhaps also equipped with that map). Does that mean that topological manifolds are really just the etale spaces for sheaves defined on the category of open sets of $\mathbb{R}^n$?

John Baez (Nov 05 2024 at 00:56):

John Onstead said:

John Baez said:

Sure. 'Etale space' is just a weird name for 'local homeomorphism'.

While I think more about the above, I also want to consider a surprising realization I just had based on this quote. A topological space is considered a manifold if it has a local homeomorphism to $\mathbb{R}^n$, right?

Not in the same sense as we're using here! A local homeomorphism $f: X \to Y$ between topological spaces is a continuous map such that each $x \in X$ has an open neighborhood $U$ such that $f|_U : U \to f(U)$ is a homeomorphism.

So, on the one hand most manifolds don't admit a local homeomorphism to $\mathbb{R}^n$. For example there is no local homeomorphism from the circle to the line.

On the other hand, there are lots of topological spaces that admit a local homeomorphism to $\mathbb{R}^n$ but aren't manifolds. For example, the etale space of the sheaf of continuous functions on $\mathbb{R}$ is not a manifold.

John Onstead (Nov 05 2024 at 01:07):

Ah, I got confused by the terminology. Usually we say that two spaces are homeomorphic if there exists a homeomorphism between them. So naturally I reasoned that a space is locally homeomorphic to another if there exists a local homeomorphism. Naming conventions in math are never straightforward!
In any case, I thought I remembered seeing somewhere that a manifold can be defined via sheaves in some way and I thought this was how it was done, but maybe I'm still not at that point where I can understand that yet! (And I know Smooth manifolds are smooth sets and smooth sets are indeed sheaves on a site, thus also making smooth manifolds sheaves, but I'm referring here to topological manifolds specifically)

John Baez (Nov 05 2024 at 01:07):

Mike Shulman said:

For toposes it feels more tautological, because a local homeomorphism of toposes $\mathcal{Y} \to \mathcal{X}$ is essentially defined to be one that identifies its domain $\mathcal{Y}$ with a slice topos $\mathcal{X}/F$ of its codomain. So "of course" the category of such local homeomorphisms, for fixed $\mathcal{X}$, is equivalent to $\mathcal{X}$ itself qua category, and hence to the category of sheaves on any site presentation of $\mathcal{X}$.

I wish I understood this. Suppose $\mathcal{Y}$ and $\mathcal{X}$ were topoi of sheaves on topological spaces and $\mathcal{Y} \to \mathcal{X}$ came from a continuous map between those spaces. Then I guess that map is a local homeomorphism in the usual sense if $\mathcal{Y}$ is equivalent to a slice category $\mathcal{X}/F$ somehow. But I don't see why this slice category is like the the category of sheaves on the etale space of some sheaf. (Is that what's supposed to be happening?)

Mike Shulman (Nov 05 2024 at 02:10):

John Onstead said:

So perhaps I should have tried to define what a "stalk" is for a presheaf for a general category (site)... This seems tricky because, in a sense, the direct limit definition of a stalk relies on the posetal nature of the category of open sets, but maybe it can be done?

There is a straightforward definition of the stalks of a sheaf on an arbitrary site, i.e. an object of an arbitrary topos. But first you have to define what a "point" is of an arbitrary topos. The definition is that a "point" of a topos $\mathcal{X}$ is a geometric morphism $\mathrm{Set} \to \mathcal{X}$. Since $\mathrm{Set} = \mathrm{Sh}(*)$ is the topos of sheaves on a 1-point space, and the category of (sober) spaces embeds fully-faithfully in the category of topoi, this is a faithful generalization (up to isomorphism) of the notion of "point" of a topological space. Now the definition of a stalk is immediate: given a point $x:\mathrm{Set} \to \mathcal{X}$, the stalk of $F\in\mathcal{X}$ at $x$ is $x^*(F)$, where $x^* : \mathcal{X} \to \mathrm{Set}$ is the inverse image part of the geometric morphism $x$.

However, you're also right that a sheaf on an arbitrary site/topos is not in general determined by its stalks, because a general topos may not have very many points. In fact, there are nontrivial toposes that have no points at all. So in the context of general toposes (and even general locales), the "local to global" nature of a sheaf is better understood as passing from infomation over smaller "open parts" to their "union", rather than from information over "points" to the set they constitute.

Mike Shulman (Nov 05 2024 at 02:13):

John Onstead said:

And I know Smooth manifolds are smooth sets and smooth sets are indeed sheaves on a site, thus also making smooth manifolds sheaves, but I'm referring here to topological manifolds specifically.

I think just as smooth manifolds are a subcategory of the topos of sheaves on the site of open subsets of $\mathbb{R}^n$ and smooth maps, topological manifolds are a subcategory of the topos of sheaves on the site of open subsets of $\mathbb{R}^n$ and continuous maps.

Mike Shulman (Nov 05 2024 at 02:20):

John Baez said:

Suppose $\mathcal{Y}$ and $\mathcal{X}$ were topoi of sheaves on topological spaces and $\mathcal{Y} \to \mathcal{X}$ came from a continuous map between those spaces. Then I guess that map is a local homeomorphism in the usual sense if $\mathcal{Y}$ is equivalent to a slice category $\mathcal{X}/F$ somehow.

Yes, that's right. A continuous map $Y\to X$ of spaces is a local homeomorphism if and only if the corresponding geometric morphism $\mathrm{Sh}(Y) \to \mathrm{Sh}(X)$ is equivalent, in the 2-category $\mathrm{Topos}/\mathcal{X}$, to the geometric morphism $\mathrm{Sh}(X)/F \to \mathrm{Sh}(X)$ induced by some object $F\in \mathrm{Sh}(X)$. And in this case, $F$ is the sheaf of sections of $Y$, and thus $Y$ (since it is etale over $X$) is the etale space corresponding to $F$.

But I don't see why this slice category is like the category of sheaves on the etale space of some sheaf. (Is that what's supposed to be happening?)

I think you're asking, given an object $F\in \mathrm{Sh}(X)$ of a sheaf topos on a topological space, why is the slice category $\mathrm{Sh}(X)/F$ equivalent to the category of sheaves on the etale space of $F$?

If you're willing to accept as known that $\mathrm{Sh}(X)$ is equivalent to the category $\mathrm{LH}/X$ of etale spaces over $X$, then this has a simple proof: $(\mathrm{LH}/X)/Y \simeq \mathrm{LH}/Y$, just as always for slice categories. (We have to use the fact that local homemorphisms are stable under composition, and that any map in $\mathrm{Top}/X$ between etale spaces is itself a local homeomorphism.)

Mike Shulman (Nov 05 2024 at 02:24):

In case anyone wants to read more about this stuff, there's a discussion in the case of locales in section C1.3 of Sketches of an Elephant.

John Onstead (Nov 05 2024 at 12:18):

Mike Shulman said:

I think just as smooth manifolds are a subcategory of the topos of sheaves on the site of open subsets of $\mathbb{R}^n$ and smooth maps, topological manifolds are a subcategory of the topos of sheaves on the site of open subsets of $\mathbb{R}^n$ and continuous maps.

Ah, that makes sense!

Mike Shulman said:

Now the definition of a stalk is immediate: given a point $x:\mathrm{Set} \to \mathcal{X}$, the stalk of $F\in\mathcal{X}$ at $x$ is $x^*(F)$, where $x^* : \mathcal{X} \to \mathrm{Set}$ is the inverse image part of the geometric morphism $x$.

That's cool- I never really stopped to think about a topos as having "points" before (well, aside from the objects of the topos itself, since objects are like the "points" of a category). The inverse image part is interesting, I'll work out later how that might relate to the usual direct limit definition in topological spaces.

Mike Shulman said:

However, you're also right that a sheaf on an arbitrary site/topos is not in general determined by its stalks, because a general topos may not have very many points. In fact, there are nontrivial toposes that have no points at all. So in the context of general toposes (and even general locales), the "local to global" nature of a sheaf is better understood as passing from infomation over smaller "open parts" to their "union", rather than from information over "points" to the set they constitute.

This is very insightful! So the local to global problems where sheaves are used concern extending information defined on certain kinds of local parts of an object known as "open parts", and in some cases where points exist, the information defined at a point fits in with this general scheme via the stalk construction.

John Onstead (Nov 05 2024 at 12:21):

So all roads lead to trying to define what exactly is meant by an "open part" of an object, since it seems that's relevant for sheaves. Conversely, it means that sheaves might fail to work properly when the parts of an object being considered are not "open"- so let's think about one such case!
Returning to the category Top, our question might be to ask: why opens? That is, why use the category of open sets over a space to define sheaves on that space? The topological reason is clear due to the special role that open sets play in topology, but from a more general abstract categorical POV, this choice of decomposition of a space into its open sets seems somewhat arbitrary. For instance, why can't we define a sheaf on a topological space to be a certain kind of contravariant set valued functor on the category of subspaces over that topological space? (Here, "category of subspace" could mean the general subobject poset, or more strongly could refer to regular monomorphisms). Maybe the answer to this is as simple as there being no "canonical" site structure on this category, but I'm not so sure about that. Both the category of opens and subspaces are posets with the morphisms as inclusions- therefore, shouldn't one be able to define a coverage on the category of subspaces in a direct analogy to how one is defined on opens? So my question is: why can't we define sheaves on the category of subspaces (or any other similar category), and must define them on the category of opens? What's so special about opens that sheaves fail for any other choice of space decomposition?

Mike Shulman (Nov 05 2024 at 15:06):

objects are like the "points" of a category

In the case of toposes, it's usually better to think of the objects of the topos as being more analogous to the open subsets of a topological space.

Mike Shulman (Nov 05 2024 at 15:30):

For instance, why can't we define a sheaf on a topological space to be a certain kind of contravariant set valued functor on the category of subspaces over that topological space? (Here, "category of subspace" could mean the general subobject poset, or more strongly could refer to regular monomorphisms).

The regular monos of topological spaces are those where the domain has the induced topology. Since any subset of a space has a unique induced topology, the poset of regular monos is equivalent to the powerset of the underlying set, so that would be equivalent to just equipping that set with the discrete topology.

Mike Shulman (Nov 05 2024 at 15:31):

Other choices might give something nontrivial, but you'll still lose the fact you have with open covers that the category of (sober) topological spaces and continuous functions embeds fully-faithfully in the category of toposes and geometric morphisms. Open sets are tied to that kind of intrinsically by the definition of "continuous function".

Mike Shulman (Nov 05 2024 at 17:21):

In general, I would say that (in this context, anyway) an "open part" of an object is just, by definition, a morphism belonging to a covering family in some site. (-:

Mike Shulman (Nov 05 2024 at 17:56):

Mike Shulman said:

In the case of toposes, it's usually better to think of the objects of the topos as being more analogous to the open subsets of a topological space.

To elaborate a little more on this: you can define $(n,1)$-toposes for any $n\in\mathbb{N}$ very analogously to ordinary (1-)toposes. An $(n,1)$-site is an $(n,1)$-category equipped with a Grothendieck topology, and the corresponding $(n,1)$-topos of $(n,1)$-sheaves is the $(n,1)$-category of functors from the site to the $(n,1)$-category of $(n-1)$-groupoids satisfying a gluing condition.

Now when $n=0$, a $(0,1)$-category is a poset, and a $(0,1)$-topos is precisely a locale, such as the locale (whose underlying frame consists) of open subsets of some topological space. So in this case the objects of the $(0,1)$-topos are precisely the "open parts" of a space in the classical sense, hence it makes some sense for higher $n$ to think of the objects of an $(n,1)$-topos qua $(n,1)$-category as its "open parts".

John Onstead (Nov 05 2024 at 21:45):

Mike Shulman said:

In the case of toposes, it's usually better to think of the objects of the topos as being more analogous to the open subsets of a topological space.

I see!

Mike Shulman said:

(sober) topological spaces and continuous functions embeds fully-faithfully in the category of toposes and geometric morphisms

This is unrelated but I thought I'd ask it here. I've never quite understood why the space needs to be sober to embed into locales and by extension toposes. Apparently, the sober spaces are the only ones where information is not lost when passing to the locale. But this doesn't make sense to me because topological spaces are literally defined in terms of open sets, so therefore, you'd expect the category of open sets (and thus the category of sheaves on that site) to contain all the information about the topological space. Yet this is only true for sober spaces? Why is it that the thing we use to completely define a topological space in the first place somehow only actually contains all the info of a space when it's sober and not in general?

John Onstead (Nov 05 2024 at 21:46):

Mike Shulman said:

The regular monos of topological spaces are those where the domain has the induced topology. Since any subset of a space has a unique induced topology, the poset of regular monos is equivalent to the powerset of the underlying set, so that would be equivalent to just equipping that set with the discrete topology.

I'm not sure I'm following the reasoning here. What does the fact that the poset of regular monos is equivalent to the powerset have to say about trying to define a sheaf on the category of regular monos? Does it disqualify the category from being able to have a Grothendieck topology defined on it in some way? And if so, why?

Mike Shulman (Nov 05 2024 at 21:52):

John Onstead said:

What does the fact that the poset of regular monos is equivalent to the powerset have to say about trying to define a sheaf on the category of regular monos? Does it disqualify the category from being able to have a Grothendieck topology defined on it in some way? And if so, why?

No, it just means that it doesn't carry any information about the topology of the space. The obvious Grothendieck topology to equip it with is the one induced by unions of subspaces. In this case, the resulting topos will be equivalent to $Set^X$, where $X$ is the underlying set of your topological space.

Mike Shulman (Nov 05 2024 at 21:54):

John Onstead said:

Apparently, the sober spaces are the only ones where information is not lost when passing to the locale. But this doesn't make sense to me because topological spaces are literally defined in terms of open sets, so therefore, you'd expect the category of open sets (and thus the category of sheaves on that site) to contain all the information about the topological space.

The definition of topological space refers to the lattice of open sets, but a topological space isn't just an abstract lattice (otherwise it would be a locale); it's a set $X$ together with a sublattice of the powerset of $X$.

For a very concrete example, take any set $X$ and equip it with the indiscrete topology. Then there are exactly two open sets no matter what $X$ is (as long as it's nonempty), so every nonempty indiscrete space has the same lattice of open sets (up to isomorphism), and hence the same induced locale and topos of sheaves. But of course indiscrete spaces with different cardinalities are not homeomorphic as topological spaces.

John Onstead (Nov 05 2024 at 22:41):

Mike Shulman said:

For a very concrete example, take any set $X$ and equip it with the indiscrete topology. Then there are exactly two open sets no matter what $X$ is (as long as it's nonempty), so every nonempty indiscrete space has the same lattice of open sets (up to isomorphism), and hence the same induced locale and topos of sheaves. But of course indiscrete spaces with different cardinalities are not homeomorphic as topological spaces.

Ah I see, that's a good example! Thanks!

Mike Shulman said:

No, it just means that it doesn't carry any information about the topology of the space. The obvious Grothendieck topology to equip it with is the one induced by unions of subspaces. In this case, the resulting topos will be equivalent to $Set^X$, where $X$ is the underlying set of your topological space.

Oh I see. So the problem isn't that you can't define the Grothendieck topology, it's that the resulting category of sheaves, in a sense, isn't very "interesting" and also doesn't relate to the topology of the space. I guess I thought it would since I believed the lattice of opens was a subcategory of the subspace lattice, but maybe this was wrong?

John Baez (Nov 05 2024 at 22:46):

I think the lattice of opens of a topological space $X$ is a subcategory of the lattice of subspaces of $X$, but I think Mike is reminding us that the latter is isomorphic to the lattice of all subsets of $X$, which is independent of the topology on $X.$ The reason is that a subspace is just a subset equipped with its subspace topology.

John Baez (Nov 05 2024 at 22:55):

I have a question of my own. Etale spaces are often non-Hausdorff. Hausdorffness implies sobriety. Do etale spaces tend to be sober?

It may depend on what kind of etale spaces you tend to be interested in. So, for example, say I take the sheaf of continuous or smooth or analytic functions $f: \mathbb{C} \to \mathbb{C}$ and form its etale space. Is this space sober? (That's 3 questions, and I bet the first two have the same answer.)

John Baez (Nov 05 2024 at 23:07):

The reason why I'm asking is that we started out talking about how sheaves can be seen as etale spaces, and then we started talking about sheaves over locales, so I'm wondering how this all fits together.

John Onstead (Nov 05 2024 at 23:31):

John Baez said:

I think the lattice of opens of a topological space $X$ is a subcategory of the lattice of subspaces of $X$, but I think Mike is reminding us that the latter is isomorphic to the lattice of all subsets of $X$, which is independent of the topology on $X.$ The reason is that a subspace is just a subset equipped with its subspace topology.

I see! So if we want to use a sheaf and the machinery of sheaf theory to attach and describe local data on an object of some category, we need to find a category "over" that object (a decomposition of that object) that is "just right" (it satisfies some sort of "Goldilocks property") If the decomposition is too general, like in the case of considering all subspaces, then the resulting sheaf theory is uninteresting and trivial and it ignores the structure of the object. But I imagine if we make the decomposition too specific, then it limits the range of things we can do. So we have to find some notion of decomposition that is "just right", in which case sheaves over that decomposition can both describe local data in a way that is faithful to the structure of the object, AND can be comprehensive enough to describe how local data in a number of scenarios can fit together into the global picture. It just so happens for topological spaces that the notion of "open set" fits this criterion. But of course I'm always interested in generalizing so my next thought will be trying to see how this "Goldilocks analysis" can be done internal to a generic category, not just Top. If it can't be done in a generic category, then maybe the class of categories it can be done for will meet the criteria of a "Top-like category" that was being discussed above!

Mike Shulman (Nov 06 2024 at 00:10):

John Baez said:

I have a question of my own. Etale spaces are often non-Hausdorff. Hausdorffness implies sobriety. Do etale spaces tend to be sober?

I believe that if a local homeomorphism has sober codomain, then its domain is also sober. However, your question made me realize that I don't know offhand a reference for a proof of this! In the comments after C1.3.2 in Sketches of an Elephant, Johnstone writes

In this chapter, we shall write $\mathbf{LH}$ for the category of locales and local homeomorphisms between them. In Part A, we used this name for the category of spaces and local homeomorphisms, but the difference is not a substantial one: although we have 'enlarged' the category $\mathbf{LH}$, Lemma 1.3.2(v) tells us that we have not changed the category $\mathbf{LH}/X$ when $X$ is a space.

However, what Lemma 1.3.2(v) says is that if the codomain of a local homeomorphism of locales is spatial, then so is its domain. Since a spatial locale is the same as a sober space, this implies that the category of locale-local-homemorphisms over a sober space is a subcategory of the category of space-local-homeomorphisms over it. But saying that it's the whole thing requires knowing that the domain of any local homeomorphism over a sober space is sober, i.e. a positive answer to your question, and I don't know where that is proven.

Mike Shulman (Nov 06 2024 at 00:15):

John Onstead said:

So we have to find some notion of decomposition that is "just right", in which case sheaves over that decomposition can both describe local data in a way that is faithful to the structure of the object, AND can be comprehensive enough to describe how local data in a number of scenarios can fit together into the global picture.

This is a good perspective. However, for a general category, I don't think it will be the case that there is only one "notion of decomposition" that is "just right" for all purposes. For instance, algebraic geometers make use of a number of different Grothendieck topologies on schemes, such as the Zariski topology, the etale topology, the fppf topology, the fpqc topology, and more.

John Onstead (Nov 06 2024 at 00:46):

Mike Shulman said:

This is a good perspective. However, for a general category, I don't think it will be the case that there is only one "notion of decomposition" that is "just right" for all purposes. For instance, algebraic geometers make use of a number of different Grothendieck topologies on schemes, such as the Zariski topology, the etale topology, the fppf topology, the fpqc topology, and more.

That's really interesting! I'm still not too familiar with schemes, but this seems to imply they have a rich structure. Perhaps at some point I'll be able to understand them better!
There's something I'm confused by however... the main Wikipedia page lists the topologies as existing on the whole category of schemes. Yet when I visit the individual page for Etale topology for instance, it gives the topology as existing on the category of etale morphisms over an individual object. So which one is it: are Grothendieck topologies for schemes defined on the whole entire category of schemes, or are they defined "within" the category, that is, individually on each category of some choice of decomposition over an object?

Mike Shulman (Nov 06 2024 at 01:01):

Many, perhaps most, of those topologies have both a "big" version, on the category of all schemes, and a "little" version, on some category of "parts" of a fixed scheme. This is also true for topological spaces and open covers: the usual topology on the open subsets of a given space is a "little" topology, but there's also a topology on the category of all spaces where the covers are jointly surjective families of open inclusions. There's a general notion (not really precise) of [[big and little toposes]] that are related by a family of local geometric morphisms.

John Onstead (Nov 06 2024 at 02:00):

Mike Shulman said:

Many, perhaps most, of those topologies have both a "big" version, on the category of all schemes, and a "little" version, on some category of "parts" of a fixed scheme. This is also true for topological spaces and open covers: the usual topology on the open subsets of a given space is a "little" topology, but there's also a topology on the category of all spaces where the covers are jointly surjective families of open inclusions. There's a general notion (not really precise) of [[big and little toposes]] that are related by a family of local geometric morphisms.

Oh! That's really interesting. Now I see how you arrived at your earlier conclusion that an "open part" of an object is a "morphism belonging to a covering family in some site"- when you define this appropriate site on Top as a whole, the elements of covering families are precisely the open sets. I guess this answers my question from above about "Top-like categories"- a category acts like Top if it has a Grothendieck topology defined on it! And indeed this "Top-like" nature is "extra structure" with a category able to act "like Top" in more than one way, such as we saw with the category of sheaves.

John Onstead (Nov 06 2024 at 02:01):

Though that does lead me to wonder how to connect the "big" and "little" notions together. For instance, how to take a "big" site and construct a "little" site for some object within it. I've taken a look at this page on the nCat Cafe and it seems for a given space X there are two ways of defining sheaves. The first is to define the category where the objects are the morphisms belonging to the covering families of X in the big site- that is, the category of "opens" over X (Open sets for Top, etale morphisms for the etale site on Scheme, etc.) equipped with the "little topology". The second is to define the slice category Top/X or Scheme/X, which somehow inherits a topology of its own from the big site, which becomes X's "big topology". Here's my theory on how they are related. Since the little site of "opens" always contains morphisms as objects, there's always an embedding from this into the slice category over X. Perhaps the "little topology" is then precisely some sort of restriction of the "big topology" from the slice category over X to the category of "opens" over X. If I'm right, then any big topology (or more generally, any Grothendieck topology at all) automatically induces both a big and small topology for each object, so for every site you can define a notion of both "big" and "small" sheaf topos for each of its objects!

Mike Shulman (Nov 06 2024 at 03:16):

I don't think a big topology automatically induces little topologies, since you have to pick which morphisms to include in your little sites. But if you read the link I gave to [[big and little toposes]], you'll see (end of section 3) that your intuition about the relationship is right: the little topos of each object is a reflective and coreflective subcategory of the big topos of that same object, giving a [[local geometric morphism]] from the big topos to the little one.

John Onstead (Nov 06 2024 at 12:29):

That's interesting, I must have missed that on my first read! I also find it fascinating that the little topos is a reflective subcategory of the big topos, I certainly did not expect that!
But anyways, thanks for your help on this, I'm finally starting to see the "big picture" so to speak when it comes to sheaves and sites. To review, it seems that sheaves most generally solve local-global problems internal to categories equipped with some coverage, where in general we want this coverage to be "interesting enough" like what we get with open sets on Top. It also seems the kinds of local-global problems sheaves help solve are those that concern when local data is assigned to parts of an object, as well as when certain kinds of statements are made about this local data (such as in the case above with geometric formulas). As long as all this seems reasonable, I think that wraps up my main line of inquiry here, though I'd still like to discuss more specifics!

John Onstead (Nov 06 2024 at 12:34):

For instance, I wanted to use this new knowledge and revisit some of the discussion above, specifically dealing with Leinster's trick. I know I asked a similar question about this above, but it was vague as I wasn't quite sure what was going on, now I want to rephrase with my new insight to make things more concrete. Given some coverage on a category $C$ and a corresponding notion of little site on objects $A$ of $C$ which we might call $\mathrm{LSite}(A)$, we know there exists a canonical functor $\mathrm{LSite}(A) \to C/A$. If we apply Leinster's trick from before (and also assume that $C/A$ has small colimits), we get an adjunction $\mathrm{Psh(LSite}(A)) \to C/A$. As mentioned above, it's unlikely that this adjunction will actually correspond with some sheafification in full generality. However, now I'm interested in the class of all coverages such that this does actually correspond with sheafification. Would you happen to know what kind of properties a coverage of $C$ would need to have such that this holds true for all little sites of all objects within it? I'm sure the "open set" coverage on $\mathrm{Top}$ can't be the only example of this phenomenon!

Mike Shulman (Nov 06 2024 at 15:46):

No, not in general. Of course the property depends on your choice of little sites.

One general context in which this is true, at least for one particular choice of little sites, but that doesn't include Top, is when $C$ is already a topos. For any object $A$ of a topos $C$, the category $C/A$ is again a topos, and so it has some site presentation, which we could call the little site of $A$, so that your adjunction would exhibit $C/A$ as the reflective subcategory of sheaves on this site.

John Baez (Nov 06 2024 at 16:28):

Mike Shulman said:

I believe that if a local homeomorphism has sober codomain, then its domain is also sober. However, your question made me realize that I don't know offhand a reference for a proof of this!

I really like this 'conservation of sobriety' result, especially given the massive failure of local homeomorphisms with Hausdorff codomain to have Hausdorff domain. Maybe we should all be working with locales and not worry about sober spaces, but if we've been raised on topological spaces it's hard not to be curious about issues like this.

I have a mental picture of the 'walking failure of an etale space to be Hausdorff', which is the etale space coming from the sheaf of sections of this bundle over the real line:

It's probably not really universal, but it seems to capture the reason etale spaces are so often non-Hausdorff: has two distinct points that don't have disjoint open neighborhood, coming from the point in the base at which two sections are equal even though they don't agree in any neighborhood of that point.

I don't really understand sobriety (hah, maybe this is why people say "soberness"): I know a space is sober iff every nonempty irreducible closed subset is the closure of a unique point, but I don't see why the above space is sober. I should probably just think about it; I haven't tried hard yet.

Mike Shulman (Nov 06 2024 at 17:20):

Soberness is hard to understand for me too: I can't look at a space and immediately see that it's sober the way I can usually for the simpler separation properties. But I do have the impression that while $T_2$ (Hausdorff) and $T_0$ are the two "familiar" separation properties that soberness lies between (it's incomparable to $T_1$), it's really a much weaker property than Hausdorffness. For example, the prime spectrum of any commutative ring is sober, though it is very far from Hausdorff. (How do I know that it's sober? Because it's the space of points of a locale! This is the best way I know of to tell that a space is sober.)

On that note, maybe here is a roundabout argument for the 'conservation of sobriety' result, using Johnstone's lemma. Suppose $E\to X$ is a local homeomorphism of spaces. Then the sheaf of sections of $E$ gives rise to a local homeomorphism of locales $E'\to X$, and by Johnstone's lemma $E'$ is also spatial. But by the adjunction between (pre)sheaves and spaces, there's a canonical map of spaces $E'\to E$ (at least I think it's that direction), and this map induces an isomorphism of sheaves of sections. Thus, since we know that sheaves on a space are equivalent to local homeomorphisms of spaces over it, this map is an isomorphism. So $E$ is sober, since it's isomorphic to the space of points of the locale $E'$.

Mike Shulman (Nov 06 2024 at 17:37):

Going back to this:
John Onstead said:

Given some coverage on a category $C$ and a corresponding notion of little site on objects $A$ of $C$ which we might call $\mathrm{LSite}(A)$, we know there exists a canonical functor $\mathrm{LSite}(A) \to C/A$. If we apply Leinster's trick from before (and also assume that $C/A$ has small colimits), we get an adjunction $\mathrm{Psh(LSite}(A)) \to C/A$. As mentioned above, it's unlikely that this adjunction will actually correspond with some sheafification in full generality. However, now I'm interested in the class of all coverages such that this does actually correspond with sheafification. Would you happen to know what kind of properties a coverage of $C$ would need to have such that this holds true for all little sites of all objects within it? I'm sure the "open set" coverage on $\mathrm{Top}$ can't be the only example of this phenomenon!

My intution is that this is probably actually quite a rare property, and the fact that it holds in Top is sort of an accident. To explain the latter, let me point out that there's another example where it almost holds for tautological reasons: when $C$ is the category $\rm Topos$ of toposes. In this case, every topos $A\in \rm Topos$ has a canonical "little" site, namely the topos $A$ itself qua category, with the topology generated by jointly epimorphic families. The functor ${\rm LSite}(A) = A \to {\rm Topos}/A$ sends each $X\in A$ to the slice topos $A/X$.

Since the colimits induced by covering families are [[van Kampen colimits]], they are taken to limits of slice categories, which is to say colimits in ${\rm Topos}$. Therefore, the "nerve" of any topos over $A$ is a sheaf. Conversely, the "realization" of any representable presheaf is always just its image under the original functor; so we get an induced equivalence between $A = {\rm Sh}(A)$ and the subcategory of ${\rm Topos}/A$ spanned by the slice toposes $A/X$, i.e. the "local homeomorphisms of topoi".

There are two reasons why this is all a lie.

The first is categorical dimension: $\rm Topos$ is a 2-category, not a 1-category, so its "nerve" doesn't land in presheaves of sets but rather presheaves of categories. Similarly, the nerve for $n$-topoi lands in presheaves of $n$-categories. But we can take this seriously, and for any $n$ and any $n$-topos $A$, get a hoped-for equivalence between the $(n+1)$-topos induced by $A$ (consisting of sheaves of $n$-categories on $A$) and the local homeomorphisms of $n$-topoi over $A$. At $n=\infty$ we can hope for it to stabilize.

The second is set-theoretic size: $\rm Topos$ is a locally large (2-)category, so its nerve doesn't land in presheaves of small anything. But it doesn't have large colimits, so you can't just boost up to large presheaves on the left. It's still true that any topos is equivalent to the category of local homeomorphisms of toposes over itself, but I don't know of a way to get that from a nerve-realization adjunction.

Except when $n=(0,1)$, in which case the size issues go away: the category of $(0,1)$-toposes, i.e. locales, is locally small. So, putting this together with the resolution of the first problem, we get an equivalence between the 1-topos of sheaves of sets on a locale and the category of local homeomorphisms of locales over it. So the "classical" version for locales is really the "tautological" case. And I think it's kind of an accident that we can pass this over to topological spaces: the definition of topological spaces is just so close to that of locales, that the "conservation of sobriety/spatiality" result means that we get a similar equivalence for spaces. But there are very few other categories that I would expect to be "so close" to a kind of topos.

John Onstead (Nov 06 2024 at 21:11):

Mike Shulman said:

Except when $n=(0,1)$, in which case the size issues go away: the category of $(0,1)$-toposes, i.e. locales, is locally small. So, putting this together with the resolution of the first problem, we get an equivalence between the 1-topos of sheaves of sets on a locale and the category of local homeomorphisms of locales over it. So the "classical" version for locales is really the "tautological" case. And I think it's kind of an accident that we can pass this over to topological spaces: the definition of topological spaces is just so close to that of locales, that the "conservation of sobriety/spatiality" result means that we get a similar equivalence for spaces. But there are very few other categories that I would expect to be "so close" to a kind of topos.

That's very interesting! So by this argument the similarity between topological spaces and locales (by their definitions, though maybe also by the adjunction between them?) is the reason for Leinster's nerve/realization construction working. So it wouldn't work then if I, for instance, tried to do this with schemes and the etale morphism coverage. Maybe that would be a good exercise- trying to prove that that restricting the nerve/realization adjunction $\mathrm{Psh(Etale}(X)) \to \mathrm{Scheme}/X$ doesn't yield sheaves on $\mathrm{Etale}(X)$! I might give this a try, after all it will help me learn more about schemes in the process!

I did return to the original post by Leinster and discovered he actually posted, alongside it, a pdf with similar information. He does mention generalizing the work he did to more general sites at the bottom. But in a cruel twist, he of course leaves that as "an exercise to the reader", just as you would expect. But him leaving this as an exercise almost seems to imply that he personally believes the construction might be widely possible, but your analysis above shows that is not true. Which leaves me wondering what Leinster believed the solution to his own exercise to be! I know he pops on very occasionally to the server, maybe when he comes back we can ask?

Mike Shulman (Nov 06 2024 at 21:27):

He doesn't say "exercise", he says "challenge". That suggests to me that maybe he didn't know of an answer.

Mike Shulman (Nov 06 2024 at 21:28):

But also, I don't necessarily interpret "do something similar for sites" as meaning "find a way to replace Top by a different category".

John Onstead (Nov 06 2024 at 23:41):

Mike Shulman said:

But also, I don't necessarily interpret "do something similar for sites" as meaning "find a way to replace Top by a different category".

You're probably right about that, the exercise directions were kind of vague.

John Onstead (Nov 06 2024 at 23:41):

When searching online the other day for how to internalize a notion of "open subset" (prior to understanding how coverages axiomatize this), I stumbled across this page on nlab: open morphism. I'm wondering what the connection is between the ideas, or if there's any general connection at all to what we were discussing. Interestingly, some of the axioms for defining a class of "open maps" in a topos given on the page might are similar to axioms for a Grothendieck topology, though I didn't look into it too much. But if this is true then I certainly think there might be some connection there!

Mike Shulman (Nov 07 2024 at 00:42):

Yes, there's a connection; often when equipping a category with a topology, the morphisms appearing in the covering families are some kind of "open morphism".

Morgan Rogers (he/him) (Nov 07 2024 at 09:14):

John Onstead said:

Here's my next question... in order to learn more about the full range of problems sheaves can be applied to, I'll need to go beyond topology. (We'll return to topology later since I still have many questions to ask about that.) As mentioned above, sheaves on topological spaces can be understood in terms of Leinster's adjunction restriction construction, and this in turn gives rise to the notions of sheafification and the duality between etale spaces/local homeomorphisms and sheaves.

What I want to know is: how general is this situation? For instance, the most general setting for discussion of sheaves is that of sheaves over a site (a category equipped with a notion of coverage or Grothendieck topology). Can I always guarantee that an analogue of sheafifcation and the etale space-sheaf correspondence holds in this most general of cases? In particular, given some (small) site $C$, is there always some functor $C \to D$ (into some $D$ with of course small colimits) such that the corresponding $\mathrm{Psh}(C) \to D$ is an adjunction that restricts to an equivalence of form $\mathrm{Sh}(C) \to E$?

You should check out the thesis of Riccardo Zanfa. You can generalize the adjunction, but you need to keep track of more information when the category is not a poset.

Morgan Rogers (he/him) (Nov 07 2024 at 09:38):

Mike Shulman said:

In general, I would say that (in this context, anyway) an "open part" of an object is just, by definition, a morphism belonging to a covering family in some site. (-:

You can put a Grothendieck topology on the poset of all subspaces of X which recovers the ordinary topos of sheaves on X by taking where a cover of a subset S is a collection of subspaces whose interiors cover the interior of S. It's a bit silly because any subspace gets identified with its interior when we move to the category of sheaves, but this example illustrates that the objects of the site don't have to be open.

Morgan Rogers (he/him) (Nov 07 2024 at 10:07):

Most sites that one considers in practice have a subcanonical Grothendieck topology, which is a property ensuring that the site embeds fully faithfully into the category of sheaves. But it's always possible to define bigger sites yielding equivalent sheaf toposes, for which some of the objects will be forced to become isomorphic when passing to sheaves.

John Onstead (Nov 07 2024 at 12:45):

Morgan Rogers (he/him) said:

You should check out the thesis of Riccardo Zanfa. You can generalize the adjunction, but you need to keep track of more information when the category is not a poset.

Thanks for the resource, it seems promising so far! I'll have to look it over again a few times since it's a little technical. But from my cursory glance it seems for a posetal site, there's always an adjunction between the presheaf category and a slice category that plays the role of sheafification. But interestingly, this particular slice category is always in the category of locales. The paper seems to converge with Mike Shulman's analysis above, since it agrees with the above conclusion that the only reason we can "transfer" this adjunction from involving a slice in Locale to the more familiar one in Top is "by accident" of the close similarity between Top and Locale. It even says "Thus, even though one classically defines the sheafification as the local sections of the etale bundle, these considerations show explicitly that there is no need to work in the topological context, since all the relevant information lives at the localic level".

Fernando Yamauti (Nov 07 2024 at 17:32):

Mike Shulman said:

Going back to this:
John Onstead said:

Given some coverage on a category $C$ and a corresponding notion of little site on objects $A$ of $C$ which we might call $\mathrm{LSite}(A)$, we know there exists a canonical functor $\mathrm{LSite}(A) \to C/A$. If we apply Leinster's trick from before (and also assume that $C/A$ has small colimits), we get an adjunction $\mathrm{Psh(LSite}(A)) \to C/A$. As mentioned above, it's unlikely that this adjunction will actually correspond with some sheafification in full generality. However, now I'm interested in the class of all coverages such that this does actually correspond with sheafification. Would you happen to know what kind of properties a coverage of $C$ would need to have such that this holds true for all little sites of all objects within it? I'm sure the "open set" coverage on $\mathrm{Top}$ can't be the only example of this phenomenon!

My intution is that this is probably actually quite a rare property, and the fact that it holds in Top is sort of an accident. To explain the latter, let me point out that there's another example where it almost holds for tautological reasons: when $C$ is the category $\rm Topos$ of toposes. In this case, every topos $A\in \rm Topos$ has a canonical "little" site, namely the topos $A$ itself qua category, with the topology generated by jointly epimorphic families. The functor ${\rm LSite}(A) = A \to {\rm Topos}/A$ sends each $X\in A$ to the slice topos $A/X$.

Since the colimits induced by covering families are [[van Kampen colimits]], they are taken to limits of slice categories, which is to say colimits in ${\rm Topos}$. Therefore, the "nerve" of any topos over $A$ is a sheaf. Conversely, the "realization" of any representable presheaf is always just its image under the original functor; so we get an induced equivalence between $A = {\rm Sh}(A)$ and the subcategory of ${\rm Topos}/A$ spanned by the slice toposes $A/X$, i.e. the "local homeomorphisms of topoi".

There are two reasons why this is all a lie.

The first is categorical dimension: $\rm Topos$ is a 2-category, not a 1-category, so its "nerve" doesn't land in presheaves of sets but rather presheaves of categories. Similarly, the nerve for $n$-topoi lands in presheaves of $n$-categories. But we can take this seriously, and for any $n$ and any $n$-topos $A$, get a hoped-for equivalence between the $(n+1)$-topos induced by $A$ (consisting of sheaves of $n$-categories on $A$) and the local homeomorphisms of $n$-topoi over $A$. At $n=\infty$ we can hope for it to stabilize.

The second is set-theoretic size: $\rm Topos$ is a locally large (2-)category, so its nerve doesn't land in presheaves of small anything. But it doesn't have large colimits, so you can't just boost up to large presheaves on the left. It's still true that any topos is equivalent to the category of local homeomorphisms of toposes over itself, but I don't know of a way to get that from a nerve-realization adjunction.

Except when $n=(0,1)$, in which case the size issues go away: the category of $(0,1)$-toposes, i.e. locales, is locally small. So, putting this together with the resolution of the first problem, we get an equivalence between the 1-topos of sheaves of sets on a locale and the category of local homeomorphisms of locales over it. So the "classical" version for locales is really the "tautological" case. And I think it's kind of an accident that we can pass this over to topological spaces: the definition of topological spaces is just so close to that of locales, that the "conservation of sobriety/spatiality" result means that we get a similar equivalence for spaces. But there are very few other categories that I would expect to be "so close" to a kind of topos.

Sorry for bringing back this comment now. I'm confused on how you can recover the $1$-localic $2$-topos associated to some $1$-topos $A$ using the étale geom morphisms. I think that the full subcat of étale topoi over $A$ is actually a $1$-cat. No? Maybe I'm misunderstanding the procedure.

John Baez (Nov 07 2024 at 18:26):

By the way, you don't need to quote so much text: if you quote just the necessary portion, it's a bit easier for readers.

Mike Shulman (Nov 07 2024 at 18:29):

Ah, yes, in order for that to work you would need a more expansive notion of "local homeomorphism". Normally a local homeomorphism of 1-topoi is defined to be just the projection from a slice topos, and so as you say the category of such would recover only the 1-topos itself. But the analogue of this in the world of (0,1)-topoi, looking at "slice (0,1)-topoi" over a locale, which is to say open subsets, would likewise recover only the locale itself rather than its associated 1-topos. To get the 1-topos associated to a (0,1)-topos, we need "local homeomorphisms" defined in the usual locale-theoretic way as the domain being covered by opens on which the map is an isomorphism. Similarly, to get the 2-topos associated to a 1-topos, we need a more general notion of "2-local-homeomorphism".

The obvious definition is that a geometric morphism $F\to E$ is 2-etale (or more precisely (2,1)-etale) if it identifies $F$ with the topos of presheaves on some internal groupoid in $E$. This isn't especially satisfying as it feels sort of tautological and isn't an obvious generalization of a local homeomorphism of locales, but I don't know a better definition.

John Onstead (Nov 07 2024 at 23:33):

I now think I have a good understanding of the generalization of the "presheaf-bundle" adjunction. It's very interesting, but of course as pointed out above it's only for topological spaces that these "bundles" correspond to containing information about stalks and germs (since a general locale might not "have enough points"). So now that we covered generalizations of the bundle correspondence and generalizations of stalks to generic sites, I wanted to know more about if the notion of a "germ" of a map can be generalized to an arbitrary category of sets with structure. In those categories, there's always, in a sense, "enough points" since unlike locales, we define these objects in terms of sets and so they all have underlying "points" that we can analyze the behavior of a map "near". Maybe we'd define some decomposition (most likely some sub-poset of the subobject poset; again like in our discussion above with open sets we'd want to choose one with "interesting" behavior) of an object $A$, a presheaf $\mathrm{Maps}(A,B)$ into $\mathrm{Set}$ from that category (that plays the role of a presheaf of maps $A \to B$ that sends every "part" of $A$ to the set of maps from that part to $B$), and then do a directed limit just like in the topological case (since we don't need the presheaf to be a sheaf, as covered above, to do this directed limit).

John Onstead (Nov 07 2024 at 23:33):

For instance, I'm really curious to know if there is a such thing as a germ of a group homomorphism (especially since points in Grp are given by morphisms not from some point object, but from the integers), a germ of a map between measure spaces, a germ of a map between (pre)convergence spaces (and if that resembles germs in Top), or even the germ of a functor between categories (and whether this analyzes the behavior of a functor near an object or a morphism). Or maybe these notions don't mean anything and you need a notion of neighborhood- ultimately associated with a topological space- to even define what it means for a function to have a "local behavior"? But if that's the case, what happens if we try to take the direct limit of a presheaf out of a (sub-poset of the) subobject poset for one of the aforementioned categories- what would this accomplish?

Mike Shulman (Nov 08 2024 at 15:56):

For a limit over a sub-poset of the subobject poset to be considered a germ "at" some particular point, all the subobjects in the sub-poset should probably contain that point. The sub-poset may as well be upwards closed. And for germs to behave like germs usually do, I would expect the colimit to be directed. In other words, the sub-poset is a [[filter]]. Now if every point of your object is equipped with a filter of subsets containing it, you have a [[pretopological space]]. So basically the only way to talk about germs is if the structure of the objects of your category "almost" induces a topology.

John Onstead (Nov 08 2024 at 19:16):

Wow, that's a really cool (and unexpected) result! Maybe I was hoping (at least secretly) that it made sense to talk about the "germ" of a morphism in general, but I also think it's really interesting that just putting a few basic reasonable restrictions on the sub-posets of the subobject poset leads inevitably back to (pre) topological spaces, out of all the mathematical objects out there.
I think that's about it for this topic! I have some related questions, but I'll save that for another topic sometime in the future. Thanks for all your help on this!

John Baez (Nov 08 2024 at 20:40):

I hadn't known about [[pretopological spaces]]. It's interesting that the only example given of a pretopological space that's not topological is so artificial. Are there any 'useful' examples?

Mike Shulman (Nov 08 2024 at 21:27):

I don't know about useful examples. I do have the general feeling that there's a lot of freedom in defining a pretopology vs a topology, since you only have to give a filter at each point with no relationship between them. So artificial examples abound, e.g. there's a pretopology on $\mathbb{R}$ where the neighborhoods of rational numbers are those from the Euclidean topology, the neighborhoods of algebraic numbers are those from the cofinite topology, and those of transcendental numbers are those from the discrete topology.

John Baez (Nov 09 2024 at 01:53):

That sounds incredibly useful. :smirk:

John Baez (Nov 09 2024 at 01:56):

It is, however, easier to understand than the example proffered at the nLab:

Here is an example of a nontopological pretopological space, although admittedly it is a bit artificial. (This is based on Section 15.6 of [[HAF]].) Consider a [[metric space]] $S$; according to the usual pretopology on $S$, $U$ is a neighbourhood of $x$ if there is a positive number $\epsilon$ such that $U$ contains the ball $\{ y \;|\; d(x,y) \lt \epsilon \}$. Now given a [[natural number]] $n$, we will give $S^n$ the _plus pretopology_: $U$ is a neighbourhood of $\vec{x} = (x_1,\ldots,x_n)$ if there is a positive number $\epsilon$ such that $U$ contains the $l^0$-ball $\{ \vec{y} \;|\; \inf_i d(x_i,y_i) \lt \epsilon \}$. (If $S$ is a line and $n = 2$, then this neighbourhood is a plus sign '+' with $(x_1,x_2)$ at the centre and cross bars of length $2 \epsilon$.) Then $S^n$ is a pretopological space, but it is topological only if $n \leq 1$ or $S$ is a [[subsingleton]].

I can't even force myself to read this. When I was a kid I enjoyed Counterexamples in Topology, but now when I see constructions like this my eyes instantly glaze over. It's like reading a very elaborate recipe for a dish I'd never want to eat, like "souffle with flambeed prunes and ground glass".

So, I've added a simplified version of your example to the nLab.

Hypatia du Bois-Marie (Nov 09 2024 at 19:41):

John Baez said:

... nLab.

I'm surprised that $T=\mathbb{F},V=2$ isn't mentioned lol. [Lawvere 2014]

John Baez (Nov 09 2024 at 19:43):

What's $\mathbb{F}$?

Hypatia du Bois-Marie (Nov 09 2024 at 19:43):

F is the filter monad $:\mathbf{Set}\to\mathbf{Set}$

Hypatia du Bois-Marie (Nov 09 2024 at 19:43):

while $\mathbf{Top}$ is $T=\mathbb{\beta},V=2$, with the ultrafilter monad.

Hypatia du Bois-Marie (Nov 09 2024 at 19:48):

these $(T,V)$-categories are given by $\alpha:X\times TX\to V$, where $T$ is a monad on $X$ and $V$ a "quantale" (weird lattices; the $(2,\leq,\wedge,1)$ above basically is $\{\text{true}, \text{false}\}$, but other quantales can be considered, such as with $V=[0,\infty]$ and $T=\operatorname{id}$ gives you metric spaces

Hypatia du Bois-Marie (Nov 09 2024 at 19:52):

(i just read about these weird little things from a Chinese math post yesterday

Ryuya Hora (Nov 10 2024 at 01:02):

Sorry for going back to this question:
John Baez said:

I hadn't known about [[pretopological spaces]]. It's interesting that the only example given of a pretopological space that's not topological is so artificial. Are there any 'useful' examples?

I want to provide an example, which is interesting at least for me. My claim is that pretopologies are ``dynamical version of topology", and naturally arise in the context of discrete dynamical systems.

In terms of the interior operator, a topology on a set $X$ is a lex comonad on its powerset $\mathrm{Int}\colon \mathcal{P}(X)\to \mathcal{P}(X)$, and a pretopology is a topology without idempotency (i.e. an co-pointed endofunctor on the powerset $\mathcal{P}(X)$).

Such non-idempotent interior operator naturally arise in cellular automaton.
Example: Conway's game of life is a discrete dynamical system over a discrete space $\mathbb{Z}^2$, with the notion of "neighborhood." In this context, "neighborhoods" define a pretopology, not a topology. We define $\mathrm{Int}(S)$ for $S\subset \mathbb{Z}^2$ to be

$\{s\in S\mid \text{the next state of }s\text{ is determined by the states of elements of }S\}$.

Formally, we define $\mathrm{Int}(S)\coloneqq \{(x,y) \in \mathbb{Z}^2\mid \forall i,j\in \{-1,0,1\},\; (x+i,y+j)\in S \}$. Here's the non-idempotency: $\mathrm{Int}^n(S)$ is the set of points, whose $n$-seconds later state is determined by $S$.

Ryuya Hora (Nov 10 2024 at 01:06):

I would like to add two comments regarding connections with topos theory.

1. This example is NOT INTERESTING in our context of "generalization of germs," since each point $(x,y)$ has the minimum neighborhood $\{(x+i,y+j) \in \mathbb{Z}^2\mid i,j\in \{-1,0,1\} \}$.

2. It induces a topos. Once I've been wondering what is the ``implicit geometry" behind Conway's game of life (and Mine-sweeper, which looks similar). It quickly became apparent that the usual notion of topology is not suitable. It is needed to use a notion of space, which can deal with both a space $\mathbb{Z}^2$ and the discrete time monoid $\mathbb{N}$-actions. So i tried to define a topos in which Conway's game of life is an object. My answer is considering the pretopological space $(\mathbb{Z}^2, \mathrm{Int})$. With the interior operator, every pretopological space can be regarded as a category internal to the topos $\mathrm{PSh}(\mathbf{B}\mathbb{N})$, which induces a relative topos $\mathcal{E}\to \mathrm{PSh}(\mathbf{B}\mathbb{N})$. For the pretopology $(\mathbb{Z}^2, \mathrm{Int})$, the resulting topos $\mathcal{E}$ is equivalent to the presheaf topos $\mathrm{PSh}(\mathcal{P} (\mathbb{Z}^2)\rtimes_{\mathrm{Int}} \mathbf{B}\mathbb{N})$ in which Conway's game of life naturally becomes an object! (I'm not 100% satisfed wiht this, since this is just presheaf, and there are no notion of coverings and sheaves for now.)

Hypatia du Bois-Marie (Nov 10 2024 at 01:37):

Ryuya Hora said:

... NOT INTERESTING ... It quickly became apparent that the usual notion of topology is not suitable. It is needed to use a notion of space, which can deal with both a space $\mathbb{Z}^2$ and the discrete time monoid $\mathbb{N}$-actions. ...

Now that you said that, I actually think it is very interesting. For example, that Zhihu OP above mentioned Sheaves on quantales as generalized metric spaces; might be a starting point of stating something much general about these, for example, in that Lawvere 2014 long paper there are multiple combinations of monads and quantales that were left open (as in Lawvere possibly did not understand what phenomena they represent))...

Hypatia du Bois-Marie (Nov 10 2024 at 01:38):

(and I think etymologically speaking "quantale" is supposed to say something about "locale" (as a generalization, by definition)..

John Baez (Nov 10 2024 at 17:26):

Yes, 'quantale' is short for 'quantum locale', since it generalizes the concept of locale in a way that's useful in quantum logic.

John Baez (Nov 10 2024 at 17:34):

Ryuya Hora said:

I want to provide an example, which is interesting at least for me.

Thanks! It took me a while to get up the courage to read that example, but now I get it. Removing the discussion of cellular automata (which I actually enjoyed), let me check to see if I understood the pretopology you described. You've got a lattice $\mathbb{Z}^2$ and you define a neighborhood of any point $(x,y) \in \mathbb{Z}^2$ to be any set $U \subseteq \mathbb{Z}^2$ that contains all these points:

$(x+i, y+j)$ where $i,j \in \{-1,0,1\}$

John Baez (Nov 10 2024 at 17:37):

This obeys all the axioms that neighborhoods should in a [[pretopological space]], but it lacks the extra property needed by neighborhoods in a topological space: namely, that a neighborhood $U$ of a point is also a neighborhood of any other point in $U$.

John Baez (Nov 10 2024 at 17:37):

Nice!

Ryuya Hora (Nov 11 2024 at 03:55):

John Baez said:

let me check to see if I understood the pretopology you described.

Yes, that's exactly what I wanted to say. Thank you for reading! And I appreciate you adding it to nLab as well.