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Stream: theory: philosophy

Topic: The generic triangle

John Baez (Oct 03 2024 at 15:55):

On Mathstodon, Boarders wrote:

Platonism vs nominalism: an old fashioned debate in philosophy is that of universals - talking of triangle as a property commits oneself to the existence of the abstract concept of triangle or perhaps a “generic triangle”.

In this case, we can, to some extent, resolve the question mathematically in a precise way - there is a mathematical object called the moduli stack of triangles, the generic triangle is the generic point of this moduli stack. We can answer various precise questions about this generic triangle such as what automorphism group it has etc. It also specializes to various more particular triangles with a precise notion of specialize due to algebraic geometers. It seems to me this is a better formal apparatus for thinking about mathematical universals than any kind of vague waffle available, but the idea has not managed to escape the purview of mathematicians.

David Corfield (Oct 03 2024 at 16:48):

There's a huge industry in philosophy in working out what's meant by when we say "Let $n$ be a natural number". A common position has it that $n$ refers to an "arbitrary natural number" which is a different thing from the individual natural numbers and has only generic properties.

Andrej Bauer at his curtest on the subject here:

What I am saying is that "a free variable is a projection" is a great deal better at explaining various perceived mysteries about free variables than fancy words from philosophy books. But your mileage may vary.

A two-pronged attack with Egbert Rijke writing:

The generalised elements of an object $A$ are morphism into $A$. Under this point of view, the generic element of $A$ is the identity morphism on $A$.

John Onstead (Oct 04 2024 at 02:43):

Category theory actually gives us the most powerful ever version of finding "the generic X". This process is known as "externalization" and it involves going from a specific instance of X to viewing it as a functor out of a particular category, known as the "generic X" or "walking X". In a sense, the "walking X" is the platonic ideal of X, the archetypal or prototypical X. Making statements about the walking X then you can specialize to all instances of X across mathematics. There's many examples ranging from the "walking morphism" A -> B which is the archetypal morphism, to the Lawvere theory of groups which is the archetypal group object. This is also related to the notion of classifying object, which is the formal dual of the walking object since every morphism into it represents an instance of X.

Julius Hamilton (Nov 16 2024 at 16:47):

That’s interesting. I feel like I’m often attracted to “maximum generality”, but I feel like quite a few people caution one that this is sort of a fool’s errand. I feel like I’ve heard quite a few people mention that there is no “single best generalization” of certain ideas; that the purpose of abstraction is to focus on what specifically is relevant to some task at hand; that mathematics has a kind of relativism where we can model structures in different languages but not claim one is more central than any other.

Mike Shulman (Nov 16 2024 at 17:01):

"Generic" is different from "maximum generality".

John Baez (Nov 17 2024 at 00:26):

Yes, the "generic X" is a perfectly well-defined thing, unique up to equivalence, after we've chosen X (the "theory") and the context in which to work with X (the "doctrine"). If you seek greater generality, you will tend to keep fiddling around with the doctrine and the theory.

(I'm not very attracted to maximum generality: I mainly want to understand things and also use math to help the world. Often these quests are aided by increasing generality - but also very often it's important to decrease generality and think very hard about a specific example! As Polya said: "if there's a question you can't solve, there's probably an easier question you can solve" - so switch to that and see if it helps. Often to find a good easier question you should decrease the level of generality. It's better to add extra assumptions than sit there stuck.)

Joe Moeller (Nov 19 2024 at 22:47):

I think there's a pretty straightforward analogy that's helpful here: abstraction is like elevation. When you realize two concrete facts are examples of one abstract fact, you've put two sticks together and you're standing at their apex. This abstract point could be stregthened by supporting it with further struts on the ground: examples. The higher you are, the more you could see (if you dare to look down). So a naively appealing idea is to just build straight upwards, so you could see everything. That's not likely to produce a very stable structure, and if other structures haven't been built in the surrounding areas, you'll likely stop being able to make out any details on the ground anyway. Obviously sometimes it pays to be daring and build upwards a bit.

Kevin Carlson (Nov 19 2024 at 22:49):

Great metaphor!

Julius Hamilton (Nov 25 2024 at 16:31):

David Corfield said:

What I am saying is that "a free variable is a projection"…

The generalised elements of an object $A$ are morphisms into $A$. Under this point of view, the generic element of $A$ is the identity morphism on $A$.

Why is a free variable a projection?

Julius Hamilton (Nov 25 2024 at 16:33):

John Baez said:

As Polya said: "if there's a question you can't solve, there's probably an easier question you can solve" - so switch to that and see if it helps. Often to find a good easier question you should decrease the level of generality. It's better to add extra assumptions than sit there stuck.)

That is an idea I will definitely explore.

Ryan Wisnesky (Nov 25 2024 at 16:55):

if you look at a term in context, say, x:Nat, y:String |- x:Nat, then the categorical semantics is a morphism Nat*String->Nat, that is in fact the first projection of the context (the variable x). Whereas if you had x:Nat, y:String |- 5:Nat, you'd have a constant morphism, not a projection.

David Corfield (Nov 26 2024 at 08:01):

Right, so when you say 'Let x be a natural number,...', you are working in a context $x: Nat \vdash ...$. Say you want to establish some property of a number in general, and you arrive at $x: Nat \vdash p: P(x)$. This corresponds to a morphism in the slice over $Nat$, $p: Nat \to P$, respecting the maps down to $Nat$. If $P$ is a proposition, this will only be possible if $P$ holds for all $x: Nat$.

Sidney Congard (Jan 06 2025 at 17:01):

It seems however that, when speaking about "the generic number" x, some philosophers do not want to be able to derive e.g. that the generic x is even or odd (as it's neither). May there be something related to a kind of parametric polymorphism, to obtain only "uniform" properties, not those obtained by case analysis? (I don't have further ideas to sharpen this distinction though)

David Corfield (Jan 06 2025 at 19:33):

Wouldn't we expect it to be a feature of generic entities that distributivity over 'or' doesn't hold? So we can say

The generic number is either odd or even,

but not

The generic number is odd or the generic number is even.

In the negative form,

The generic number is neither odd nor even,

is fine in the sense

The generic number is not odd nor is it even,

but not in the sense

The generic number is neither odd nor even.

Sidney Congard (Jan 06 2025 at 19:51):

In this setup, "the generic X" seems replaceable by "any X", but idk if everyone want to bite the bullet (but I kinda do so it's hard speaking in their stead)

John Baez (Jan 06 2025 at 20:41):

I'd hope the logical rules governing the modality 'generically' would match those for open dense sets, since in topology 'P is generically true' means P is true on an open dense set.

John Onstead (Jan 08 2025 at 01:18):

John Baez said:

In this case, we can, to some extent, resolve the question mathematically in a precise way - there is a mathematical object called the moduli stack of triangles, the generic triangle is the generic point of this moduli stack. We can answer various precise questions about this generic triangle such as what automorphism group it has etc. It also specializes to various more particular triangles with a precise notion of specialize due to algebraic geometers. It seems to me this is a better formal apparatus for thinking about mathematical universals than any kind of vague waffle available, but the idea has not managed to escape the purview of mathematicians.

I actually wanted to revisit this point now that I learned more about the Yoneda lemma, co-Yoneda lemma, and how it connects to representability. Let's say I wanted to find the "generic triangle" as an object of a category that represents triangles (IE, such that the sets of morphisms out of it represent the triangles found in other objects). My immediate first guess was to think of the simplex category, in which you can think of each object as a "generic simplex". You can then build a simplicial set by "gluing together" these generic simplices from each dimension. In this case, the 2-simplex is actually the "generic triangle", since by the Yoneda lemma, every morphism from its representable simplicial set to any other simplicial set "picks out" a triangle. Via the geometric realization, we can also pick out triangles in topological spaces, again with the 2-simplex as the representing object. (Of course, every morphism from the 2-simplex into something like the cartesian plane can be deformed by homeomorphism to a circle or really any other similar shape)

John Onstead (Jan 08 2025 at 01:21):

So for me, the 2-simplex object in the simplex category is the "de facto" generic triangle. But this really makes me wonder how this notion of "generic triangle" connects with the one Boarders was writing about. Some time ago I did try to make an effort to understand moduli spaces and stacks, but I think I must've given up. A stack is like a groupoid, right? So this would involve determining what properties a "generic point" of a moduli stack has as an object of this corresponding groupoid (if that's even a valid thing to ask). In any case, my question would be: how might Boarder's notion of "generic point" and the 2-simplex as a representing object for triangles relate as notions of a "generic triangle" (whether that be on a philosophical or mathematical level)?

John Baez (Jan 08 2025 at 05:27):

John Onstead said:

A stack is like a groupoid, right?

There's a confusing diversity of ways of thinking about stacks, but a stack is like an appropriately categorified kind of sheaf. So, to get the definition of a stack, you can take the definition of a sheaf of sets, replace the sets by categories, require the laws of a sheaf only up to natural isomorphism, and make those isomorphisms obey their own coherence laws.

If the categories are groupoids, you get a "stack in groupoids". And these are like generalized groupoids - in the same sort of way that sheaves are like generalized sets, and stacks are like generalized categories.

In algebraic geometry the most commonly considered stacks are stacks in groupoids, so there it is a very good idea to think of a stack as like a blend of a groupoid and a scheme (which is the algebraic geometer's favorite kind of space).

John Onstead (Jan 08 2025 at 06:30):

John Baez said:

So, to get the definition of a stack, you can take the definition of a sheaf of sets, replace the sets by categories, require the laws of a sheaf only up to natural isomorphism, and make those isomorphisms obey their own coherence laws.

Ok, so a stack in groupoids is a functor $C^{op} \to \mathrm{Grpd}$ that's like a sheaf. So the category of stacks on a site $C$ is a lex reflective localization of $[C^{op}, \mathrm{Grpd}]$.
In that case, for the moduli stack of triangles, what is the precise site $C$, what are the objects of $C$, and what groupoids does the stack functor send each object to?

John Baez (Jan 08 2025 at 06:37):

John Onstead said:

Ok, so a stack in groupoids is a functor $C^{op} \to \mathrm{Grpd}$ that's like a sheaf.

Basically yeah. A 2-functor, since $\mathrm{Grpd}$ is a 2-category. (Maybe that's what you meant by
"functor".)

So the category of stacks on $C$ is a lex reflective localization of $[C^{op}, \mathrm{Grpd}]$.

Maybe, but that's not how I think about them. I don't know enough about stacks to know if that's equivalent to the definitions I've seen. In fact I'm far from an expert on stacks, I just kinda get some of the basic ideas.

In that case, for the moduli stack of triangles, what is the precise site $C$, what are the objects of $C$, and what groupoids does the stack functor send each object to?

That depends on whether you want a topological stack of triangles, a differentiable stack, an algebraic stack, or some other sort of stack. Each has its own site, or sites.

So, before you get all stacky, you have to decide whether you want to think about a topological space of triangles, a manifold of triangles (or a smooth space), or an algebraic variety of them (or a scheme). Then you have to decide what kind of symmetries between triangles you want to consider.

But I am happy to try to make these choices if you want. In fact, that's probably the only way we'll get a version of the "stack of triangles" that I have a chance of actually understanding.

Peter Arndt (Jan 08 2025 at 14:49):

I have a copy of a short text (from an abandoned book project of Kai Behrend, Brian Conrad, Dan Edidin, William Fulton, Barbara Fantechi, Lothar Göttsche and Andrew Kresch) discussing a moduli stack of triangles. I can post it here, but I don't want to spoil the fun of making your own choices...

John Baez (Jan 08 2025 at 15:45):

Interesting! Kai Behrend has a 127-page Introduction to algebraic stacks that uses various moduli stacks of triangles as examples.

Peter Arndt (Jan 08 2025 at 20:24):

Oh, that goes well beyond what I have. I meant Section 3 here.

John Onstead (Jan 08 2025 at 20:45):

John Baez said:

Interesting! Kai Behrend has a 127-page Introduction to algebraic stacks that uses various moduli stacks of triangles as examples.

I've found that same resource yesterday when thinking about this! This article seems to present stacks as being "categories fibered in groupoids". In other words, it's a special type of "groupoid fibration" satisfying some extra properties like locality and gluing, just like a sheaf! However, this is a fibration and not a functor into groupoids unlike a sheaf, though there might be some sort of Grothendieck construction that relates these.

The thing in the article I found the most interesting was the notion of a "universal family" (not "versal family", since those arent unique). This seems to me to be the best candidate for being the "generic point" of a moduli stack. Please correct me if I'm wrong and the generic point isn't what this author calls a "universal family"! But I think there's good reason to believe it is- the "universal family" is one that any other object (family of things) can be exhibited as a pullback of. This means there's at least some sort of canonical morphism from every object to it, right? And if this morphism is unique, it would make the universal family truly universal- as a terminal object of the groupoid fibration! If this is true, it reminds me a bit of Set- one can think of sets as "families of points", and the terminal object is indeed the "generic point". So it would make sense, in a category where objects are "families of triangles", for the generic triangle to be the terminal object. But I could be going off in a very wrong direction here, these are just my current thoughts!

John Onstead (Jan 09 2025 at 11:17):

Ok, I've tried to read through the paper again but my head was left spinning! I think I need to take an entire series of entire graduate level courses in algebraic geometry before I have any hope of understanding this article. But obviously I can't do that here, so I've formulated a few very precise questions to help me get started:

1. Is "universal family" and "generic point" in the context of a stack the exact same thing- that is, are these just synonyms?
2. Say you are given a stack defined to be a "groupoid fibration"/"category fibred in groupoids". Is the domain of the fibration (presumably resulting from some Grothendieck construction) necessarily a groupoid? Or is it possible to have a stack valued in groupoids but end up with a non-groupoid category as the domain of this fibration?
3. Given a groupoid fibration for a fine stack, is it, or is it not, true that the "universal family", viewed as an object of the domain of the fibration, is in fact the terminal object of the domain of the fibration?
4. Let's cover the specific case of the stack of triangles in the topological sense, since that's what is done on page 73 of the article. They define such a stack to be a fibration over Top, meaning it comes from a functor from Top valued in groupoids. My question now is: what exactly is this functor doing? It seems to send a topological space to the groupoid of families of triangles "inside" or "parameterized by" that space. What is that supposed to mean? How is a "triangle", let alone a family of them, meant to be defined "in" some arbitrary topological space? And lastly, which specific object in Top has the universal family of triangles as an object in the fiber over it, and why?

John Baez (Jan 09 2025 at 16:40):

@John Onstead - I was not recommending that you read Behrend's paper; indeed this sort of thing is why I said "I'm far from an expert on stacks, I just kinda get some of the basic ideas."

Differential and topological stacks are easier to understand than algebraic stacks, at least for people like me who understand manifolds and topological spaces better than schemes. Part of the difficulty of algebraic stacks is that they blend the subtlety of stacks with the subtleties of algebraic geometry.

Though I want to understand them better, I think the algebraic stacks as usually defined (and their important special cases, Artin stacks and Deligne-Mumford stacks) are too complicated for a lot of things that I want to do. I like Jim Dolan's approach where instead of an algebraic stack we use a symmetric monoidal cocomplete $k$-linear category for some commutative ring $k$ - a kind of categorification of a ring.

John Baez (Jan 09 2025 at 17:09):

So, I'll answer your questions in the only way I can, which is by using my intuition rather than actual knowledge.

John Onstead said:

1. Is "universal family" and "generic point" in the context of a stack the exact same thing- that is, are these just synonyms?

I've never heard about the generic point of a stack, though it may make perfect sense.

Let me do the generic point of an affine scheme. An affine scheme $X = \mathrm{Spec}(R)$ is just a commutative ring $R$ viewed as an object in the opposite category. The 'points' of $X$ are the prime ideals $I \subseteq R$. This generalizes the earlier idea that points are maximal ideals $I$, each of which gives a homomorphism to a field $R \to R/I$. You should imagine this homomorphism as restricting a function on $X$ to a point in $X$.

The prime ideals include points that look much larger than the points coming from maximal ideals. In particular $\{0\} \subseteq R$ is a prime ideal, and the corresponding point is so large and diffuse it's dense in $X$. This is the 'generic point' of $X$. The corresponding homomorphism $R \to R/\{0\}$ is just the identity homomorphism!

So this is very Yoneda-esque, where a 'generalized element' of an object $x$ is just a morphism $f: y \to x,$ and the generic element is the identity $1_x: x \to x$.

All this is perhaps a prerequisite for, the concept of 'universal family'. To talk about a universal family of things, or a moduli space of things, or a moduli stack of things, we need to talk about those things. Above we have a space $X$ (specifically an affine scheme), but it's not yet a space 'of things'.

We can consider any space to be a space of things, where the things are just its points. However when we talk about something like 'the moduli stack of triangles' we are supposed to start by understanding the things - i.e., clarifying what we mean by triangle, since this is a very flexible concept - and then using our understanding to define a 'family of things', and a 'universal family of things', and a 'moduli stack of things', etc.

John Baez (Jan 09 2025 at 17:22):

Say you are given a stack defined to be a "groupoid fibration"/"category fibred in groupoids". Is the domain of the fibration (presumably resulting from some Grothendieck construction) necessarily a groupoid?

I never think about stacks this way, but I think the answer is "no": the Stacks Project says:

In this section we explain how to think about categories fibred in groupoids and we see how they are basically the same as functors with values in the (2,1)-category of groupoids.

We can have a (pseudo)functor $f: \mathsf{C}^{\text{op}} \to \mathbf{Gpd}$ when $\mathsf{C}$ is a category with non-invertible morphisms, and then the Grothendieck construction of $f$ will have non-invertible morphisms.

What matters more to me is getting a feel for the examples we typically care about here! I think this is the sort of example I understand. For any topological space, there's a stack of vector bundles over that space. So let $\mathsf{C}$ be the poset of opens of some topological space, and given $O \in \mathsf{C}$ let $f(O)$ be the groupoid of vector bundles over $O$. This gives a functor $f: \mathsf{C}^{\text{op}} \to \mathbf{Gpd}$.

John Baez (Jan 09 2025 at 17:31):

Let's cover the specific case of the stack of triangles in the topological sense, since that's what is done on page 73 of the article. They define such a stack to be a fibration over Top, meaning it comes from a functor from Top valued in groupoids. My question now is: what exactly is this functor doing? It seems to send a topological space to the groupoid of families of triangles "inside" or "parameterized by" that space.

Wait! "Inside" is completely different from "parametrized by". Suppose we only care about triangles in the plane. They're all inside the plane. We can still care about a 1-parameter family of triangles $T_t$ where $t \in \mathbb{R}$. Then these triangles are parametrized by the line.

So here's my answer to "what is exactly is this functor doing?" It may not be right, because I haven't even looked at page 73, but here's what my answer is:

We start with concept of triangles in the plane and isomorphisms between these. For any topological space $X$ we get a groupoid $f(X)$ where an object is a triangle parametrized by $X$, i.e. roughly a map sending each point of $X$ to a triangle in a smooth (or actually algebraic) sort of way. This gives a (pseudo)functor $f: \mathsf{Top}^{\text{op}} \to \mathbf{Gpd}$.

John Onstead (Jan 09 2025 at 21:43):

John Baez said:

I've never heard about the generic point of a stack, though it may make perfect sense.

The "generic point" was introduced in the original Mathstodon post that inspired this whole discussion! I'd like to be able to ask the original poster more clarifying information (such as describing which variant of the "moduli stack of triangles" being referred to, since there seems there's multiple possibilities of how such a thing can be defined), but I don't have a Mathstodon account.

John Baez said:

What matters more to me is getting a feel for the examples we typically care about here! I think this is the sort of example I understand. For any topological space, there's a stack of vector bundles over that space.

Maybe that would be a better thing to cover first, since I already have some familiarity with vector bundles from the previous conversation on them. For instance, trying to figure out what the "universal family" of the stack of vector bundles would be (or if this stack even is a fine stack, since if it isn't then there will not be such a universal family). Also, it would be interesting to wonder what the "underlying topological space" is of the stack of vector bundles (IE, the "moduli space" of vector bundles), and what a generic point in this space might be (if that's even something worth asking).

John Onstead (Jan 09 2025 at 21:46):

John Baez said:

So here's my answer to "what is exactly is this functor doing?" It may not be right, because I haven't even looked at page 73, but here's what my answer is:

trianglesstack.png

No idea what any of these words mean (degree 3 covering? A continuous map such that the triangle inequality is satisfied? Where's the actual triangles!?)

John Baez (Jan 09 2025 at 22:08):

John Onstead said:

John Baez said:

I've never heard about the generic point of a stack, though it may make perfect sense.

The "generic point" was introduced in the original Mathstodon post that inspired this whole discussion!

Okay. I said what the generic point of a scheme is, and I also kind of know what it's like. Since an algebraic stack can be seen as a '2-scheme', or more precisely a blend of the concepts of scheme and groupoid, I wouldn't be surprised if some people on Mathstodon run around talking about the generic point of a stack and expecting we'll understand. Hopefully it's the obvious generalization.

What matters more to me is getting a feel for the examples we typically care about here! I think this is the sort of example I understand. For any topological space, there's a stack of vector bundles over that space.

Maybe that would be a better thing to cover first, since I already have some familiarity with vector bundles from the previous conversation on them.

Note I was talking about the stack of all vector bundles on some particular topological space $X$. This is a stack on the site of open subsets of $X$ that assigns to each open set $O \subseteq X$ the groupoid of all vector bundles over that open set.

There's also the stack of vector bundles, which is a stack on the site $\mathsf{Top}$, which assigns to each object $X \in \mathsf{Top}$ the groupoid of all vector bundles over $X$.

I used to find this dual usage of stacks incredibly confusing. Now I just find it confusing. Anyway, it's good to be clear which one you're talking about.

There's a discussion of the second one on Stack Exchange (naturally).

John Baez (Jan 09 2025 at 22:11):

I have tons more intuition for the universal family of triangles (or elliptic curves) than the universal family of vector bundles, which I've never heard anyone discuss.

I guess the idea would be that you take the stack of vector bundles, and think of it as being like a space. (You can always think of a stack as being like a 'space where the points have automorphisms' - this is another conceptual maneuver people do.) Then, you show there's a tautological vector bundle over this space, which is the 'universal family of vector bundles'.

John Onstead (Jan 10 2025 at 06:31):

John Baez said:

I used to find this dual usage of stacks incredibly confusing. Now I just find it confusing. Anyway, it's good to be clear which one you're talking about.

This must be one of the instances in which it's better to "get used to" math, as per the quote by Von Neumann. In any case, the difference between sheaves on the open sets of a single space and on all topological spaces is the difference between the petit (little) and gros (big) perspectives on doing sheaf theory. So this confusion isn't something unique to stacks of vector bundles!

John Onstead (Jan 10 2025 at 06:39):

John Onstead said:

No idea what any of these words mean (degree 3 covering? A continuous map such that the triangle inequality is satisfied? Where's the actual triangles!?)

I also think I understand this a bit better. A "degree 3 covering" might mean a function $T' \to T$ such that every point in $T$ has a fiber over it consisting of exactly three points from $T'$. The map $T' \to \mathbb{R}$ that obeys the triangle identity then ensures that each set of three points in $T'$ actually behaves like a triangle. Therefore, when we say that $T$ parameterizes triangles in $T'$, we mean that by this map, every point in $T$ represents a triangle in $T'$. So $T$ itself, along with this map, actually acts like a whole set of triangles- thus a family of triangles! So the groupoid that the stack sends a space $T$ to will consist of every single morphism from every single other topological space into $T$ obeying this property of being a degree 3 covering, along with the extra structure of a map from the domain topological space to the real numbers satisfying the triangle identity. So it's a groupoid of structured bundles over $T$ that has a faithful forgetful functor into the category of all degree 3 coverings over $T$, and the latter category then has a full and faithful embedding into the slice category $\mathrm{Top}/T$. Hopefully this is a good assessment of what this stack is doing as a functor!

John Baez (Jan 10 2025 at 07:20):

John Onstead said:

John Baez said:

I used to find this dual usage of stacks incredibly confusing. Now I just find it confusing. Anyway, it's good to be clear which one you're talking about.

This must be one of the instances in which it's better to "get used to" math, as per the quote by Von Neumann. In any case, the difference between sheaves on the open sets of a single space and on all topological spaces is the difference between the petit (little) and gros (big) perspectives on doing sheaf theory. So this confusion isn't something unique to stacks of vector bundles!

Right. Back in my confused youth, the petit and gros distinction confused the hell out of me. Maybe people also talk about 'petit 2-topoi' and 'gros 2-topoi' of stacks.

John Baez (Jan 10 2025 at 07:22):

John Onstead said:

A "degree 3 covering" might mean a function $T' \to T$ such that every point in $T$ has a fiber over it consisting of exactly three points from $T'$.

Yes, I bet that's what it means. For example if I have a 'triangle bundle' over a space (a bundle where each fiber is a triangle, however we decide to define a triangle) then the fact that every triangle has three vertices gives a degree 3 covering of that space.

David Corfield (Jan 10 2025 at 10:49):

Amazing what's going on when someone points to a diagram on a blackboard and says "Consider this triangle". You can't be sure the type to which the element belongs: triangles with labelled/unlabelled vertices, ordered/unordered/cyclically ordered vertices, oriented, up to scale invariance, translation invariance, rotation invariance,... Notational devices generally give some indication.

Then there are morphisms between these types, such as your degree 3 cover above. Sometimes it seems like a trick when we pass between types, such as the proof of equal angles in an isosceles triangle that goes via the congruence of ABC and ACB.

John Baez (Jan 10 2025 at 17:01):

Thanks for pointing out the multiplicity of things we could mean by "triangle"! I'd been scared to get into it. In my short article The moduli space of acute triangles, I pick one, or actually two, for acute triangles.

Sometimes it seems like a trick when we pass between types, such as the proof of equal angles in an isosceles triangle that goes via the congruence of ABC and ACB.

In many versions of the moduli stack of triangles the isosceles triangles are 'stacky points' because of these 2-fold symmetries, and then equilateral triangles are even more stacky due to their 6-fold symmetries. These are visible in the picture in my article.

(This may be mainly for @John Onstead.)

John Onstead (Jan 11 2025 at 00:27):

John Baez said:

Thanks for pointing out the multiplicity of things we could mean by "triangle"!

I'm wondering how that's meant to work. Is it that each notion of "triangle" corresponds to a certain family of triangles, that is, an object in the "category fibered in groupoids" corresponding to the stack of triangles? In which case every notion of triangle is contained within just this single stack. Or is it that each notion of triangle gives rise to entirely different stacks defined on different categories, and the notion of "stack of triangles" on Top which we've been covering is just one type?

John Onstead (Jan 11 2025 at 00:29):

John Baez said:

Yes, I bet that's what it means. For example if I have a 'triangle bundle' over a space (a bundle where each fiber is a triangle, however we decide to define a triangle) then the fact that every triangle has three vertices gives a degree 3 covering of that space.

Great! I thought as much, since it sounded like "a cover, but three times over".

I'm interested in what the triangle stack functor does to the terminal object of $\mathrm{Top}$. At first thought, I thought it would send it to the groupoid of individual triangles, since a family parameterized by one point is just a singular instance of the object. However, I realized that no matter how many triangles you have in a space, there's only one morphism from the space to the point (via the terminal object property), meaning there isn't a way for each individual triangle in a space to be individually parameterized, at least not in this way. In addition, the map wouldn't satisfy being a degree 3 cover- only a map from maybe a 3 element space would suffice.

This leaves me with two questions. First, is there a way to study individual triangles in the moduli stack of triangles, or can you only study families and have to "downgrade" to the moduli space to study individual triangles (but then you'd be studying isomorphism classes of triangles...) Second, let's say we've taken the pullback of a degree 3 cover and a point, meaning we've found the space representing a fiber for any degree 3 cover, and thus any triangle. This obviously does have a degree 3 cover map to the terminal object, meaning it is an object of the resulting groupoid of families of triangles over the point. Does it possess any interesting properties in this setting?

John Baez (Jan 11 2025 at 04:42):

John Onstead said:

Is it that each notion of "triangle" corresponds to a certain family of triangles, that is, an object in the "category fibered in groupoids" corresponding to the stack of triangles? In which case every notion of triangle is contained within just this single stack. Or is it that each notion of triangle gives rise to entirely different stacks defined on different categories [...]?

The latter. Let's get specific and do a couple examples. Suppose you define a triangle to be an ordered triple of points in the plane $\mathbb{R}^2$ and define an isomorphism from a triangle $T = (v_1, v_2, v_3)$ to a triangle $T' = (v'_1, v'_2, v'_3)$ to be a map $f: \mathbb{R}^2 \to \mathbb{R}^2$ of the form

$f(x) = A x + b$

where $A \in \text{SO}(2)$ is a rotation and $b \in \mathbb{R}^2$ describes a translation, such that

$f(v_i) = v'_i$.

This so far defines a groupoid of triangles, say $\mathsf{T}$. This is as fine as far as it goes. We already see that some triangles have larger automorphism groups than others. We can completely classify the possible automorphism groups. This is a good exercise, e.g. figure out the automorphism group of an isosceles or equilateral triangle, or a degenerate triangle where all 3 vertices are equal, and figure out what other possibilities there are.

Next, note that we can give $\mathsf{T}$ the structure of a groupoid internal to $\mathsf{Top}$ by the giving the space of objects, namely $\mathbb{R}^6$, its usual topology, and giving the space of all morphisms its obvious topology as well. (This topology is not extremely obvious, so it's worth thinking about.)

Now, there's a way to turn any groupoid internal to $\mathsf{Top}$ into a topological stack, as explained here:

• nLab, Topological stack.

so this gives one topological stack of triangles. For many purposes we don't even need to care about topological stacks to have fun here: we can work with the groupoid internal to $\mathsf{Top}$! However, the morphisms between topological stacks are more flexible than the usual notion of morphism between groupoids internal to $\mathsf{Top}$.

We can get another groupoid internal to $\mathsf{Top}$, and thus another topological stack, by using the same ideas but allowing $A$ to be a reflection and/or rotation, i.e. $A \in \text{O}(3)$. Now for example an equilateral triangle has 6 automorphisms instead of 3. We could say the triangles in the previous stack are oriented, while those in this stack are unoriented.

Now let's switch gears and give $\mathsf{T}$ the structure of a groupoid internal to $\mathsf{Diff}$, the category of smooth manifolds! This amounts to noticing that the set of objects and the set of morphisms of $\mathsf{T}$ can both be given the structure of smooth manifolds, with all the groupoid operations being smooth.

Starting from this groupoid internal to $\mathsf{Diff}$ we can get a differentiable stack.

We can also do this for unoriented triangles.

John Baez (Jan 11 2025 at 04:52):

I'm interested in what the triangle stack functor does to the terminal object of $\mathsf{Top}$.

It gives you the groupoid of "continuously varying families of triangles parametrized by a point". But this is just the groupoid of triangles! What that is depends on what we mean by 'triangle', but if we make one choice we get the groupoid called $\mathsf{T}$ above.

John Baez (Jan 11 2025 at 04:55):

First, is there a way to study individual triangles in the moduli stack of triangles... [?]

I think we just saw how: $\mathsf{T}$ is the groupoid of triangles, and its objects are triangles. It's what our chosen 'triangle stack' gives when you hand it the terminal object of $\mathsf{Top}$.

John Baez (Jan 11 2025 at 04:56):

Second, let's say we've taken the pullback of a degree 3 cover and a point, meaning we've found the space representing a fiber for any degree 3 cover, and thus any triangle. This obviously does have a degree 3 cover map to the terminal object, meaning it is an object of the resulting groupoid of families of triangles over the point. Does it possess any interesting properties in this setting?

I find this question confusing, so I hope that the stuff I just said either helps you answer it, or helps you decide there's some better question.

John Onstead (Jan 11 2025 at 14:54):

John Baez said:

The latter.

Ah ok, that makes sense. I guess it's similar to how you need to change your category when considering different level of details about the objects in the category. For instance, if you're fine with considering spaces up to homeomorphism, then you will use Top. But if you want to preserve the geometric structure of the object, you might use the category of smooth manifolds or Cartesian spaces or affine spaces or something like that.

John Baez said:

It gives you the groupoid of "continuously varying families of triangles parametrized by a point". But this is just the groupoid of triangles!

I'm a little confused by this. A family of some object parameterized by $X$ is an object of a category of structured bundles over $X$- that is, a category with a faithful functor into $\mathrm{Top}/X$, at least if we are considering defining triangles topologically. It's hard to see how the category of families of triangles parameterized by the point is the groupoid of individual triangles when you're contending with the universal property of the point, in that $\mathrm{Top}/* \cong \mathrm{Top}$, meaning there's only one morphism into the point for every space. In any case, as I mentioned above, a "triangle family" is a degree 3 cover, and the category of those is a subcategory of $\mathrm{Top}/X$ on the morphisms satisfying that property. But a morphism into the point can only satisfy that property if it corresponds to a topological space with three points. This already narrows down the number of "triangles" to 29, since there's 29 topologies on a set with 3 elements, and so 29 objects in the category of degree 3 coverings over the point. But there's obviously an infinite number of triangles, so where am I going wrong here?

John Baez (Jan 11 2025 at 18:35):

In any case, as I mentioned above, a "triangle family" is a degree 3 cover...

I don't agree with that: in my way of thinking, a triangle family parametrized by $X$ gives a degree three cover of $X$, but it's not the same thing.

More precisely, if you define a triangle family as a degree 3 cover, and you combine that with what I recently said, you're led to a kind of contradiction, as you note.

I prefer to think of a triangle family parametrized by a topological space $X$ as a map of topological stacks from $X$ (regarded as a topological stack) to the topological groupoid $\mathsf{T}$ of triangles which I defined earlier (also regarded as a topological stack).

That's a mouthful, and by the way I'm now starting to use "topological groupoid" to mean "groupoid internal to $\mathsf{Top}$". (This is a bit risky since there are a couple of different meanings.)

But in the case where $X$ is just a point (regarded as the terminal topological stack) then we see a family of triangles parametrized by a point is just an object of $\mathsf{T}$ - that is, a triangle!

John Onstead (Jan 11 2025 at 21:41):

Ok, I think I see. The topological groupoid is kind of like the moduli space in that it "classifies" families of triangles. Then it makes sense to say a triangle is parameterized by the point since a morphism from the point into any topological space picks out a point in that space.

But as with any moduli space, the classifying space only serves to reinterpret any morphism of some type into an object as a morphism out of it. And the article certainly defines a family of triangles as a morphism into an object, that is, as a kind of bundle. But I think I know where I might have gone wrong after looking things over again. Yes, the triangle family needs to be a degree 3 cover, but it needs to come with a morphism from the domain of the cover to the positive real numbers. And of course there's an infinite number of those, and hence an infinite number of triangles.

But this is something I'm still a little confused on. Since the map is into the positive real numbers, I'm imagining it as something that assigns a "side length" to the triangles, in which case the three points aren't meant to be the vertices but rather the sides of the triangle. But if a degree 3 cover is meant to parameterize triangles "in" the domain, then the points have to be interpreted as vertices and not side lengths. That's because a "side" isn't a single point but a whole line of them, and in any case those might only exist in euclidean spaces, not in any general topological space. So what exactly is going on here: are the points in a fiber of a degree 3 cover sides or vertices of a triangle? And if they are vertices, then why not have the map instead into the cartesian plane where any set of three points has an immediate notion of "side length" between them via the distance function there (and to boot, forms an actual honest triangle)?

John Baez (Jan 11 2025 at 23:34):

By the way, I'm not reading Kai Behrend's article now and I don't plan to read it, so if you ask me questions about it I won't exactly answer them: instead I'll talk about my own views about various moduli stacks of triangles. This is not because I disagree with him, or anything like that! It's just that it's better for me to think about this myself rather than read a paper about it.

So, when you say "the map is into the positive real numbers", I don't know what that means. I had defined a topological groupoid $\mathsf{T}$ of triangles, where the objects are actual triangles in the plane; then a map from a topological space $X$ (viewed as a degenerate sort of topological groupoid) into $\mathsf{T}$ picks out for each point of $X$ an actual triangle, and this seems like a very intuitive and reasonable concept of "family of triangles parametrized by $X$".

(What I just said ignores how maps between topological stacks are more supple than the obvious maps between topological groupids, but never mind: I'm just explaining why I can't answer your question, but am still happy.)

John Onstead (Jan 12 2025 at 04:00):

John Baez said:

By the way, I'm not reading Kai Behrend's article now and I don't plan to read it, so if you ask me questions about it I won't exactly answer them

Sorry about that! I did find another part of the paper where he goes into a little more detail about what he means by a map into the positive real numbers, but in any case I'll stop asking about it.

John Baez said:

So, when you say "the map is into the positive real numbers", I don't know what that means. I had defined a topological groupoid $\mathsf{T}$ of triangles, where the objects are actual triangles in the plane; then a map from a topological space $X$ (viewed as a degenerate sort of topological groupoid) into $\mathsf{T}$ picks out for each point of $X$ an actual triangle, and this seems like a very intuitive and reasonable concept of "family of triangles parametrized by $X$".

This approach kind of reminds me of the beginning of our discussion on fiber bundles. There, we were discussing how to switch between viewing a bundle as a function into a set that assigns every element of a fiber over a point to that point, and as a function out of a set that assigns each point to the fiber over it in some "set of fibers". I think the conclusion was that we could do this in Top if we enlarged to a quasitopos like pretopological spaces and then did some internal hom constructions. But now I'm realizing it may have been better to think instead in terms of the topological groupoid that represents the moduli stack of bundles, and think of a bundle as a map into that!

Using your point of view, I will ask this question: for which object in Top, and for which morphism from it into this topological groupoid, is that morphism exactly the universal family of triangles?

John Baez (Jan 12 2025 at 07:21):

This approach kind of reminds me of the beginning of our discussion on fiber bundles. There, we were discussing how to switch between viewing a bundle as a function into a set that assigns every element of a fiber over a point to that point, and as a function out of a set that assigns each point to the fiber over it in some "set of fibers". I think the conclusion was that we could do this in Top if we enlarged to a quasitopos like pretopological spaces and then did some internal hom constructions. But now I'm realizing it may have been better to think instead in terms of the topological groupoid that represents the moduli stack of bundles, and think of a bundle as a map into that!

I think you're right - or at least, a bunch of mathematicians do it that way. This is especially true for things like principal $G$-bundles in Diff: a smooth principal $G$-bundle $P \to X$ should give a map $X \to BG$ where $BG$ (a notation that people use for too many different things!) now means the moduli stack of principal $G$-bundles, which is a differentiable stack. (We should probably call it a 'smooth stack'.) The machinery of differentiable stacks should let us take the fact that the map $X \to BG$ is smooth and extract from this the fact that "the fiber $P_x$ of $P \to X$ over the point $x \in X$ depends smoothly on $x$".

All this should work for topological principal $G$-bundles, and it should generalize to locally trivial bundles and maybe even fully general bundles - though I've never seen anyone talk about a moduli stack of those, because it's probably 'large' in the set-theoretic sense! Maybe some 'modal homotopy type theorists' could handle it.

John Baez (Jan 12 2025 at 07:42):

Using your point of view, I will ask this question: for which object in Top, and for which morphism from it into this topological groupoid, is that morphism exactly the universal family of triangles?

This may be the sort of case where a universal family doesn't exist in the sense you describe - there's a big discussion about situations where a universal family does exist or does not exist. But it's still worth stating one obvious initial guess: the space of objects of $\mathsf{T}$. Remember, $\mathsf{T}$ is the topological groupoid where the objects are triangles in the plane and morphisms are symmetries between triangles. So an obvious guess for the universal family of triangles is

$i: \mathrm{Ob}(\mathsf{T}) \to \mathsf{T}$

(the inclusion of $\mathrm{Ob}(\mathsf{T})$ viewed as a topological groupoid with only identity morphisms into $\mathsf{T}$.)

This initial guess is probably wrong, but it's still worth stating, because I think it's very interesting to think about families of triangles that do or do not factor through this one. If they all factor through this one with no problem, then this one deserves to be called the universal family.

John Onstead (Jan 12 2025 at 14:56):

John Baez said:

This may be the sort of case where a universal family doesn't exist in the sense you describe - there's a big discussion about situations where a universal family does exist or does not exist.

I know you might not like this, but I do have to mention the article I read does assert that the stack of triangles is a "fine stack" and so does have a universal family!

John Onstead (Jan 12 2025 at 14:58):

John Baez said:

If they all factor through this one with no problem, then this one deserves to be called the universal family.

Ok, I'm trying to understand what this means. So a family of triangles parameterized by some space is, in a sense, some particular set of triangles. But it seems that the universal family of triangles is parameterized by the space that's none other than a "moduli space" of triangles itself (the object $T$). This space essentially contains all the triangles as points. So would that mean that the universal family of triangles, as a family of triangles parameterized by the space of all triangles itself, be the family of all triangles? That is, a family of triangles that's not just some particular set of triangles, but the set of all triangles, in family form?

Mostly though it seems the universal family is "universal" since it satisfies a sort of pullback property. That every family of triangles can be seen as a pullback of some sort of this universal one. I'm not quite sure of what that means. Maybe this condition is indeed confirming that the universal family of triangles includes all triangles, since in some cases a pullback does allow you to find subsets. Could it be that the "all families of triangles are unique pullbacks of the universal family" condition is merely stating "all families of triangles are subsets of the set of all triangles"?

John Baez (Jan 12 2025 at 18:53):

John Onstead said:

John Baez said:

If they all factor through this one with no problem, then this one deserves to be called the universal family.

Ok, I'm trying to understand what this means. So a family of triangles parameterized by some space is, in a sense, some particular set of triangles.

It's a lot more structured than a mere set of triangles, just like any function $f: X \to Y$ has a lot more structure than the mere range of that function.

A family of triangles parameterized by some space $X$ is - roughly, later we will have to revisit this and improve it using stacks! - a continuous map from $X$ to the topological space of all triangles. You should imagine you have a specific set of knobs that you can turn to change the shape of a triangle. $X$ is the space of all knob positions.

But it seems that the universal family of triangles is parameterized by the space that's none other than a "moduli space" of triangles itself (the object $T$). This space essentially contains all the triangles as points. So would that mean that the universal family of triangles, as a family of triangles parameterized by the space of all triangles itself, be the family of all triangles? That is, a family of triangles that's not just some particular set of triangles, but the set of all triangles, in family form?

Right. That's why it's universal. But again, it's more than a mere set.

Any space is parametrized by itself via the identity map.

Could it be that the "all families of triangles are unique pullbacks of the universal family" condition is merely stating "all families of triangles are subsets of the set of all triangles"?

It's not merely saying that. If you start talking about mere subsets, you're throwing away the parametrization and everything involving topology. You're taking a continuous map $f: X \to Y$ and replacing it with the set $\mathrm{ran} f$, which is far less interesting.

John Onstead (Jan 13 2025 at 02:03):

John Baez said:

It's not merely saying that. If you start talking about mere subsets, you're throwing away the parametrization and everything involving topology.

I see. So then what is the pullback condition saying specifically? What does it even mean that one can express a family of triangles as a pullback of the universal family?
For instance, in the category Top. If you have a morphism $X \to T$ (with $T$ the topological groupoid for triangles) and you also have the universal family map $U \to T$, then the pullback of this cospan $P$ gives a projection $P \to X$. What exactly is this doing, and what kind of object is $P$? And more importantly, what is this pullback doing in the category in the domain of the fibration (I'm not too clear on what a Cartesian morphism is, and what it has to do with pullbacks, so please enlighten me on how pullbacks in Top might affect what goes on in any category fibered over Top)

John Baez (Jan 13 2025 at 02:14):

What's $U$?

The way I've been trying explain the universal family and other families of triangles to you does not rely on pullbacks or fibered categories, as far as I can tell. It's possible that this is one of those commonly occurring games where we can turn "maps in" into "maps out", and one view involves pullbacks while the other does not.

John Onstead (Jan 13 2025 at 05:10):

John Baez said:

The way I've been trying explain the universal family and other families of triangles to you does not rely on pullbacks or fibered categories, as far as I can tell. It's possible that this is one of those commonly occurring games where we can turn "maps in" into "maps out", and one view involves pullbacks while the other does not.

Ah, I see.

John Onstead (Jan 13 2025 at 05:10):

I've enjoyed learning at least a bit about moduli spaces and algebraic geometry here, but ultimately it would take too much time to cover fully. At the same time, I still want to understand what Boarders meant in the original mathstodon post, since I want my initial question (about the philosophy of how the algebraic geometry approach to genericness via generic points compares to the categorical approach via representability) answered. Unfortunately, all we can do here is speculate about that, since as we've seen there's so much vagueness across the board, from defining the stack of triangles itself, to determining what a generic point is supposed to mean outside the context of schemes, and so on. Thus to answer my question in a satisfactory way, I think I need to know the author's "original intent" so to speak.

To that extent I've drafted up an email I want to send directly to Boarders, but I'm not sure where to direct it. Under the username it gives something that looks like an email address (boarders@mathstodon.xyz) but as I've never used mastodon, I don't know if this is an actual genuine email that redirects messages to the user's own personal email (or potentially their mastodon account). That's my question for now: is this an actual email, and if not, do you know where I can find the actual email address? And also, if you want, I can share my drafted email below if you want to review it, or if you wanted to add on questions of your own. Thanks!

John Baez (Jan 13 2025 at 18:20):

It's not an email address. Join us at Mathstodon and you can send Boarders a direct message asking your question, or simply respond to their post.

Indeed, I invite all of you to join Mathstodon. I post lots of fun math and physics stuff there at

https://mathstodon.xyz/@johncarlosbaez

You don't need to join Mathstodon to read those posts, but to interact with people on Mathstodon you need to either join it or join some other instance of Mastodon. One advantage of Mathstodon over other instances is that you can use $\LaTeX$.

John Onstead (Jan 13 2025 at 22:38):

I signed up, but I suppose my application is under review. Hopefully they'll approve!

John Baez (Jan 13 2025 at 22:40):

Great! I think it's routine.

John Baez (Jan 13 2025 at 22:50):

By the way, I would be perfectly happy to keep talking about the stack of triangles if I get to talk about it my way. I think we were getting close to the interesting part, but we didn't reach the interesting part. I tried to explain why triangles form a topological groupoid $\mathsf{T}$ and why to a first approximation a family of triangles parametrized by a topological space $X$ is a map of topological groupoids $X \to \mathsf{T}$. However we didn't get to the point where I show you why this isn't good enough: we really need a more general concept of map between topological groupoids, not the obvious one... and when we switch to this improved concept of map, then we get the 2-category of topological stacks.

So in the end, in this approach, topological stacks are just topological groupoids, but the maps between them are more subtle than you might think.

John Baez (Jan 13 2025 at 22:53):

There's nothing particularly special about triangles here, except that $\mathsf{T}$ is very easy to understand (for me anyway). We could also do the topological stack of $n$-dimensional vector bundles, or the topological stack of principal $G$-bundles, which are more important examples. However these have a somewhat different flavor because they are "more stacky": the relevant topological groupoid, or at least the one that's really easy to describe, has fewer objects and more automorphisms per object, so it's less like a topological space.

John Onstead (Jan 14 2025 at 01:44):

I made it onto the Mathstodon server! Hopefully people can see my posts!

John Baez (Jan 14 2025 at 02:35):

John is here, btw:

https://mathstodon.xyz/@johnonstead

Graham Manuell (Jan 14 2025 at 07:25):

John Baez said:

we really need a more general concept of map between topological groupoids, not the obvious one...

While they probably aren't what a naive person would first guess, I don't think the morphisms are actually that mysterious. Even in normal constructive mathematics you want to be using [[anafunctors]] instead of functors, so it really isn't too much of a surprise that this is the case for internal categories too.

John Baez (Jan 14 2025 at 18:31):

If you detected a sense of 'mystery', it's because I was attempting to build up a bit of suspense to keep @John Onstead interested. He had already indicated he was getting ready to give up discussing this stuff:

I've enjoyed learning at least a bit about moduli spaces and algebraic geometry here, but ultimately it would take too much time to cover fully.

But I'd actually been trying to explain topological stacks, since both John and I understand topology better than algebraic geometry. I was planning to lead up to anafunctors by looking at examples how we use the moduli stack of triangles, or the moduli stack of vector bundles: why functors between topological groupoids don't suffice, and how anafunctors save the day.

John Onstead (Jan 16 2025 at 07:12):

John Baez said:

But I'd actually been trying to explain topological stacks, since both John and I understand topology better than algebraic geometry. I was planning to lead up to anafunctors by looking at examples how we use the moduli stack of triangles, or the moduli stack of vector bundles: why functors between topological groupoids don't suffice, and how anafunctors save the day.

I wouldn't mind learning about anafunctors. I don't know much about them since I heard they usually come up when the axiom of choice does not apply, and I don't have too much of an interest in constructive mathematics. So I am wondering why they're useful, when to use them, and of course why they are needed for talking about moduli stacks.

David Michael Roberts (Jan 16 2025 at 12:11):

Happy to answer any questions, but currently down with a certain virus... so it might be rather asynchronous

David Michael Roberts (Jan 16 2025 at 12:18):

The main point to start with is that doing things geometrically means you can't find a continuous section of a continuous surjection. So one cannot make continuous (or worse, smooth) families of choices of lifts, as when it can be done for ordinary categories or groupoids

Madeleine Birchfield (Jan 16 2025 at 14:55):

John Onstead said:

I heard they usually come up when the axiom of choice does not apply, and I don't have too much of an interest in constructive mathematics.

They're still of interest to the classical mathematician who doesn't accept the full axiom of choice, say because they work in the Solovay model or assume determinancy.

John Baez (Jan 16 2025 at 18:09):

I don't really know much about constructive mathematics, but anafunctors are perfect if we want to understand how

• for a topological group $G$, principal $G$-bundles over a topological space $X$ are described by maps from $X$ (viewed as a topological groupoid) into the topological groupoid $BG$.

• for a Lie group $G$, principal $G$-bundles over a manifold $X$ are described by maps from $X$ (viewed as a Lie groupoid) into the Lie groupoid $BG$.

John Baez (Jan 16 2025 at 18:12):

and a bunch of other stuff. I'm no expert on stacks, but my preferred viewpoint on topological (resp. differentiable, etc.) stacks is to think of them as internal groupoids but using internal anafunctors as morphisms.

John Baez (Jan 16 2025 at 18:15):

As a nod to the constructivists: this is precisely because the axiom of choice doesn't hold in the categories $\mathsf{Top}$ or $\mathsf{Diff}$.: not every epimorphism splits in these categories, because these categories have 'cohesiveness': intuitively, the points in a topological space or manifold 'stick together' in a way that can prevent us from - for example - finding a global section of a bundle.

John Onstead (Jan 16 2025 at 19:50):

John Baez said:

• for a Lie group $G$, principal $G$-bundles over a manifold $X$ are described by maps from $X$ (viewed as a Lie groupoid) into the Lie groupoid $BG$.

This is similar to the discussion we were having about bundles and connections a few months ago. But anafunctors never came up there and I don't believe I saw them in any of the articles I reviewed (though I could have just missed it). I just assumed these were, for instance, the morphisms of the category of internal categories in Smooth or Top.

John Baez said:

and a bunch of other stuff. I'm no expert on stacks, but my preferred viewpoint on topological (resp. differentiable, etc.) stacks is to think of them as internal groupoids but using internal anafunctors as morphisms.

I understand you prefer to talk about stacks in terms of internal groupoids, but I wanted to ask this since I'm a little more used to the functorial than groupoid perspective on stacks. Are anafunctors between internal groupoids viewed as stacks the same thing as natural transformations between groupoid-valued functors from Top viewed as stacks?

Also, is there any concrete examples of two stacks, viewed as topological groupoids, where we need to use anafunctors to construct a desired map between them?

David Michael Roberts (Jan 16 2025 at 20:25):

Yes! Consider the group G=Z/2, and the space X=the circle. Then the stack of all G-bundles on the category of manifolds corresponds to the groupoid |BG with ine object and Z/2 for its automorphism group. The G-bundles on X, where X is viewed as a topological/Lie groupoid with only identity arrows, correspond not to functors from X to |BG, because there is only one of those, but to anafunctors

David Michael Roberts (Jan 16 2025 at 20:36):

There are two Z/2-bundles on the circle up to isomorphism, and two equivalence classes of anafunctors from the circle to |BZ/2.

Moreover (nontrivial in less degenerate cases) the group of automorphisms of a bundle is isomorphic to the group of automorphisms of the corresponding anafunctor.

David Michael Roberts (Jan 16 2025 at 20:42):

For any G--bundle pi:P->X there are in fact many anafunctors that correspond to it, but there is in some sense a 'best' one: the span

X <-- Č(P) --> |BG

Where Č(P) is the groupoid with object space P, and a unique arrow from p1 to p2 iff pi(p1)=pi(p2).

The left functor is pi on objects, and the right functor sends such a pair (p1,p2) to the unique element of G mapping p1 to p2 under the G-action.

David Michael Roberts (Jan 16 2025 at 20:44):

But one can also turn a collection of transition functions over a given open cover of X into an anafunctor. Can you figure out what it might be?

John Baez (Jan 16 2025 at 21:05):

@John Onstead

This is similar to the discussion we were having about bundles and connections a few months ago. But anafunctors never came up....

Yeah, we were doing more basic stuff so they hadn't come up yet. But they appear very naturally when you continue thinking about how we define bundles using open covers and transition functions.

The reason is that any open cover of a space gives a topological groupoid that wants to be equivalent to the original space - but it's not if you use functors internal to Top. It is if you use internal anafunctors. And this not hard to see, once we get the definitions laid out.

John Baez (Jan 16 2025 at 21:08):

I understand you prefer to talk about stacks in terms of internal groupoids, but I wanted to ask this since I'm a little more used to the functorial than groupoid perspective on stacks. Are anafunctors between internal groupoids viewed as stacks the same thing as natural transformations between groupoid-valued functors from Top viewed as stacks?

I hope so; maybe someone who knows more about stacks can answer that. There should be a few differently defined but equivalent 2-categories of topological stacks.

David Michael Roberts (Jan 16 2025 at 21:13):

The short answer is yes.

David Michael Roberts (Jan 16 2025 at 21:15):

Moreover the modifications between such transformations between pseudofuntors to Gpd are the same thing as transformations between anafunctors.

John Baez (Jan 16 2025 at 21:16):

Also, is there any concrete examples of two stacks, viewed as topological groupoids, where we need to use anafunctors to construct a desired map between them?

Yes! I was hinting at this moments ago. Now that I'm sitting at my computer I can explain it better:

Any open cover of a topological space $X$ gives a surjective submersion $\pi : U \to X$. We can define a topological groupoid $\tilde{U}$ with points of $U$ as objects and with a unique isomorphism $f: u_1 \to u_2$ if $\pi(u_1) = \pi(u_2)$, and none otherwise.

There's an obvious continuous functor $\tilde{U} \to X$ coming from $\pi$ - do you see what it is?

But there's usually no continuous functor going back the other way from $X$ to $\tilde{U}$ because you sometimes have to discontinously hop from one choice of $u$ with $\pi(u) = x$ to another as you move $x$ around. See what I mean? (This 'discontinuous hopping' problem is the failure of the axiom of choice in $\mathsf{Top}$.)

There is, however, an internal anafunctor going back from $X$ to $\tilde{U}$! (In the end you'll be amused to see how tautological this is.)

And thanks to this, we can show $X$ and $\tilde{U}$ are equivalent as topological stacks!

So this is one reason anafunctors are so useful: any open cover of a space gives a topological stack that's equivalent to that space.

Kevin Carlson (Jan 16 2025 at 21:39):

@David Michael Roberts Surely it's pseudonatural transformations, though, not strictly natural ones?

John Baez (Jan 16 2025 at 21:48):

Oh, I hadn't even noticed that. And maybe we should also say "pseudofunctors" where John O. said "functors" here:

Are anafunctors between internal groupoids viewed as stacks the same thing as natural transformations between groupoid-valued functors from Top viewed as stacks?

There's a certain noble breed of person who says "functor" for a pseudofunctor between 2-categories, confident that history is on their side, and I wouldn't be surprised if such people said "transformation" for a pseudonatural transformation between these, but saying "natural transformation" might be pushing it.

David Michael Roberts (Jan 16 2025 at 22:10):

@Kevin Carlson sure. I just meant we should think of Gpd not as a 1-category, and do all the relevant things to get the stacks/indexed category equivalence to work

David Michael Roberts (Jan 16 2025 at 22:11):

I was being lazy!

David Michael Roberts (Jan 16 2025 at 22:11):

There's really a biequivalence of bicategories floating around

John Onstead (Jan 16 2025 at 22:16):

David Michael Roberts said:

Yes! Consider the group G=Z/2, and the space X=the circle. Then the stack of all G-bundles on the category of manifolds corresponds to the groupoid |BG with ine object and Z/2 for its automorphism group. The G-bundles on X, where X is viewed as a topological/Lie groupoid with only identity arrows, correspond not to functors from X to |BG, because there is only one of those, but to anafunctors

Thanks! It always helps to have a running example I can refer back to.

John Onstead (Jan 16 2025 at 22:19):

David Michael Roberts said:

But one can also turn a collection of transition functions over a given open cover of X into an anafunctor. Can you figure out what it might be?

John Baez said:

Any open cover of a topological space $X$ gives a surjective submersion $\pi : U \to X$. We can define a topological groupoid $\tilde{U}$ with points of $U$ as objects and with a unique isomorphism $f: u_1 \to u_2$ if $\pi(u_1) = \pi(u_2)$, and none otherwise.

There's an obvious continuous functor $\tilde{U} \to X$ coming from $\pi$ - do you see what it is?

Looks like the first question was answered!
As for what this functor is, it probably maps the points in exactly the same way as the underlying surjective submersion. As for the morphisms, it "collapses" the isomorphisms into identities. At least I think!

John Onstead (Jan 16 2025 at 22:25):

David Michael Roberts said:

There's really a biequivalence of bicategories floating around

This is the biequivalence between the bicategory of stack functors on Top and natural transformations between them, and the bicategory of topological groupoids and anafunctors between them?

This makes me wonder how to define the category of internal categories with anafunctors between them. I know the category of internal categories with internal functors between them, defined in any category with pullbacks, is the category of models of the essentially algebraic theory of categories internal to that category with pullbacks. Is there a similar way of defining the anafunctor category, perhaps through some other category of models or some universal property? The only thing I can think of currently is to try and define the category of internal categories with internal profunctors between them and take the subcategory on representable profunctors.

John Onstead (Jan 16 2025 at 22:27):

John Baez said:

But there's usually no continuous functor

Also, as a quick aside, I was confused what limit preserving functors had to do with this situation. But then I realized that a "continuous functor" is just the notion of functor internal to Top. Math terminology can be confusing!

David Michael Roberts (Jan 16 2025 at 23:18):

@John Onstead The standard reference is, I blush to say, http://www.tac.mta.ca/tac/volumes/26/29/26-29abs.html

David Michael Roberts (Jan 16 2025 at 23:21):

You can't just take the 'stack functors on Top', but those that admit a 'presentation', namely a suitably nice epimorphism from a stack represented by a topological space (that is, a representable sheaf considered as a stack). So, denoting the yoneda embedding top Top into [Top^op,Gpd] (the 2-category, mind), then for a stack F: Top^op --> Gpd of groupoids, you need a space X_0 and an appropriate map of stacks y(X_0) --> F.

David Michael Roberts (Jan 16 2025 at 23:24):

These are the 'topological stacks', and are exactly the ones that arise (up to appropriate equivalence) by taking the naive presheaf of groupoids associated to an internal groupoid $X = (X_1\rightrightarrows X_0)$ that looks like $U \mapsto Hom(U,X)$, where this hom is the groupoid of internal functors from the space $U$ (considered as a topological groupoid) to the topological groupoid $X$ and internal natural transformations..... and then stackifying it.

John Onstead (Jan 17 2025 at 00:07):

David Michael Roberts said:

The standard reference is, I blush to say, http://www.tac.mta.ca/tac/volumes/26/29/26-29abs.html

Wow, this is really interesting. I haven't gone through everything yet, but from what I've seen it looks like defining a notion of "anafunctor" in full generality ties into sites, localizations, and categorical homotopy theory. I was certainly not expecting this connection!

John Onstead (Jan 17 2025 at 00:11):

David Michael Roberts said:

You can't just take the 'stack functors on Top', but those that admit a 'presentation', namely a suitably nice epimorphism from a stack represented by a topological space (that is, a representable sheaf considered as a stack). So, denoting the yoneda embedding top Top into [Top^op,Gpd] (the 2-category, mind), then for a stack F: Top^op --> Gpd of groupoids, you need a space X_0 and an appropriate map of stacks y(X_0) --> F.

I see! This also helps to explain another thing I was wondering about, which was how the two pictures of a stack, as the functor into groupoids from Top, and as an internal groupoid in Top, are interrelated.

David Michael Roberts (Jan 17 2025 at 00:11):

The concept of anafunctor, however, can be defined in more general 2-categories with certain properties, but let's not get too far afield!

John Baez (Jan 17 2025 at 02:52):

John Onstead said:

John Baez said:

Any open cover of a topological space $X$ gives a surjective submersion $\pi : U \to X$. We can define a topological groupoid $\tilde{U}$ with points of $U$ as objects and with a unique isomorphism $f: u_1 \to u_2$ if $\pi(u_1) = \pi(u_2)$, and none otherwise.

There's an obvious continuous functor $p: \tilde{U} \to X$ coming from $\pi$ - do you see what it is?

As for what this functor is, it probably maps the points in exactly the same way as the underlying surjective submersion.

Right!

As for the morphisms, it "collapses" the isomorphisms into identities. At least I think!

Right! There's nothing else it can possibly do, once we

• agree on what this functor does to objects

and

• agree that when I reuse the same notation $X$ for a topological space to mean a topological groupoid, I mean the topological groupid with $X$ as its space of objects and only identity morphisms (so that $X$ is also its space of morphisms).

Note that this functor $p : \tilde{U} \to X$ is essentially surjective, full and faithful. So it's an equivalence of categories. More precisely: using the axiom of choice we can choose a functor $q: X \to \tilde{U}$ going back, such that $p q$ and $q p$ are naturally isomorphic to identity functors.

But we can't, in general, choose the functor $q$ to be continuous (or even continuous on objects)! This is why we need (continuous) anafunctors to save the day.

John Baez (Jan 17 2025 at 02:55):

John Onstead said:

John Baez said:

But there's usually no continuous functor

Also, as a quick aside, I was confused what limit preserving functors had to do with this situation. But then I realized that a "continuous functor" is just the notion of functor internal to Top. Math terminology can be confusing!

Yeah! I guess we can blame the person who first used the term "limit" in category theory. By the way, who were they, and is it too late to hold them to account for the confusion you went through? :smiling_devil:

Kevin Carlson (Jan 17 2025 at 03:00):

It’s certainly used in the original Eilenberg Mac Lane paper, though often decorated with “projective” and “direct” instead of “” and “co”. Of course this language is inherited from other parts of algebra going further back. I don’t know any of the earlier history but suffice to say, yes, it’s much too late!

Toby Bartels (Jan 17 2025 at 03:08):

John Baez invited me into this conversation, and I don't know if he just thought that I would find it interesting or if he wanted me to say something about internal anafunctors.

I'll mention some history: Internal categories go back to Grothendieck in 1960, and anafunctors go back to Makkai in 1996, but as far as I can tell, internal anafunctors didn't appear until my PhD in 2006. In the special case of internalization to Diff, however, categories internal to Diff go back to Ehresmann in 1959, and anafunctors between Lie groupoids go back to Hilsum and Skandalis in 1987. (I didn't mention Hilsum–Skandalis maps in my dissertation, because I didn't talk to the right people to find out about them until a few months afterwards.) I went into teaching, but David Roberts (who I see has been commenting here) went on to develop internal anafunctors further.

My PhD dissertation is at https://tobybartels.name/2bundles/. David has already cited his work here, so you should probably just read whatever he says.

David Michael Roberts (Jan 17 2025 at 03:16):

Yes, I should have said that Toby is the person who first internalised anafunctors, but I think my thesis (in the case of topological groupoids) and then the paper I linked was where the link to stacks was fleshed out (relying on work of Pronk in _her_ thesis and published in 1996)

David Michael Roberts (Jan 17 2025 at 03:24):

The history of the topic is very multithreaded, and lots of people contributed things that ultimately all turned out to be equivalent, under different names.

David Michael Roberts (Jan 17 2025 at 03:27):

Rather amusingly, you can still see my comment from about 2007 or so where I expressed the worry that Toby's approach and Pronk's approach might not be equivalent.

John Onstead (Jan 17 2025 at 05:17):

John Baez said:

But we can't, in general, choose the functor q to be continuous (or even continuous on objects)! This is why we need (continuous) anafunctors to save the day.

Ah, this helps to clarify things further. So the problem isn't with choosing the map going the opposite direction on the "underlying category" of the topological groupoids. The problem is choosing this map in such a way that it also preserves the topological structure. Which is only possible with anafunctors, since there's no such "regular" internal functor that would do it.

Toby Bartels said:

My PhD dissertation is at https://tobybartels.name/2bundles/. David has already cited his work here, so you should probably just read whatever he says.

Thanks! I'll be sure to read it along with the rest of David Michael Roberts' article.
I find the history very interesting, especially how certain concepts like internal categories or internal anafunctors were described, at least for some special cases, before later research would fully develop that concept.

David Michael Roberts said:

Rather amusingly, you can still see my comment from about 2007 or so where I expressed the worry that Toby's approach and Pronk's approach might not be equivalent.

I'm glad it worked out!

Toby Bartels (Jan 17 2025 at 08:46):

The original ‘limits’ in mathematics were limits of increasing sequences of real numbers (specifically partial sums of positive geometric series). You might think ‹Yes of course, these are limits in the sense of topology, not limits in the sense of category theory, which should have a different name›. But in fact they are limits in the sense of category theory: they are suprema, that is joins, that is coproducts in a thin category. Both the topological and category-theoretic notions of limit are generalizations of this original example.

Category theory took the name from abstract algebra, and the ‘limits’ there weren't the usual finitary examples that beginning category theorists learn: pullbacks, equalizers, etc. They were (as we would say today) colimits of functors from the poset N of natural numbers viewed as a thin category, and limits of cofunctors from N (so functors from N^op). These were (and still are) called ‘direct limits’ and ‘inverse limits’, and you can see how people thought of them as limits of sequences. Again, limits of increasing (or decreasing) infinite sequences of (possibly extended) real numbers are a special case.

Limits in general topology were generalized in a different direction; even if they take values in an ordered space, if the sequence is not increasing or decreasing, then the limit is not a category-theoretic limit (or colimit). But even in analysis (which is perhaps the branch of mathematics least connected to category theory), we have upper and lower limits (lim sup and lim inf), which are better described in order-theoretic terms (secretly category-theoretic) than topological.

John Onstead (Jan 17 2025 at 09:11):

Toby Bartels said:

The original ‘limits’ in mathematics were limits of increasing sequences of real numbers

I suppose this is in reference to the reason for naming "continuous functors". So I guess, just as a continuous function preserves topological limits, a continuous functor preserves categorical limits. Now the naming scheme makes at least a little more sense!

Toby Bartels said:

Limits in general topology were generalized in a different direction

I learned a lot about this on a recent thread, with a lot of implications. For instance, topological spaces- and more general kinds of space- can be defined by a convergence binary relation on ultrafilters or filters. But in an interesting twist, this particular notion of "limit", unlike the simpler elementary examples, cannot be given by any imaginable categorical construction internal to Top (meaning "taking a limit" of this kind is not something that happens in Top). In a way it makes sense- outside of Hausdorff spaces the limits are not unique, while categorical limits are always unique via the universal property. The process of "taking a limit" instead happens in Set with the defining convergence relation for the topological space (or with any equivalent characterization of a space from which this relation can be derived).

Mike Shulman (Jan 17 2025 at 16:13):

The problematic step, terminology-wise, was generalizing the word "limit" from the directed case to that of arbitrary diagram shapes like products and pullbacks where there is clearly no "limiting process" going on in the intuitive sense. I think Freyd tried to get people to call the general notion a "root" instead, although I don't find that very evocative either. Unfortunately, as has been mentioned, that ship has now sailed and "limits" it is. But it's probably a good idea to avoid the ambiguous term "continuous functor" when there are topologies floating around; to be specific we can say "Top-internal functor" or "limit-preserving functor" depending on what we mean.

Kevin Carlson (Jan 17 2025 at 18:10):

Well, “limit-preserving” could technically still mean “topologically continuous”…

Mike Shulman (Jan 17 2025 at 18:23):

Technically, perhaps, but I think anyone who actually used it that way would be subject to well-deserved opprobrium.

John Baez (Jan 17 2025 at 18:27):

I've got my opprobrium lined up and ready to go.

Toby Bartels (Jan 17 2025 at 18:56):

John Onstead said in part:

I suppose this is in reference to the reason for naming "continuous functors". So I guess, just as a continuous function preserves topological limits, a continuous functor preserves categorical limits. Now the naming scheme makes at least a little more sense!

Specifically in response to the discussion between John Baez and Kevin Carlson about how ‘limit’ is a problematic term. (I meant it to be a direct reply to Kevin's post, but I guess that there's no way to see that if I don't quote any of it.)

[O]utside of Hausdorff spaces the limits are not unique, while categorical limits are always unique via the universal property.

Well, unique up to unique isomorphism. So even in preregular spaces, limits are unique up to unique isomorphism in the 2-category of topological (or convergence) spaces, continuous maps, and the Alexandrov/specialization (pre)order. (But then, preregular spaces are equivalent to Hausdorff spaces in this 2-category. At least to find an essentially non-Hausdorff space, you need to go for a more sophisticated example such as Sierpiński space or a Zariski topology rather than the sledgehammer of an indiscrete space.)

David Michael Roberts (Jan 18 2025 at 08:31):

I wanted to make a joke about multiple opprobria, but I couldn't think of one that felt even vaguely grammatically plausible, even if in a fake way

John Baez (Jan 18 2025 at 18:17):

John Onstead said:

John Baez said:

But we can't, in general, choose the functor q to be continuous (or even continuous on objects)! This is why we need (continuous) anafunctors to save the day.

Ah, this helps to clarify things further. So the problem isn't with choosing the map going the opposite direction on the "underlying category" of the topological groupoids. The problem is choosing this map in such a way that it also preserves the topological structure. Which is only possible with anafunctors, since there's no such "regular" internal functor that would do it.

Exactly! Maybe my work is done here. The nLab article [[anafunctor]] explains the details pretty well, not only for anafunctors but for what we really care about here - internal anafunctors.

Morgan Rogers (he/him) (Jan 21 2025 at 09:47):

John Onstead said:

there's so much vagueness across the board, from defining the stack of triangles itself, to determining what a generic point is supposed to mean outside the context of schemes

The definitions didn't seem vague to me, and I'm glad the discussion continued. I found it was helpful for me to think about what a map from the interval to the classifying space of triangles that John described consists of to get a feel for what an interval-indexed family of triangles is.