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Stream: learning: questions

Topic: Why prefer Bundles and Fibers to indexed sets?

Alex Kreitzberg (Jul 28 2025 at 16:05):

There is an equivalence between seeing sets indexed by natural numbers as coming from the natural numbers

$\mathbb{N} \rightarrow \text{Set}$

Or going into the natural numbers

$E \rightarrow \mathbb{N}$

With different operations being more or less difficult depending on which representation we're using.

Nevertheless I get a strong impression the fiber/bundle/over category point of view is generally preferred.

I remember in school they would be quick to mention functions couldn't have multiple outputs, even though it wasn't hard to imagine all the outputs as contained in subsets. Eventually this insistence was somewhat clarified when I eventually learned complex square roots are a double cover. Though I'll admit I don't fully understand the big payoff of the fibered point of view over the $\pm$ notation. But I believe there is one!

Even in type theory, it's common to notate a dependent type vec : Nat -> Type, connotating sets mapped from numbers; but, then the semantics of judgements like n : Nat ⊢ Vec n Type are often explained as slices $\text{Vec} \rightarrow \text{Nat}$.

I think the Yoneda lemma, because of presheafs, is the only theorem statement I'm aware of that emphasizes set valued maps from the indexing set. (Though as I write that, it occurs to me often the whole point of Yoneda is to say it's safe to think of certain sets as arrows).

Am I noticing a real tendency towards bundles, and if yes, what are the key advantages to that point of view? Is it a special case of category theory expressing everything in terms of arrows?

Mike Shulman (Jul 28 2025 at 16:25):

I think the main advantage is that you can talk about "bundles" internal to any category, whereas talking about "families" requires extra structure on the category such as a [[category with families]]. So "of course" category theorists prefer the bundle perspective. Type theorists may feel differently. (-:

Peva Blanchard (Jul 28 2025 at 16:32):

It is not a direct answer to your question, but there is a nice interplay between étale bundles and sheaves; see John Baez series on topos theory. In particular, pulling back a sheaf on $Y$ along $f :: X \to Y$ is easier to express from the bundle point of view.

John Baez (Jul 28 2025 at 16:35):

I'd say the ability to switch viewpoints between indexed entities and fibered entities is is more important than deciding which one is "better" and why. They're good in different ways.

For example, it's nice that indexed categories $f: \mathsf{C} \to \mathbf{Cat}$ are equivalent to fibered categories $p : \mathsf{D} \to \mathsf{C}$

It's also nice that characteristic functions $\chi: X \to \{T,F\}$ correspond to subsets $i: S \to X$.

There are many other variations on this theme. In all cases it's helpful that the first viewpoint is contravariant (for example in the set $X$) while the second is covariant, giving us two sets of tools we can deploy.

Adittya Chaudhuri (Jul 28 2025 at 17:11):

I may be misunderstanding the context, but I often feel the bundle point of view are advantagoues to pack the information of coherence laws that may arise naturally (due to choices of natural isomorphisms) when we work with the index point of view: $f \colon C \to \mathsf{Cat}$, i.e when the fibres are itself categories. Also, I feel "internalisation of the indexing point of view" may be a bit tricky, which I think is a "bit more straightforward" when we take the bundle perspective. For example $VB$-groupoids (Vector bundle object in the category of Lie groupoids) are more easier to define than 2-term representations upto homotopy of Lie groupoids as done here.

Alex Kreitzberg (Jul 28 2025 at 17:20):

Thanks for the answers so far, I am satisfied with "they're both useful" but that there's a feeling for some that "often indexing requires widgets that maybe we can't afford."

My takeaway is I'll get as comfortable as I can with switching and try to guess at why folks make their choices in specific contexts.

Ruby Khondaker (she/her) (Jul 28 2025 at 17:37):

I think one nice thing about the bundle pov is that it can be easier to express what it means to have an indexed family that “varies coherently”

For example, in differential geometry, you often want bundles where the fibers vary smoothly. In the bundle pov this is easy - just ask for the projection to be smooth. For the other pov, you’d need to put some kind of smooth structure on, say, the class of all vector spaces…

John Baez (Jul 28 2025 at 17:44):

And that's why people talk about the stack of vector spaces. A vector bundle on a manifold is a smooth map from that manifold to the stack of vector spaces.

So the question becomes: can you afford stacks? :laughing:

Ruby Khondaker (she/her) (Jul 28 2025 at 17:46):

Oh I’d never come across that notion before! I suppose that stack lets you use the non-bundle pov for vector bundles?

John Baez (Jul 28 2025 at 17:47):

Yes, I added a sentence to my comment to make that clear.

John Baez (Jul 28 2025 at 17:49):

Have you heard of the Stacks Project? Its size may put you off stacks, but it's big because they're trying to explain stacks in algebraic geometry, and first explain a huge amount of algebraic geometry. Right now I was talking about stacks in differential geometry - so-called differential stacks, rather than algebraic stacks.

Nonetheless the basic goal is the same: create a way to equip categories with the extra geometrical structure needed to make the indexed viewpoint equivalent to the fibered viewpoint in the context of geometry. It's useful enough that people want to do it.

Alex Kreitzberg (Jul 28 2025 at 18:51):

I found this motivation for stacks deeply satisfying, and decided to see if I could recontextualize the explanation from "The Princeton companion to Mathematics" because I remember not really "getting it".

This is what the text had to say on Stacks:

Finally, if we marry a scheme to an orbifold, the outcome is a stack. The study of stacks is strongly recommended to people who would have been flagellants in earlier times.

This both amused and annoyed me. I can see why I didn't understand the concept from this book.

I find myself more defensive of the indexed viewpoint now after this thread's discussion XD

Kevin Carlson (Jul 28 2025 at 19:47):

Yeah, that's the sort of joke that reveals the author doesn't understand anything and would like to propagate their lack of knowledge as far as possible, unfortunately. The connection with orbifolds is that, in a stack, points have symmetries, as in the stack of vector bundles every vector bundle has its symmetry group; orbifolds are a particularly simple case where most points have no nontrivial symmetries, although I don't think they're a particularly characteristic example and certainly there's no need to know a word about scheme theory to study stacks that aren't algebraic.

Ruby Khondaker (she/her) (Jul 28 2025 at 19:54):

John Baez said:

Have you heard of the Stacks Project? Its size may put you off stacks, but it's big because they're trying to explain stacks in algebraic geometry, and first explain a huge amount of algebraic geometry. Right now I was talking about stacks in differential geometry - so-called differential stacks, rather than algebraic stacks.

Nonetheless the basic goal is the same: create a way to equip categories with the extra geometrical structure needed to make the indexed viewpoint equivalent to the fibered viewpoint in the context of geometry. It's useful enough that people want to do it.

I had heard of the stacks project but I had no idea this was its goal - that seems so cool!! Is it still ongoing?

John Baez (Jul 28 2025 at 20:13):

Yes. Digressing somewhat, here is a bit from Wei Ho of the Institute for Advanced Studies about the Stacks Project, from an interview at Quanta magazine:

You’re also involved in the Stacks Project, which is an extensive online resource. What’s unique about it?

The sheer volume and accessibility of it. It’s this massive — more than 7,500 pages if you printed it out — online collaborative project. But realistically, [the Columbia University mathematician] Aise Johan de Jong writes nearly all of it. It’s a rigorous, carefully written resource for algebraic geometers. It’s an amazing thing he’s done for the community.

Every week or two, it grows. It’s a trusted reference for nearly anything. It covers a huge amount of algebraic geometry that you would need to look at like 20 textbooks for.

It’s living in the sense that things can be added and edited. If there are mistakes, they will be caught.

The other thing that’s kind of interesting about it is the tag system. Even though this document is constantly growing, you can still reference a specific tag forever. There are over 21,000 permanent tags for particular results you might want to quote. Pieter Belmans built the whole back end, which has been used in other projects as well. Other people have adapted the technology of it.

The problem is — and Johan knows this — he’s eventually not going to be able to keep writing this. Someday, if we want this to continue, it needs other people to be more involved.

What role do your workshops play in the Stacks project?

The point is to start getting younger people involved. We’re having them write bits and pieces that may eventually get incorporated into it. There are some tensions here, because for the website to remain correct and high quality as a resource, it needs to be moderated carefully. So Johan still needs to do a lot of the work in putting things into it. It cannot be like Wikipedia where anyone can touch it. That’s a little unfortunate but has to happen if you want this to work.

Someone can move this elsewhere if they want.

John Baez (Jul 28 2025 at 20:17):

Kevin wrote:

certainly there's no need to know a word about scheme theory to study stacks that aren't algebraic.

Yeah, that passage from the Princeton Companion was ignorant as well as obnoxious.

For anyone who wants to learn a bit about orbifolds and differentiable stacks, here's an expository article by a friend of mine:

• Eugene Lerman, Orbifolds as stacks?

Josselin Poiret (Jul 29 2025 at 08:46):

Mike Shulman said:

I think the main advantage is that you can talk about "bundles" internal to any category, whereas talking about "families" requires extra structure on the category such as a [[category with families]]. So "of course" category theorists prefer the bundle perspective. Type theorists may feel differently. (-:

I have a different opinion that stems from the same reasoning: “families” are more general because what you'd call “bundles” are but one instance of them. You can choose finer notions of families than what the category has as bundles. This is in some sense what fibration categories do, in a more restricted sense.

Josselin Poiret (Jul 29 2025 at 09:12):

regarding the initial question, I'd say that you first need to define what bundles you're interested in: for example, with categories, the basic bundle is that of a displayed category. You can ask for stronger bundles, with transport above morphisms in covariant (resp. contravariant) maps with Grothendieck opfibrations (resp. fibrations). In all these cases there are indexing viewpoints, but they are quite different. Also, notice that the indexing viewpoint for e.g. Grothendieck opfibrations involves 2-categories already, exemplifying the idea that classifying objects are often more complicated than the objects they classify. On the other hand, just the fact that a classifying object exists might make studying it directly interesting!

Vincent Moreau (Jul 29 2025 at 12:39):

@Alex Kreitzberg Let me give two constructions that are nicely expressed in a fibered way. All of this is very simple and should be folklore, but I felt an immense joy when understanding these!

Vincent Moreau (Jul 29 2025 at 12:41):

Suppose you have a function $p : \Sigma \to \mathbb{N}$ in $\mathbf{Set}$, yielding a family of sets $\Sigma_{n} := p^{-1}\{n\}$ whose elements we think of as operations of arity $n$. To equip this family with the structure of an operad, one may choose to consider the things that one can plug in some $\sigma \in \Sigma_n$, namely tuples $\vec{\tau} := (\tau_1, \dots, \tau_n)$. These tuples assemble into the family

$$p^* \quad:\quad \Sigma^* \ \longrightarrow\ \N^*$$

where $(-)^*$ is the monad of monoids, with $p^*$ sending some $\vec{\tau}$ on its "arity" as a tuple, i.e. a list of natural numbers which we can then sum using $\N^* \to \N$. More generally, any functor $F : \mathbf{C} \to \mathbf{C}$ sends a family over the base $B$ to a family over the base $FB$. For example, we can take $F$ to be the ultrafilter monad and $B$ to carry a compact Hausdorff topology. This is in the spirit of generalized multicategories, where we consider spans instead of a single morphism $p$.

Vincent Moreau (Jul 29 2025 at 12:43):

For the second construction, suppose that your ambiant category $\mathbf{C}$ is monoidal and $B$ has the structure of a monoid. Then, we can define some "abstract convolution" on families over $B$ by

$$p\ :\ E \to B \quad,\quad p'\ :\ E' \to B \qquad\rightsquigarrow\qquad m \circ (p \otimes p') \ :\ E \otimes E' \to B$$

where $m : B \otimes B \to B$ is the multiplication of the monoid structure. This is because composition of morphisms represents a form of dependent sum. When $\mathbf{C}$ is $\mathbf{Set}$ with its cartesian monoidal structure, we recover in this way the usual convolution of families, i.e. Day tensor product of presheaves over $B$ seen as a discrete monoidal category!

Matteo Capucci (he/him) (Aug 03 2025 at 12:51):

Josselin Poiret said:

I have a different opinion that stems from the same reasoning: “families” are more general because what you'd call “bundles” are but one instance of them

Uhm how so? Any pullback-stable class of maps can reasonably be taken to encode a notion of displayed 'type family' automatically respecting the ambient structure, but in general it seems quite hard to externalize such a structure. Conversely, not every indexed notion of 'type family' admits comprehension/straightening into a display map of sorts. So I would say in general neither concept is more general than the other.
The obvious umbrella is to see both bundles and families as instances of certain fibrations of categories... or indexed categories lol and then we are back to the same problem one level up.

Josselin Poiret (Aug 03 2025 at 13:17):

Matteo Capucci (he/him) said:

but in general it seems quite hard to externalize such a structure

That's precisely what a CwF is meant to capture though.

Matteo Capucci (he/him) said:

not every indexed notion of 'type family' admits comprehension/straightening into a display map of sorts.

As long as your CwF admits Σ-types, you do have what you would usually call a display map. If you additionally have a universe, you do get straightening/unstraightening (this is tautological, I'm just translating vocabulary here).

Mike Shulman (Aug 03 2025 at 19:21):

Yes, families are more general. (You need a coherence theorem to make the bundles in an arbitrary category with pullbacks into a CwF, but you could also use a weaker notion of "family" like a comprehension category.) My point was that category theorists are not usually so interested in this extra generality and prefer to avoid tte attendant complexity.

Matteo Capucci (he/him) (Aug 04 2025 at 10:08):

Josselin Poiret said:

Matteo Capucci (he/him) said:

but in general it seems quite hard to externalize such a structure

That's precisely what a CwF is meant to capture though.

Oh, I confess I wasn't interpreting 'families' in that sense. Then I agree!

Matteo Capucci (he/him) (Aug 04 2025 at 10:08):

Josselin Poiret said:

Matteo Capucci (he/him) said:

not every indexed notion of 'type family' admits comprehension/straightening into a display map of sorts.

As long as your CwF admits Σ-types, you do have what you would usually call a display map. If you additionally have a universe, you do get straightening/unstraightening (this is tautological, I'm just translating vocabulary here).

Yep, my point is that such extra structure in not a given.

Vincent Moreau (Aug 05 2025 at 21:07):

Thinking about CwFs today, I have found the following little property: let $\mathbf{C}$ be a category, and $p : U_\bullet \to U$ any morphism that we think of as some fixed family, e.g. the universe in type theory. We define a functor

$$P\quad:\quad \mathbf{C}^{\mathrm{op}} \ \longrightarrow\ \mathbf{Fam}$$

where $\mathbf{Fam}$ is equivalent to $\mathbf{Set}^{\to}$. This is an "externalization" of the internal family $p$, as the functor $P$ sends any $X$ on the set-theoretic family $p \circ (-) : \mathbf{C}(X, U_\bullet) \to \mathbf{C}(X, U)$, whose fibers are local sections of $p$. This is akin to externalization of internal categories, or really of any internal model of an essentially algebraic theory. If $\mathbf{C}$ has a terminal object and $p$ can be pulled back along any morphism of $\mathbf{C}$, then this functor $P$ yields a CwF. Is this construction in the literature somewhere?

Vincent Moreau (Aug 05 2025 at 23:09):

Well, it seems that this construction is quite trivial from the viewpoint of natural models: a morphism $p : U_\bullet \to U$ induces a natural transformation between presheaves $p \circ (-) : \mathbf{C}(-, U_\bullet) \to \mathbf{C}(-, U)$ which is representable in the sense of Grothendieck as soon as $\mathbf{C}$ has pullbacks of $p$ along any morphism!