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Stream: learning: questions

Topic: Groupoids - How are things the same but different?

Alex Kreitzberg (Oct 04 2025 at 23:15):

Trying to understand HoTT seems to have deeply clarified Groupoids for me, specifically that they can be thought of as a generalization of an equivalence relation.

Thinking back on many concepts I've been trying to formalize categorically, it seems like groupoids account for, or at least clarify, most of them. But I'm a bit confused how things are "the same" but "different". So I'm going to try and give examples of this, but also I'd like to just chat a bit about groupoids.

Awhile ago I was trying to understand accordions with category theory. The discussion can be summarized by saying it's useful to model musical concepts with G-Torsors, or "generalized musical intervals" in David Lewin's terminology. By thinking of G-Torsors as a translation groupoid - I could imagine musical concepts as objects, and the arrows transforming between them as given by group transformations. These symmetries could then be given a physical interpretation for how the musical objects would be played, via an accordion layout.

One aspect that bugged me about this, was that an actual accordion had a finite number of buttons, but $\mathbb{Z}^2$ -Torsors wanted an infinite number of buttons. Applying a translation to the highest note falls off the edge of the accordion.

The article "Groupoids: Unifying Internal and External Symmetry" illustrated how action groupoids (or transformation groupoids) restricted to a bounded set, allows one to define symmetrical features via automorphisms of the ambient space.

An infinite grid, its symmetries exist at intersections, and midpoints of lines:
Infinite tile grid

A finite grid requires groupoids to find more distinct sorts of symmetries:
image.png

For example, Its possible to detect the corners of a tiled rectangle, by noticing certain reflections preserve the corner edges.

This gives me ideas for exploring accordions, but maybe I should switch gears to something more familiar. I also wanted to model chess moves, which incidentally also requires the pieces move on a grid. Moving from and to either the corners, edges, or interior squares corresponds to very different familes of moves, which I would say are roughly indicated by the symmetries indicated on the finite rectangle.

However if I choose my isomorphisms to be knight moves of the would be groupoid, the knight can travel to any square. It's still a puzzle to ask which paths there are between two points on the board.

Very similar to this, Baez gave the 16 puzzle as an example of a groupoid, where all the positions of the game correspond to different objects, with moves between states given by isomorphisms. If a game can be won from a given position, it's usually thought of as "equivalent" to a winning position. Conway invented the Surreal numbers by carefully identifying positions of similar "winnability". So I suspect this groupoid perspective is "correct" in that a puzzle that hasn't been finished yet, is "the same" as a finished puzzle. This is a bit confusing to me.

Normally, when I think of something as the same, I think of a sort of congruence. If I know how to play a song on the accordion, and I shift my hands, it's still the same song. I'd even go so far as to suggest a path of transitions between various dance positions determines a dance, which different people can perform in different places by simply changing where they start.

Dancing over diagram

Regardless, in HoTT, I believe it's still meaningful to provide a solution to a 16 puzzle in an identity type, to show the puzzle is actually solvable. How should I think about "equivalance" in a context where it's important to provide a proof that things are equivalent, like in solutions to the 16 puzzles?

Context for a second question, I'm used to groups being defined as transformations which preserve certain structures or properties of a set we want to explore. With the classic groups being given by isometries. So interesting stuff doesn't happen until you have a symmetrical figure that maps to itself.

If we lift our perspective to groupoids, thinking of one specified triangle as having lengths $(1, 2, 3)$ , then every isometry on this triangle will give an entire groupoids worth of triangles across the plane. Geometric operations on this triangle, like circumscribing with a circle, in a sense should circumscribe all the triangles at the same time. That is, we have a groupoids worth of circles under the same transformations, given by the previous geometric construction. All triangles with various possible side lengths, are connected components of their own groupoids, in general the groupoids will be contractible; however, the non-contractible groupoids correspond to the symmetrical triangles, baking in potential ambiguities.

Somehow it feels like there should be more to the story, than a groupoid of non-symmetrical triangles being contractible to a point. I'm assuming "the rest of the story" is clarified by generalizing to categories. Maps from the contractible groupoid of one triangle, should be different than maps from the triangle of a different contractible groupoid. But I'm not sure how to go about this.

So to restate my questions, what's happening that makes groupoids more natural than groups, even though they're equivalent? Why do the skeletons seem to be worse? How can things be "equal" but different?

Secondly, I have some narrative in my head of how Klein organized geometry around his Erlangen program. I think it's mostly right, but now I'm feeling there could be a real, or false but simpler, history that takes this narrative further. Is there a natural sequence of ideas to motivate classifying geometry by groupoids instead of groups - and then somehow by categories instead of groupoids? Assuming I really understand groupoids well, how do they fit into the story around the "Erlangen program"?

In any case, I have a much deeper feeling for how important groupoids are, and in what way they are "better than" sets with equality. So I'd appreciate any further discussions or suggestions to take this understanding further.

(I have this sense, "in mathematics", that there is a very well understood arrow "groupoids → categories" and a mostly understood arrow "groupoids → ∞-groupoids/HoTT", but that it's still an active research program to work out the "pushout" of "∞-categories" in a really nice way. But right now I'm still trying to sort out those first two arrows).

Chris Grossack (she/they) (Oct 04 2025 at 23:50):

As you say, you can think of a groupoid as being a bunch of things, some of which are the same and some of which aren't. I think two good examples to think about are

The category of finite sets, with bijections
The category of finite dimensional vector spaces, with invertible linear maps.
Eg, the sets {1,2} and {a,b} aren't literally the same, but for many purposes they are, and this groupoid reflects that fact.

I would push back a little bit on saying that "groupoids are equivalent to groups". Every connected groupoid is equivalent to a group, but part of the fun is that groupoids don't have to be connected! For instance, example (1) is the disjoint union of all the symmetric groups $S_n$ . This actually matters quite a lot! For instance, there's a new "parabolic induction" tensor product that you can only see if you look at representations of all the $S_n$ simultaneously. This leads to a very pretty connection with braid diagrams (that gets even prettier if you work with example (2), over a finite field).

Mike Shulman (Oct 04 2025 at 23:53):

Arguably only pointed connected groupoids are equivalent to groups. Of course every connected groupoid can be made pointed somehow, but in general there's no canonical or unique choice of a point. An unpointed connected groupoid only determines a group "up to conjugacy".

Chris Grossack (she/they) (Oct 04 2025 at 23:54):

Yeah, that's an important clarification. Especially since "group up to conjugacy" is a useful way that groupoids naturally show up

Alex Kreitzberg (Oct 05 2025 at 04:11):

Could you say a little bit more about the "parabolic induction" tensor? I see how it's a dramatic demonstration of how useful all the connected components are.

Isn't saying a groupoid is a group "up to conjugacy" the same thing as saying "No really think about the groupoid!"? How do you have a group up to conjugacy without just having all the objects of the groupoid?

Alex Kreitzberg (Oct 05 2025 at 05:03):

I spent a couple days trying to find compelling groupoid resources about games and just now found Dan Pipioni linked to a blog post (https://cornellmath.wordpress.com/2008/01/27/puzzles-groups-and-groupoids/) as a reply to this math overflow question (https://mathoverflow.net/questions/1114/whats-a-groupoid-whats-a-good-example-of-a-groupoid) :face_palm:

I didn't fully understand the significance of the 15 puzzle as a groupoid example. It was way more subtle than I gave it credit for. The blog gave a compelling way to think of different objects in the 15 puzzle groupoid as the same. I'll think about whether I can work this back to my knight move example as well as how this relates to conjugation.

Morgan Rogers (he/him) (Oct 05 2025 at 11:31):

Alex Kreitzberg said:

Could you say a little bit more about the "parabolic induction" tensor? I see how it's a dramatic demonstration of how useful all the connected components are.

I haven't heard the name "parabolic induction tensor" before (does that come from representation theory @Chris Grossack (she/they) ?), but there are a few tensor products on the groupoid of finite sets and bijections. Notably, the restrictions of products and coproducts of finite sets provide tensor products (note that these no longer have the respective universal properties!) which capture how some symmetries of a set can be constructed by decomposing the set and putting together symmetries of the parts.

John Baez (Oct 05 2025 at 12:33):

Since Chris is talking about tensoring representations of different symmetric groups, I figure they must be talking about how $\mathsf{S}$ , the groupoid of all finite sets and bijections, is symmetric monoidal under disjoint union. Thanks to this $[\mathsf{S},\mathsf{Vect}]$ , the category of representations of the groupoid $\mathsf{S}$ , gets a [[Day convolution]] symmetric monoidal structure.

$\mathsf{S}$ is the coproduct of connected components, one for each $n \in \mathbb{N}$ . Call the $n$ th connected component $B S_n$ , since it's equivalent to the one-object groupoid where the automorphisms of that object are the symmetric group $S_n$ :

$\mathsf{S} \simeq \sum_{n \in \mathbb{N}} B S_n$

Thus

$[\mathsf{S}, \mathsf{Vect}] \simeq \prod_{n \in \mathbb{N}} [B S_n , \mathsf{Vect} ]$

and $[B S_n , \mathsf{Vect} ]$ is usually called the category of representations of $S_n$ .

The Day convolution monoidal structure lets you tensor a representation of $S_n$ and one of $S_m$ and get one of $S_{n+m}$ . This is usually called 'induction', since you can define it an [[induced representation]].

Jorge Soto-Andrade (Oct 05 2025 at 16:11):

I would add that parabolic induction was discovered in the 70s by Andrey Zelevinsky and published in Zelevinsky. Representations of Finite Classical Groups (LNM0869, Springer, 1981), also for the case of the q-analogue $GL(n, \mathbb F_q)$ of $S_n$ .
Also, I think, Chris, that another nice example of groupoid to have in mind, besides Path Groupoids, is the Action Groupoid $A(X,G)$ , associated to any (non necessarily transitive) $G$ -set $X$ , used by Alain Connes among others. Its sets of objects is $X$ and its arrows are the triples $(x,g,y)$ where $x,y \in X$ , $g\in G$ and $g.x=y$ . This groupoid allows you to do "geometric induction", without choosing a fixed subgroup $H$ of $G$ , from a representation (quite often a linear character) of $A(X,G)$ to $G$ . This is most useful to construct "Gelfand Models" for the linear complex representations of $G$ . More details in this arXiv preprint In the case of $G = S_n$ the relevant $G$ -set is the set of all involutions in $S_n$ on which $S_n$ acts by conjugacy. The action groupoid has an easy to guess nice linear character...

Mike Shulman (Oct 05 2025 at 17:21):

Alex Kreitzberg said:

How do you have a group up to conjugacy without just having all the objects of the groupoid?

Define the 2-category GrpConj of "groups up to conjugacy" as follows:

Its objects are groups.
Its morphisms are group homomorphisms.
Its 2-cells from $\phi:G\to H$ to $\psi:G\to H$ are elements $h\in H$ such that $h \cdot \phi(g) = \psi(g)\cdot h$ for all $g\in G$ .

Though defined without any reference to groupoids, this 2-category is equivalent to the 2-category of (unpointed!) connected groupoids. Specifically, there is a 2-functor $\mathrm{GrpConj} \to \mathrm{Groupoid}$ taking a group to its corresponding 1-object groupoid. The 2-cells in GrpConj are chosen exactly so as to make this 2-functor 2-fully-faithful (an isomorphism on hom-categories), so it is an equivalence onto its essential image, which is the 2-category of connected groupoids.

Alex Kreitzberg (Oct 06 2025 at 19:35):

I'm still processing all of this and relating it to my examples, I wanted to give a couple comments to indicate I'm listening.

In Shulman's explanation, Functors between groups as categories "are just" homomorphisms, and natural transformations "are just" conjugations (of homomorphisms). That latter part was new to me.

For any arrow $f : x \rightarrow y$ and two Functors $F, G : C \rightarrow D$ , the naturality condition gives $\alpha_y \circ F f = G f \circ \alpha_x$ . When $C, D$ become single object groupoids $G, H$ , the functors $F, G$ become homomorphisms $\phi, \psi : G \rightarrow H$ , the arrow $f$ becomes an arbitrary $g : G \rightarrow G$ , and the natural transformation is given by just one arrow, $h: H \rightarrow H$ from $H$ . Which gives $h \cdot \phi(g) = \psi(g) \cdot h$ for all $g \in G$ .

The point I missed, is instead of viewing groupoids as a member of Cat, I should view them as a member of ~~2-Cat~~ the 2-category Cat.

So to relate this to a concrete example. If I have two congruent equilateral triangles in a plane, then I have $D_3$ choices of how to lay one on top of the other, which is encoded in a group homomorphism. But, even better, in the 2 Cat we can tell two homomorphisms are "the same" when they both express a reflection over some axis.

Baez's and Soto-Andrade's comments aren't the same but "rhyme". There's a lot I'm unpacking, but its clear I need to understand representations better. Here's a bit about how I'm thinking about this.

To understand the 15 puzzle - I'm imagining a 4 by 4 board, with a rook that can only move one square (Or like a character in an old zelda game). There is a (2) Functor from this board as a groupoid to 15 puzzles. Each position of the board, maps to the 15 puzzles with a gap in the same spot. The numbers get jumbled up based on how the gap moves, which then defines a permutation in $S_{15}$ .

So I think, we have $A(15\text{-puzzle},\text{gameboard})$ , or $15\text{-puzzle} : [\text{gameboard}, S]$ (The former is like the category of elements for the later I think). Maybe it'd be cool to consider permutation linear transformations.

I can see the point of Andrade's discussion, that the action groupoid allows a definition of a process we call "induction" (that Baez also referenced) which doesn't require an arbitrary choice of subgroup.

The equivalence $[\sum_{n \in \mathbb{N}} B S_n, \mathsf{Vect}] \simeq \prod_{n \in \mathbb{N}} [B S_n , \mathsf{Vect} ]$ , is also a nice way to see how infinite products and sums in groupoids are interesting.

I want to explicitly reference a "guiding principle" indicated in Alan Weinstein's paper "Groupoids: Unifying Internal and External Symmetry":

Almost every interesting equivalence relation on a space $B$ arises in a natural way as the orbit equivalence relation of some groupoid $G$ over $B$ . Instead of dealing directly with the orbit space $B/G$ as an object in the category $S_\text{map}$ of sets and mappings, one should consider instead the groupoid $G$ itself as an object in the category $G_\text{htp}$ of groupoids and homotopy classes of morphisms.

My motivation for this thread is internalizing this perspective. I'm not sure what the best road is for doing this.

Mike Shulman (Oct 06 2025 at 21:30):

Alex Kreitzberg said:

instead of viewing groupoids as a member of Cat, I should view them as a member of 2-Cat.

No, I would say that rather than viewing groupoids as objects of the 1-category Cat, you should view them as objects of the 2-category Cat. 2-Cat is something different, its objects are 2-categories.

John Baez (Oct 07 2025 at 09:30):

Nobody should ever say "groupoids as a member of Cat".

It's always dangerous to mix singular and plural - like groupoids as a member - because it promotes ambiguity, and the reader, trying to resolve this ambiguity, can reach for different solutions. Do you mean "the category of groupoids as a member of Cat" or "groupoids as members of Cat"?
It's also not usual to say "members" for objects of a category or 2-category - for me at least it denotes elements of a set. So reading "groupoids as a member of Cat", I'm also imagining a set called Cat, but I have to discard this as an unlikely interpretation.

A moment of haste when writing can produce minutes of annoyed head-scratching for dozens of readers.

Alex Kreitzberg (Oct 07 2025 at 17:46):

The singular and plural mix seems to reflect my own confusion actually, good opportunity to fix my semantics by checking my syntax :sweat_smile:.

The "member of" in my mind was an attempt at "type of", but the later phrase feels weird to write informally. I don't think that fixes the sentence though.

Here's another attempt.

"A group is a single object groupoid" is actually a great definition, because groupoids should be read as having natural transformations.

So I'm trying to write in my own words "don't forget the 2-cells!" Maybe that's enough.

Using Shulman's writing as an example, keeping the spirit of my original sentence:

"A groupoid is not just an object of the 1-category Cat, it's even an object of the 2-category Cat!"

I think I wanted to say "A groupoid is a two category" or more obviously wrong "A category is a 2-category", but I didn't understand the error.

Something like "Don't forget the 2-cells!" Is the right way to say this.

Alex Kreitzberg (Oct 07 2025 at 17:56):

I want to assemble a "plural" of "groups" into a "singular" "category". In a sentence like

"A group is a single object groupoid; when assembling groups into a category, don't forget the 2-cells!"

Alex Kreitzberg (Oct 07 2025 at 17:56):

Is that still ambiguous?

Alex Kreitzberg (Oct 07 2025 at 18:26):

It's like "a murder of crows", but honestly I can still feel the risk because we don't usually talk about a "a crime of murders" or whatever. The towers of sorts in category theory should be written about with care. I'll keep this in mind.

I guess the real issue is the word "member", it made that sentence fall apart.

John Baez (Oct 08 2025 at 09:32):

Two things make the phrase "groupoids as a member of Cat" feel to me like fingernails rubbing on a blackboard:

mixing singular and plural
saying "member of Cat".

John Baez (Oct 08 2025 at 09:33):

Each one requires me to choose a way to correct the phrase, so I can get a phrase I can understand.

John Baez (Oct 08 2025 at 09:37):

"A group is a single object groupoid; when assembling groups into a category, don't forget the 2-cells!"

Is that still ambiguous?

No, that's not ambiguous. It's even true if you stretch the meaning of "is" a bit. To clarify this word "is" youcan say something like "any group $G$ gives a one-object groupoid $BG$ , and conversely any one-object groupoid arises in this way up to isomorphism." One can polish this statement further.

But experts don't like to talk about the number of objects in a category since it violates the principal of equivalence. (That's the woke way to say it's evil.) So instead of talking about one-object groupoids they prefer to talk about pointed connected groupoids.

Theorem. A groupoid is equivalent to $B G$ for some group $G$ iff it's a pointed connected groupoid.

John Baez (Oct 08 2025 at 09:38):

There are better things to say....

Alex Kreitzberg (Oct 08 2025 at 20:56):

I don't know if I'm able to express this correctly and tersely.

The 2-category with groups as objects, homomorphisms as 1-morphisms, and conjugations between homomorphisms as 2-morphisms

is equivalent to

the 2-category of connected groupoids.

I also want to say:

The category of groups is equivalent to the category of pointed connected groupoids

But I've been convinced my handle on the basic definitions is worse than I thought, so I looked up [[pointed connected groupoid]]. Happily, I found the book Symmetry by Marc Bezem, Ulrik Buchholtz, Pierre Cagne, Bjørn Ian Dundas, and Daniel R. Grayson.

This book seems like an extended answer to this thread topic from the point of view of HoTT - maybe that's what I want. For example it gives some nice context for pointed connected groupoids on the fifth page in the margin:

pointed connected groupoid motivation

I'd really like to understand groupoids from a "mathematical" lens before I understand them from a "computer science" lens, if that's practical.

John Baez (Oct 09 2025 at 12:01):

Alex Kreitzberg said:

I don't know if I'm able to express this correctly and tersely.

The 2-category with groups as objects, homomorphisms as 1-morphisms, and conjugations between homomorphisms as 2-morphisms

is equivalent to

the 2-category of connected groupoids.

Umm, either that's true or we need to stick the word "pointed" in "the 2-category of connected groupids".

John Baez (Oct 09 2025 at 12:03):

The category of groups is equivalent to the category of pointed connected groupoids.

Maybe I'll throw this one back at you before I think harder about it: what are you taking the morphisms to be in the category of pointed connected groupoids?

(I think there's a fairly obvious answer, but let's make sure we agree on it!)