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Stream: theory: category theory

Topic: groupoids from partial group actions

John Baez (Jan 15 2024 at 16:56):

In studying Indian music I ran into a math fact I hadn't noticed. You can get a groupoid, not only from a group acting on a set, but from a partially defined group action on a set.

Let's say a partial action of a group $G$ on a set $X$ is a partially defined map

$G \times X \to X$
$(g,x) \mapsto gx$

such that:

$1x$ is defined and equal to $x$ for all $x \in X$
If one of $(gh)x$ and $g(hx)$ are defined, then so is the other, and they are equal.

John Baez (Jan 15 2024 at 17:00):

From such a thing we get a groupoid where objects are elements of $X$ , a morphism $g: x \to y$ is an element $g \in G$ with $gx = y$ , and the composite of morphisms $g$ and $h$ is their product $gh$ in $G$ .

John Baez (Jan 15 2024 at 17:01):

This is a straightforward generalization of the [[action groupoid]] coming from a group action, but I never noticed it. Is it known? It must be.

The way it arises in Indian music is sort of interesting. In Carnatic music there are 72 seven-note scales called Melakarta ragas, and by cyclically permuting the notes of a Melakarta raga you sometimes get another Melakarta raga. This process is called graha bhedam and it has its own Wikipedia article. It would be an action of the group $\mathbb{Z}/7$ on the set of Melakarta ragas... except sometimes when you cyclically permute the notes of a Melakarta raga you don't get another Melakarta raga, since there are constraints on which 7-note scales count as Melakara ragas. So it's really just a partial group action, in the sense defined above.

John Baez (Jan 15 2024 at 17:11):

So, there's a groupoid of Melakarta ragas, and the connected components of this groupoid are studied on the Wikipedia article.

James Deikun (Jan 15 2024 at 17:45):

This sounds like the natural notion of action for partial monoids in the sense that underlie an [[effect algebra]] or [[separation algebra]]. In general the construction would produce a partial groupoid--it's interesting that only the group needs to be total to get a total groupoid.

David Michael Roberts (Jan 15 2024 at 19:28):

If you have a group action on Y, and X is a subset of Y, you do get a groupoid with object set X, namely the full subgroupoid on X. I'd bet this is what you have defined.

David Michael Roberts (Jan 15 2024 at 19:30):

Your morphisms should include the data of x, though, otherwise you can't define the source map.

John Baez (Jan 16 2024 at 17:11):

Yes, I said a morphism from x to y is an element of G with some property, but since homsets are disjoint it's implicit that x and y are also part of the data of the morphism.

John Baez (Jan 16 2024 at 17:12):

If you have a group action on Y, and X is a subset of Y, you do get a groupoid with object set X, namely the full subgroupoid on X. I'd bet this is what you have defined.

To be precise, I've gotten a groupoid in the case where you have a partial action of G on Y that is not necessarily obtained by restricting an ordinary action of G on some set X to some subset Y $\subseteq$ X.

John Baez (Jan 16 2024 at 17:14):

However, elsewhere you have nicely shown that any partial action of G on Y can be canonically extended to an action of G on some larger set X.

John Baez (Jan 16 2024 at 17:16):

So, knowing that, we don't gain any generality by considering partial actions that aren't given as restrictions of actions.

John Baez (Jan 16 2024 at 17:19):

I think of your nice construction as "the lizard that can regrow its tail when you cut it off" (a metaphor that Jim Dolan likes to use in other contexts). We can restrict any action of G on X to any subset Y $\subseteq$ X, and then "grow back" a new set X' on which G acts, with an inclusion Y $\subseteq$ X'.

John Baez (Jan 16 2024 at 17:27):

The new tail may not be as long as the old tail. But as long as Y contains at least one point of some orbit of the action of G on X, your procedure will correctly re-grow the whole orbit!