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Given the HoTT activity we’ve just seen at regular action, is there anything similarly interesting to say at transitive action for $\infty$-actions?
We have a result there
Let $\rho : G \times X \to X$ be a transitive action and suppose that $X$ is inhabited. Then $\rho$ is equivalent to the action of $G$ by multiplication on a coset space $G/H$, where the subgroup $H$ is taken as the stabilizer subgroup $H = G_x = \{ g \in G \mid g x = x \}$ of some arbitrary element $x \in X$. In particular, the transitivity of $\rho$ guarantees that the $G$-equivariant map $G/H \to X$ defined by $g H \mapsto g x$ is a bijection.
I guess Urs has done everything at stabilizer+group#ForInfinityGroupActions, but with a transitive action $\mathbf{B}Stab_\rho(x)$ becomes $X/G$.
Given that transitivity concerns connectness of quotient, there could be a generalization to $k$-connectness. There’s also $n$-transitivity to consider.
An action would be transitive if, I guess, $\prod_x \prod_y x=y$ is inhabited, for $x,y:X//G$
That would allow empty $X$. Should we do that? The main definition speaks of a single orbit, yet later we’re supposing that $X$ is inhabited.
If we opt for $isContr(||X//G||_0)$, that would have to have $X$ inhabited.
Looking about, the consensus is that $X$ is non-empty. So then we should stop after ’action’ in
Let $\rho : G \times X \to X$ be a transitive action and suppose that $X$ is inhabited,
or is there something subtle going on with ’inhabited’?
I see Mike and Toby were thrashing out what it meant at inhabited object.
David #3 that’s probably what I was thinking. I guess ’exactly one orbit’ is the thing that is used.
In terms of whether the definition of transitive action should require the set to be inhabited, I’ve seen it both ways and like the more general definition. Something in the background for this particular fact is that really there is a 1-to-1 correspondence between pointed transitive G-actions and subgroups of G, by the stabilizer and coset action constructions. I do not know whether this has been explained nicely somewhere in terms of an equivalence of categories–I first learned about it from Samuel Vidal’s thesis and later in a nice series of blog posts by Qiaochu Yuan. I am kind of offline for a few days, but it would be great to incorporate this into the article at some point.
As I said in #1, all ingredients for the full $\infty$-action story are at stabilizer group.
We never got round to settling which version (inhabited or possibly empty) to adopt. What we had was inconsistent, so I added to make it at least consistent. But then now the discussion of $k$-transitivity needs attention. Do people want a 3-transitive action on $X$ to require at least 3 elements?
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