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Here is explained that the free-loop-space object of a one-object delooping groupoid is equivalent to the action groupoid of the adjoint action (=conjugation) of on itself. I am trying to see if something like this is true for the free-loop-space object of a general groupoid object in an (,1)-category. Unfortunately I don’t see how the proof given there generalizes to the desired case.
In fact I have rarely seen the manipulation of action ∞-groupoids where the acting groupoid is not of the form BG for an ∞-group . Is there some reference for this?
Hi Stephan,
sorry, I didn’t realize that it remained mysterious, this is quite straightforward.
Let’s look at first, presented by sSet. Given any fibrant object – a Kan complex – we want to compute the homotopy pullback of the diagonal along itself. By the usual yoga, it is sufficient to replace one of the two copies with a Kan fibration and then form the ordinary pullback.
Now, effectively by definition of path space object, the map is such a replacement of the diagonal.
So we find that the free loop space of the -groupoid is presented by the ordinary pullback of simplicial sets
From this you can immediately read off what is: in degree 0 we find that its vertices are precisely the 1-cells of whose endpoints match. And so on.
Write out for yourself, or here on the forum, what the 1-cells look like, as deduced from this pullback.
If this is not what you want to see, let me know what you are after.
sorry, I didn’t realize that it remained mysterious
No, this was clear.
If this is not what you want to see, let me know what you are after.
What I wanted to do was expressing as an action groupoid for some groupoid object and an appropriate definition of action groupoid. But I am realizing that this does not make sense since any morphism in which has as domain an ∞-groupoid already may be seen as a ∞-groupoid action on its essential image.
I am not sure how this perspective would really help with what you are after.
Notice that – or rather the anchor map that you’d want for the definition of the action of the groupoid – has not even invariant meaning as soon as is higher truncated than a 1-groupoid.
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