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  1. Here is explained that the free-loop-space object LBGL BG of a one-object delooping groupoid BGBG is equivalent to the action groupoid of the adjoint action (=conjugation) ρ\rho of GG on itself. I am trying to see if something like this is true for the free-loop-space object LXLX of a general groupoid object XX in an (\infty,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 GG. Is there some reference for this?

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeApr 27th 2012
    • (edited Apr 27th 2012)

    Hi Stephan,

    sorry, I didn’t realize that it remained mysterious, this is quite straightforward.

    Let’s look at Grpd\infty Grpd first, presented by sSet. Given any fibrant object XsSetX \in sSet – a Kan complex – we want to compute the homotopy pullback of the diagonal XX×XX \to X \times X 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 (s,t):X IX×X(s,t) : X^I \to X \times X is such a replacement of the diagonal.

    So we find that the free loop space of the \infty-groupoid XX is presented by the ordinary pullback of simplicial sets

    LX X I X X×X. \array{ L X &\to& X^I \\ \downarrow && \downarrow \\ X &\to& X \times X } \,.

    From this you can immediately read off what LXL X is: in degree 0 we find that its vertices are precisely the 1-cells of XX 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.

  2. 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 LXLX as an action groupoid (LX) 0//Y(LX)_0//Y for some groupoid object YY and an appropriate definition of action groupoid. But I am realizing that this does not make sense since any morphism in (,1)Cat(\infty,1)Cat which has as domain an ∞-groupoid already may be seen as a ∞-groupoid action on its essential image.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeApr 27th 2012

    I am not sure how this perspective would really help with what you are after.

    Notice that (LX) 0(L X)_0 – or rather the anchor map (LX) 0X 0(L X)_0 \to X_0 that you’d want for the definition of the action of the groupoid XX – has not even invariant meaning as soon as XX is higher truncated than a 1-groupoid.