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    • CommentRowNumber1.
    • CommentAuthorZhen Huan
    • CommentTimeOct 5th 2024

    I have a maybe basic question but I cannot figure it out. We know on the free loop space LX of a smooth manifold X, there is a circle action induced by the rotation. If we consider any an (∞,1)-category C with (∞,1)-pullbacks instead of the category of smooth manifolds, we can define the free loop space object, as we can see here https://ncatlab.org/nlab/show/free+loop+space+object#CircleAction Can we still have a higher “rotation” on a free loop object? I don’t think the intrinsic circle action on that page is equivalent to “rotation”. Thanks.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 5th 2024
    • (edited Oct 5th 2024)

    This is the issue of passing from

    • the circle S1 regarded as a Lie group and as such as a smooth group -stack which is 0-truncated but not geometrically discrete

    to

    • its “shape” ʃS1 which is equivalently the “simplicial circle” and as such a smooth group -stack which is 1-truncated but geometrically discrete.

    The shape unit map

    ηS1:S1ʃS1

    respects these -group structures and hence records what becomes of the “naive” S1 action by rotation as one passes to homotopy types and is left “only” with the higher group action of ʃS1.

    The relation between

    • the free loop space of the form Map(S1,()), with its naive action by circle rotation

    to

    • the free loop space Map(ʃS1,ʃ()), with only a homotopy-action by ʃS1 left

    is to a large part the topic of that article “Cyclification of Orbifolds”, beginning p. 13.

    • CommentRowNumber3.
    • CommentAuthorZhen Huan
    • CommentTimeOct 5th 2024

    Thanks! Zhen