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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.
This is the issue of passing from
to
The shape unit map
ηS1:S1↦ʃS1respects 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
to
is to a large part the topic of that article “Cyclification of Orbifolds”, beginning p. 13.
Thanks! Zhen
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