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1. • CommentRowNumber1.
   • CommentAuthorZhen Huan
   • CommentTimeOct 5th 2024
   • PermaLink
   Author: Zhen Huan
   Format: MarkdownItexI 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.

   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.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeOct 5th 2024
   • (edited Oct 5th 2024)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThis is the issue of passing from - the circle $S^1$ regarded as a Lie group and as such as a smooth group $\infty$-stack which is 0-truncated but not geometrically discrete to - its "shape" $ʃ S^1$ which is equivalently the "simplicial circle" and as such a smooth group $\infty$-stack which is 1-truncated but geometrically discrete. The shape unit map $$ \eta_{S^1} : S^1 \mapsto ʃ S^1 $$ respects these $\infty$-group structures and hence records what becomes of the "naive" $S^1$ action by rotation as one passes to homotopy types and is left "only" with the higher group action of $ʃ S^1$. The relation between - the free loop space of the form $Map(S^1, (-))$, with its naive action by circle rotation to - the free loop space $Map(ʃ S^1, ʃ (-))$, with only a homotopy-action by $ʃ S^1$ left is to a large part the topic of that article "[[schreiber:Cyclification of Orbifolds|Cyclification of Orbifolds]]", beginning [p. 13](https://arxiv.org/pdf/2212.13836#page=13).

   This is the issue of passing from

   • the circle $S^1$ regarded as a Lie group and as such as a smooth group $\infty$-stack which is 0-truncated but not geometrically discrete

   to

   • its “shape” $ʃ S^1$ which is equivalently the “simplicial circle” and as such a smooth group $\infty$-stack which is 1-truncated but geometrically discrete.

   The shape unit map

   $$\eta_{S^1} : S^1 \mapsto ʃ S^1$$

   respects these $\infty$-group structures and hence records what becomes of the “naive” $S^1$ action by rotation as one passes to homotopy types and is left “only” with the higher group action of $ʃ S^1$.

   The relation between

   • the free loop space of the form $Map(S^1, (-))$, with its naive action by circle rotation

   to

   • the free loop space $Map(ʃ S^1, ʃ (-))$, with only a homotopy-action by $ʃ S^1$ left

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

3. • CommentRowNumber3.
   • CommentAuthorZhen Huan
   • CommentTimeOct 5th 2024
   • PermaLink
   Author: Zhen Huan
   Format: MarkdownItexThanks! Zhen

   Thanks! Zhen