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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeApr 17th 2018
   • PermaLink
   Author: Urs
   Format: MarkdownItexFixed some hickups in the very first sentence. <a href="https://ncatlab.org/nlab/revision/diff/free+action/12">diff</a>, <a href="https://ncatlab.org/nlab/revision/free+action/12">v12</a>, <a href="https://ncatlab.org/nlab/show/free+action">current</a>

   Fixed some hickups in the very first sentence.

   diff, v12, current

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeApr 17th 2018
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexOne must be careful with [[transitive action]] about the empty set, but all is fine here? I guess any group acting on the empty set is a free action. What's the HoTT version of freeness? Something like $IsSet(\sum_{\ast: BG} V(\ast))$.

   One must be careful with transitive action about the empty set, but all is fine here? I guess any group acting on the empty set is a free action.

   What’s the HoTT version of freeness? Something like $IsSet(\sum_{\ast: BG} V(\ast))$.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeApr 17th 2018
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI would probably state it as: for any $x$, the type of $g$ such that $g x = x$ is contractible (or, equivalently, a proposition, since it's always inhabited by $e$).

   I would probably state it as: for any $x$, the type of $g$ such that $g x = x$ is contractible (or, equivalently, a proposition, since it’s always inhabited by $e$).

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeApr 16th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded definition in terms of the [[shear map]] being a monomorphism. <a href="https://ncatlab.org/nlab/revision/diff/free+action/19">diff</a>, <a href="https://ncatlab.org/nlab/revision/free+action/19">v19</a>, <a href="https://ncatlab.org/nlab/show/free+action">current</a>

   added definition in terms of the shear map being a monomorphism.

   diff, v19, current

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeNov 1st 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexTyped up in the [[Sandbox]] the observation/proof that infinity-quotients of infinity-free actions in an infinity-topos are 0-truncated. But need to go offline right now. Will port and polish tomorrow.

   Typed up in the Sandbox the observation/proof that infinity-quotients of infinity-free actions in an infinity-topos are 0-truncated. But need to go offline right now. Will port and polish tomorrow.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeNov 2nd 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded ([here](https://ncatlab.org/nlab/show/free+action#HomotopyColimitsOfFreeActions)) statement and detailed proof that, in an $\infty$-topos, $\infty$-quotients of $\infty$-free $\infty$-actions (i.e. all higher shear maps are $(-1)$-truncated) are plain quotients of the underlying 0-truncated action. <a href="https://ncatlab.org/nlab/revision/diff/free+action/22">diff</a>, <a href="https://ncatlab.org/nlab/revision/free+action/22">v22</a>, <a href="https://ncatlab.org/nlab/show/free+action">current</a>

   added (here) statement and detailed proof that, in an $\infty$-topos, $\infty$-quotients of $\infty$-free $\infty$-actions (i.e. all higher shear maps are $(-1)$-truncated) are plain quotients of the underlying 0-truncated action.

   diff, v22, current

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeNov 2nd 2021
   • (edited Nov 2nd 2021)
   • PermaLink
   Author: Urs
   Format: MarkdownItex[ thought I could weaken the assumptions of the prop, but maybe I can't ]

   [ thought I could weaken the assumptions of the prop, but maybe I can’t ]

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeNov 2nd 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexWhat I was after in #7 does work, after all, one has to appeal to the groupoidal Segal conditions: I have stepped back and added a Definition-subsection * [Definition -- in an $\infty$-topos](https://ncatlab.org/nlab/show/free+action#DefinitionInAnInfinityTopos) with the declaration that an $\infty$-action is free if its shear map "in degree 1" is (-1)-truncated, and a proof that this implies that also all "higher shear maps" are (-1)-truncated. WIth this, the Prop. from #6 above ([here](https://ncatlab.org/nlab/show/free+action#HomotopyQuotientsOfFreeActionsAreQuotientsOfZeroTruncatedAction)) now says that as soon as the shear map $X \times G \to X \times X$ of an $\infty$-action is $(-1)$-truncated, the homotopy quotient of the $\infty$-action is the plain quotient of its 0-truncated version. <a href="https://ncatlab.org/nlab/revision/diff/free+action/25">diff</a>, <a href="https://ncatlab.org/nlab/revision/free+action/25">v25</a>, <a href="https://ncatlab.org/nlab/show/free+action">current</a>

   What I was after in #7 does work, after all, one has to appeal to the groupoidal Segal conditions:

   I have stepped back and added a Definition-subsection

   • Definition – in an $\infty$-topos

   with the declaration that an $\infty$-action is free if its shear map “in degree 1” is (-1)-truncated, and a proof that this implies that also all “higher shear maps” are (-1)-truncated.

   WIth this, the Prop. from #6 above (here) now says that as soon as the shear map $X \times G \to X \times X$ of an $\infty$-action is $(-1)$-truncated, the homotopy quotient of the $\infty$-action is the plain quotient of its 0-truncated version.

   diff, v25, current

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeNov 3rd 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexI should add a commentary to the entry on the following point, which can be confusing: The $\infty$-action on a principal $\infty$-bundle is always fiber-wise free, but not globally so unless it's base space is 0-truncated. Partially-conversely, every $\infty$-action is fiberwise free (in fact principal) over its homotopy quotient. (all internal to some $\infty$-topos).

   I should add a commentary to the entry on the following point, which can be confusing:

   The $\infty$-action on a principal $\infty$-bundle is always fiber-wise free, but not globally so unless it’s base space is 0-truncated. Partially-conversely, every $\infty$-action is fiberwise free (in fact principal) over its homotopy quotient. (all internal to some $\infty$-topos).