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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJul 20th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexFollowing discussion [here](https://nforum.ncatlab.org/discussion/14796/locus/?Focus=101148#Comment_101148), I am creating this entry make room for the traditional notion of *[[locus]]*. Have effectively rewritten the previous material. <a href="https://ncatlab.org/nlab/revision/Joyal+locus/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Joyal+locus">current</a>

   Following discussion here, I am creating this entry make room for the traditional notion of locus. Have effectively rewritten the previous material.

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeApr 8th 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexwhere it says in the entry "the collection of" I have added in parenthesis "ie.: the $\infty$-Grothendieck construction on" <a href="https://ncatlab.org/nlab/revision/diff/Joyal+locus/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/Joyal+locus/4">v4</a>, <a href="https://ncatlab.org/nlab/show/Joyal+locus">current</a>

   where it says in the entry “the collection of”

   I have added in parenthesis “ie.: the $\infty$-Grothendieck construction on”

   diff, v4, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeApr 19th 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexIt ought to be true that for a *monoidal* Joyal locus the $\infty$-topos of its parameterized objects should inherit the corresponding [[external tensor product|external]] tensor product. Do we have a formal proof of that?

   It ought to be true that for a monoidal Joyal locus the $\infty$-topos of its parameterized objects should inherit the corresponding external tensor product. Do we have a formal proof of that?

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeMay 25th 2023
   • (edited May 25th 2023)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThe passage from an $\infty$-category $\mathcal{C}$ (possibly a Joyal locus) to its $\infty$-category $\int_{\mathcal{X} \in Grpd_\infty} \mathcal{C}^{\mathcal{X}}$ of parameterized objects is "clearly" an $\infty$-analog of the [[free coproduct completion]] for ordinary categories. Concretely, it ought to be the free completion under coproducts *and* homotopy quotients by $\infty$-group actions. Has this been discussed as such?

   The passage from an $\infty$-category $\mathcal{C}$ (possibly a Joyal locus) to its $\infty$-category $\int_{\mathcal{X} \in Grpd_\infty} \mathcal{C}^{\mathcal{X}}$ of parameterized objects is “clearly” an $\infty$-analog of the free coproduct completion for ordinary categories. Concretely, it ought to be the free completion under coproducts and homotopy quotients by $\infty$-group actions. Has this been discussed as such?

5. • CommentRowNumber5.
   • CommentAuthorvarkor
   • CommentTimeMay 25th 2023
   • (edited May 25th 2023)
   • PermaLink
   Author: varkor
   Format: MarkdownItexI don't know whether this is at all relevant, but the description "coproducts and homotopy quotients by ∞-group actions" reminds me of the *quasi-coproducts* of Hu and Tholen's [Quasi-coproducts and accessible categories with wide pullbacks](https://link.springer.com/article/10.1007/BF00122686) (see section 1).

   I don’t know whether this is at all relevant, but the description “coproducts and homotopy quotients by ∞-group actions” reminds me of the quasi-coproducts of Hu and Tholen’s Quasi-coproducts and accessible categories with wide pullbacks (see section 1).

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeMay 25th 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks for the pointer. Interesting. Yes, this is somewhat related. Though for my question above we would discard the condition that group actions be free and just look at (homotopy) colimits over any diagram that is a skeletal groupoid. Concretely, for a category $\mathcal{C}$, the Grothendieck construction $$ \textstyle{\int}_{X \in Set} \mathcal{C}^{X} $$ is the free coproduct completion, and I am wondering about its enhancement to $$ \textstyle{\int}_{\mathcal{X} \in Grpd_{skl}} \mathcal{C}^{\mathcal{X}} $$ which should be something like the free completion under coproducts and homotopy quotients by group actions. Here we can drop the ${}_{skl}$-subscript up to equivalence, but maybe we need some nicety conditions on $\mathcal{C}$, namely an analogue of the Joyal-locus condition, to ensure that the Cartesian squares in $\textstyle{\int}_{\mathcal{X} \in Grpd_{skl}} \mathcal{C}^{\mathcal{X}}$ of the form $$ \array{ \mathscr{V}_{pt} &\longrightarrow& \mathscr{V}_{\mathbf{B}G} \\ \Big\downarrow && \Big\downarrow \\ 0_{pt} &\longrightarrow& 0_{\mathbf{B}G} } $$ do exhibit the objects over connected skeletal groupoids as homotopy quotients $\mathscr{V}_{\mathbf{B}G} \;\simeq\; \mathscr{V} \sslash G$ .

   Thanks for the pointer. Interesting. Yes, this is somewhat related.

   Though for my question above we would discard the condition that group actions be free and just look at (homotopy) colimits over any diagram that is a skeletal groupoid.

   Concretely, for a category $\mathcal{C}$, the Grothendieck construction

   $$\textstyle{\int}_{X \in Set} \mathcal{C}^{X}$$

   is the free coproduct completion, and I am wondering about its enhancement to

   $$\textstyle{\int}_{\mathcal{X} \in Grpd_{skl}} \mathcal{C}^{\mathcal{X}}$$

   which should be something like the free completion under coproducts and homotopy quotients by group actions.

   Here we can drop the ${}_{skl}$-subscript up to equivalence, but maybe we need some nicety conditions on $\mathcal{C}$, namely an analogue of the Joyal-locus condition, to ensure that the Cartesian squares in $\textstyle{\int}_{\mathcal{X} \in Grpd_{skl}} \mathcal{C}^{\mathcal{X}}$ of the form

   $$\array{ \mathscr{V}_{pt} &\longrightarrow& \mathscr{V}_{\mathbf{B}G} \\ \Big\downarrow && \Big\downarrow \\ 0_{pt} &\longrightarrow& 0_{\mathbf{B}G} }$$

   do exhibit the objects over connected skeletal groupoids as homotopy quotients $\mathscr{V}_{\mathbf{B}G} \;\simeq\; \mathscr{V} \sslash G$ .

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeJun 2nd 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexI take back the second line in [#6](#Comment_109947): Of course the freeness condition is just what makes the colimit a homotopy colimit, silly me. Will create *[[quasi-coproduct]]* now and add some discussion.

   I take back the second line in #6: Of course the freeness condition is just what makes the colimit a homotopy colimit, silly me.

   Will create quasi-coproduct now and add some discussion.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeJul 11th 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexmade explicit that the original observation is due to [G. Biedermann (2007)](https://ncatlab.org/nlab/show/Joyal+locus#Biedermann07) (recently had the pleasure of chatting with Georg about how this came about) <a href="https://ncatlab.org/nlab/revision/diff/Joyal+locus/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/Joyal+locus/7">v7</a>, <a href="https://ncatlab.org/nlab/show/Joyal+locus">current</a>

   made explicit that the original observation is due to G. Biedermann (2007)

   (recently had the pleasure of chatting with Georg about how this came about)

   diff, v7, current

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeOct 27th 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItex**Question** Do parameterized "Real" $H\mathbb{C}$-module spectra form an $\infty$-topos? Here I am thinking of the $\infty$-version of Atiyah-Real vector bundles: the base being $\infty$-groupoids equipped with $\mathbb{Z}\!/\!2$-involution and the $H\mathbb{C}$-module spectra covering these involutions by morphisms of underlying $H\mathbb{R}$-module spectra which are $\mathbb{C}$-anti linear.

   Question

   Do parameterized “Real” $H\mathbb{C}$-module spectra form an $\infty$-topos?

   Here I am thinking of the $\infty$-version of Atiyah-Real vector bundles: the base being $\infty$-groupoids equipped with $\mathbb{Z}\!/\!2$-involution and the $H\mathbb{C}$-module spectra covering these involutions by morphisms of underlying $H\mathbb{R}$-module spectra which are $\mathbb{C}$-anti linear.