**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.

]]>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)

]]>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.

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 $\mathscr{V}_{\mathbf{B}G} \;\simeq\; \mathscr{V} \sslash G$ .

]]>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).

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?

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?

where it says in the entry “the collection of”

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

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