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