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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJul 20th 2022

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

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeApr 8th 2023

    where it says in the entry “the collection of”

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

    diff, v4, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeApr 19th 2023

    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?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMay 25th 2023
    • (edited May 25th 2023)

    The passage from an \infty-category 𝒞\mathcal{C} (possibly a Joyal locus) to its \infty-category 𝒳Grpd 𝒞 𝒳\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?

    • CommentRowNumber5.
    • CommentAuthorvarkor
    • CommentTimeMay 25th 2023
    • (edited May 25th 2023)

    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).

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMay 25th 2023

    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 𝒞\mathcal{C}, the Grothendieck construction

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

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

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

    𝒱 pt 𝒱 BG 0 pt 0 BG \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 𝒱 BG𝒱G\mathscr{V}_{\mathbf{B}G} \;\simeq\; \mathscr{V} \sslash G .

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJun 2nd 2023

    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.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJul 11th 2023

    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

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeOct 27th 2023


    Do parameterized “Real” HH\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 /2\mathbb{Z}\!/\!2-involution and the HH\mathbb{C}-module spectra covering these involutions by morphisms of underlying HH\mathbb{R}-module spectra which are \mathbb{C}-anti linear.