Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory infinity integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • 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.