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 comma 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 finite 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 integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic 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.
    • CommentAuthorDmitri Pavlov
    • CommentTimeOct 20th 2020

    Redirect for bicategorical localization

    diff, v10, current

    • CommentRowNumber2.
    • CommentAuthorDmitri Pavlov
    • CommentTimeOct 20th 2020
    • (edited Oct 20th 2020)

    Can we obtain the bicategory of Grothendieck toposes and geometric morphisms as a bicategorical localization?

    A statement that I have in mind is that the bicategory of Grothendieck toposes and geometric morphisms should be something like the bicategorical localization of the 1-category of localic groupoids, (internal) functors, and some version of (internal) essentially surjective fully faithful functors as weak equivalences.

    Has anything like this appeared in the literature?

    I am aware of the results mentioned at classifying topos of a localic groupoid, where Joyal–Tierney and Moerdijk get us pretty close to such a statement, but not quite.

    The original work of Pronk seems to treat only etale groupoids and etendues.

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 21st 2020

    There was discussion of this on the category theory Zulip chat recently. I think the construction of Moerdijk of a localic category from a localic groupoid, where he then shows their categories of sheaves are equivalent, is something that might be able to be avoided, if we take localic categories rather than localic groupoids as the basic input. If we want the non-invertible 2-arrows, then localic groupoids and the functors between them is not sufficient. Moerdijk’s notion of morphism between localic groupoids in the localisation looks like bitorsors between the associated localic categories.

    I definitely think it should be possible, and I don’t think it’s in the literature.

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeOct 21st 2020

    My suspicion is that it should help to consider localic groupoids as a category enriched over double categories, where one direction of the hom-double-categories consists of internal natural transformations in the category of locales and the other direction consists of levelwise inequalities with respect to the locally-ordered nature of the category of locales. Neither of these two kinds of 2-cells alone can possibly carry all the information in the 2-cells of toposes, since the former is invertible while the latter is thin, but when combined there is some hope.

    • CommentRowNumber5.
    • CommentAuthorDmitri Pavlov
    • CommentTimeDec 9th 2020

    For the case of invertible 2-morphisms, this is asserted (but unfortunately proved only for homotopy 1-categories) in Ieke Moerdijk’s The classifying topos of a continuous groupoid I, Theorem 7.7. Specifically, bicategorically localizing etale-complete localic groupoids (with open source and target maps) at open essentially surjective fully faithful functors produces the bicategory of Grothendieck toposes, geometric morphisms, and natural isomorphisms.

    Part II deals with noninvertible 2-morphisms, but Moerdijk only formulates it as an equivalence of the bicategory of Grothendieck toposes, geometric morphisms, and natural transformations (not necessarily invertible) and the bicategory of localic groupoids (with open source and target maps), complete flat bibundles between their completions, and homomorphisms of bibundles. However, Part II does not say anything about bicategories of fractions.

    • CommentRowNumber6.
    • CommentAuthorDmitri Pavlov
    • CommentTimeDec 9th 2020

    Added Moerdijk’s result.

    diff, v11, current

    • CommentRowNumber7.
    • CommentAuthorDmitri Pavlov
    • CommentTimeDec 9th 2020
    • (edited Dec 9th 2020)

    Re #3:

    There was discussion of this on the category theory Zulip chat recently.

    Is there a way to see the discussion without registering? (Is there any particular reason why the chat not available for public viewing?)

    • CommentRowNumber8.
    • CommentAuthorDavidRoberts
    • CommentTimeDec 9th 2020

    There was meant to be a public archive/readable version, but I don’t know what happened with that yet. I’m messaged one of the mods.

    • CommentRowNumber9.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 7th 2021

    Is there a citeable reference for the fact that taking the bicategory of fractions of localic etale groupoids with respect to fully faithful essentially surjective functors produces a bicategory equivalent to the bicategory of etendues with invertible 2-morphisms, where etendues are understood in the generalized localic sense?

    Something like Corollary 35 in Pronk’s Etendues and stacks as bicategories of fractions, but for locales instead of topological spaces?

    • CommentRowNumber10.
    • CommentAuthorDavidRoberts
    • CommentTimeApr 7th 2021

    I suspect not, though one might be able to cobble it together using something of mine for the localic groupoid side (to get a bicategory of fractions), and then using the comparison theorem Pronk proves, pointing out any changes (or not) necessary from the topological etendue version.

    • CommentRowNumber11.
    • CommentAuthorzhaoxurui
    • CommentTimeSep 2nd 2021
    In [2CF3] of this page, it seems that \alpha need not be an iso-2-cell. But in Pronk's article, \alpha needs to be an iso-2-cell. Is this a new result? (or it is just my misunderstanding...)
    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeSep 2nd 2021
    • (edited Sep 2nd 2021)

    Thanks for the alert.

    For the record, clause 2CF3 is here.

    Since the entry does not claim any originality but entirely attributes this to Pronk

    Given such a setup, Pronk constructs…

    it seems safe to assume that this is not a claim but a glitch. If you are an expert on the matter, please feel invited to fix the entry.

    diff, v15, current