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    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 27th 2018

    I copied over Pavlovic’s definition. Is there are better way to speak of his E opE^{op}?

    v1, current

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 27th 2018

    In the setting of LSR on mode theories, could one think of trifibrations over a mode 2-category, \mathcal{M}, as classified by a map from \mathcal{M} to the 2-category of adjoint triples?

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeJun 27th 2018

    Re #2: yes.

    I’m not a fan of the “trifibration” terminology, because a “bifibration” is “a fibration in two ways”, whereas a “trifibration” is not a “fibration in three ways”. Also it leaves no terminology remaining for a fibration whose transition functors have right adjoints but not left adjoints. In FBMF I called trifibrations “*\ast-bifibrations”, with “*\ast-fibrations” for the case of only right adjoints.

  1. I copied over Pavlovic’s definition. Is there are better way to speak of his E opE^{op}?

    I think this is usually called the dual fibration – I just created a stub for that, and suggest the notation E *E^* (or E dE^d or somesuch) over E opE^{op}.

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 27th 2018

    I adopted Noam’s suggestion of E dE^d for the dual fibration and linked to it, and I added in the reference to Mike’s Framed bicategories and monoidal fibrations.

    diff, v3, current

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 27th 2018

    Re #3, so shall we adopt your naming convention?

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeJun 27th 2018

    Well, I won’t insist if I’m in the minority. What do others think?

    Regarding notation, I agree that E opE^{op} is poor since that looks like it’s referring to the opposite of the total category. What about E E^\circ or E E^\bullet?

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 27th 2018

    To be clear, is the use of *\ast in *\ast-(bi)fibration just alluding to the existence of f *f_{\ast}?

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeJun 27th 2018

    Yes.

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 28th 2018

    I guess there is a precedent for reliance on the vagaries of notation choice. There is *\ast-autonomous category based on the use of *\ast to mark dual vector spaces. (By the way, why ’autonomous’?)

    Then perhaps the choice of E *E^{\ast} in #4 would fit well.

    • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeJun 28th 2018

    Precedent, maybe, but not necessarily good precedent. (-:O

    • CommentRowNumber12.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 28th 2018

    Well that did worry me. Is there any reason we should prefer your use of *\ast? There’s no natural reason for the use of *\ast in the adjoints.

    • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeJun 28th 2018

    No natural reason, no, but at least they’re fairly well established. I suppose we could say something like “Π\Pi-fibration”, with “Σ\Sigma-fibration” being another name for a bifibration; that would also be based on notation but at least somewhat more specific than *\ast. On the other hand, it might also be taken to imply Beck-Chevalley conditions (since indexed products do have such a condition), which isn’t part of the definition we’re talking about here.

    (Re: “why autonomous”, some people used to use the bare adjective “autonomous” for a closed monoidal category, or sometimes for a compact closed monoidal category, but it seems to not have survived very far. I suppose the intuition in those cases is that with internal-hom objects the category is “sufficient unto itself” without needing hom-sets, or some such.)

    • CommentRowNumber14.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 29th 2018

    While we’re pondering the terminological choice, I added your alternative.

    I see ’trifibration’ is also used in a multivariable adjunction setting here.

    diff, v4, current

    • CommentRowNumber15.
    • CommentAuthorMike Shulman
    • CommentTimeJun 29th 2018

    I see ’trifibration’ is also used in a multivariable adjunction setting here

    In case this wasn’t clear to anyone else (it wasn’t to me until I looked at the paper), Guitart’s “trifibrations” are something totally different. They are the discrete fibrations over triple products A×B×CA\times B\times C that correspond to the common value of the functors A(a,f(b,c))B(b,g(a,c))C(c,h(a,b))A(a,f(b,c)) \cong B(b,g(a,c)) \cong C(c,h(a,b)) associated to a “mutual right” two-variable adjunction, i.e. a morphism (A,B,C)()(A,B,C) \to () in the polycategory of multivariable adjunctions. In particular, definitely not any kind of generalization of the present meaning of “bifibration”.