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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeJun 27th 2018
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   Author: David_Corfield
   Format: MarkdownItexI copied over Pavlovic's definition. Is there are better way to speak of his $E^{op}$? <a href="https://ncatlab.org/nlab/revision/trifibration/1">v1</a>, <a href="https://ncatlab.org/nlab/show/trifibration">current</a>

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

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeJun 27th 2018
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexIn 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?

   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?

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeJun 27th 2018
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   Author: Mike Shulman
   Format: MarkdownItexRe #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](http://www.tac.mta.ca/tac/volumes/20/18/20-18abs.html) I called trifibrations "$\ast$-bifibrations", with "$\ast$-fibrations" for the case of only right adjoints.

   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.

4. • CommentRowNumber4.
   • CommentAuthorNoam_Zeilberger
   • CommentTimeJun 27th 2018
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   Author: Noam_Zeilberger
   Format: MarkdownItex> I copied over Pavlovic's definition. Is there are better way to speak of his $E^{op}$? I think this is usually called the [[dual fibration]] -- I just created a stub for that, and suggest the notation $E^*$ (or $E^d$ or somesuch) over $E^{op}$.

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

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

5. • CommentRowNumber5.
   • CommentAuthorDavid_Corfield
   • CommentTimeJun 27th 2018
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   Author: David_Corfield
   Format: MarkdownItexI adopted Noam's suggestion of $E^d$ for the [[dual fibration]] and linked to it, and I added in the reference to Mike's Framed bicategories and monoidal fibrations. <a href="https://ncatlab.org/nlab/revision/diff/trifibration/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/trifibration/3">v3</a>, <a href="https://ncatlab.org/nlab/show/trifibration">current</a>

   I adopted Noam’s suggestion of $E^d$ 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

6. • CommentRowNumber6.
   • CommentAuthorDavid_Corfield
   • CommentTimeJun 27th 2018
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexRe #3, so shall we adopt your naming convention?

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

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeJun 27th 2018
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexWell, I won't insist if I'm in the minority. What do others think? Regarding notation, I agree that $E^{op}$ is poor since that looks like it's referring to the opposite of the total category. What about $E^\circ$ or $E^\bullet$?

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

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

8. • CommentRowNumber8.
   • CommentAuthorDavid_Corfield
   • CommentTimeJun 27th 2018
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   Author: David_Corfield
   Format: MarkdownItexTo be clear, is the use of $\ast$ in $\ast$-(bi)fibration just alluding to the existence of $f_{\ast}$?

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

9. • CommentRowNumber9.
   • CommentAuthorMike Shulman
   • CommentTimeJun 27th 2018
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexYes.

   Yes.

10. • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 28th 2018
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    Author: David_Corfield
    Format: MarkdownItexI 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 $E^{\ast}$ in #4 would fit well.

    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 $E^{\ast}$ in #4 would fit well.

11. • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeJun 28th 2018
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    Author: Mike Shulman
    Format: MarkdownItexPrecedent, maybe, but not necessarily *good* precedent. (-:O

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

12. • CommentRowNumber12.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 28th 2018
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexWell 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.

    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.

13. • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeJun 28th 2018
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexNo natural reason, no, but at least they're fairly well established. I suppose we could say something like "$\Pi$-fibration", with "$\Sigma$-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.)

    No natural reason, no, but at least they’re fairly well established. I suppose we could say something like “$\Pi$-fibration”, with “$\Sigma$-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.)

14. • CommentRowNumber14.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 29th 2018
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexWhile we're pondering the terminological choice, I added your alternative. I see 'trifibration' is also used in a multivariable adjunction setting [here](http://rene.guitart.pagesperso-orange.fr/textespublications/Guitart110.pdf). <a href="https://ncatlab.org/nlab/revision/diff/trifibration/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/trifibration/4">v4</a>, <a href="https://ncatlab.org/nlab/show/trifibration">current</a>

    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

15. • CommentRowNumber15.
    • CommentAuthorMike Shulman
    • CommentTimeJun 29th 2018
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItex> 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 $A\times B\times C$ that correspond to the common value of the functors $A(a,f(b,c)) \cong B(b,g(a,c)) \cong C(c,h(a,b))$ associated to a "mutual right" two-variable adjunction, i.e. a morphism $(A,B,C) \to ()$ in the polycategory of multivariable adjunctions. In particular, definitely not any kind of generalization of the present meaning of "bifibration".

    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 $A\times B\times C$ that correspond to the common value of the functors $A(a,f(b,c)) \cong B(b,g(a,c)) \cong C(c,h(a,b))$ associated to a “mutual right” two-variable adjunction, i.e. a morphism $(A,B,C) \to ()$ in the polycategory of multivariable adjunctions. In particular, definitely not any kind of generalization of the present meaning of “bifibration”.