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1. • CommentRowNumber1.
   • CommentAuthorTobyBartels
   • CommentTimeJun 28th 2010
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   Author: TobyBartels
   Format: MarkdownItexThis used to redirect to [[lax natural transformation]], which doesn't leave much room to talk about the strong case. So I created [[pseudonatural transformation]] and had things redirect there, although I didn't actually make use yet of that room.

   This used to redirect to lax natural transformation, which doesn’t leave much room to talk about the strong case. So I created pseudonatural transformation and had things redirect there, although I didn’t actually make use yet of that room.

2. • CommentRowNumber2.
   • CommentAuthorJohn Baez
   • CommentTimeJun 30th 2010
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   Author: John Baez
   Format: MarkdownItexI copied a definition of pseudonatural transformation from [[Schur functor]] over to [[pseudonatural transformation]]. Maybe someone can use it to complete the definition of [[lax natural transformation]] - right now that fizzles out just when it's getting really interesting!

   I copied a definition of pseudonatural transformation from Schur functor over to pseudonatural transformation. Maybe someone can use it to complete the definition of lax natural transformation - right now that fizzles out just when it’s getting really interesting!

3. • CommentRowNumber3.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJul 22nd 2017
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   Author: Dmitri Pavlov
   Format: MarkdownItexThe entry [[pseudonatural transformations]] states that > A pseudonatural transformation is called a pseudonatural equivalence if each component ϕ(s) is an equivalence in the 2-category C. This is equivalent to ϕ itself being an equivalence in the 2-category [S,C] of 2-functors, pseudonatural transformations, and modifications. An equivalence in [S,C] has as its components equivalences ϕ(s), which, furthermore, are _natural_ in s (as equivalences). So the above claim seems to imply that any pseudonatural transformation that is an objectwise equivalence can be made into a pseudonatural equivalence, i.e., the inverse to ϕ(s) can be chosen naturally in s. How does this claim not contradict the existence of fully faithful essentially surjective morphisms of Lie groupoids or stacks that do not have an inverse (except as a bibundle)? Such a morphism of stacks would give us a pseudonatural transformation of presheaves of groupoids that on every object is a fully faithful essentially surjective morphism of groupoids. So the above claim seems to imply that it is invertible as a morphism of presheaves of groupoids.

   The entry pseudonatural transformations states that

   A pseudonatural transformation is called a pseudonatural equivalence if each component ϕ(s) is an equivalence in the 2-category C. This is equivalent to ϕ itself being an equivalence in the 2-category [S,C] of 2-functors, pseudonatural transformations, and modifications.

   An equivalence in [S,C] has as its components equivalences ϕ(s), which, furthermore, are natural in s (as equivalences). So the above claim seems to imply that any pseudonatural transformation that is an objectwise equivalence can be made into a pseudonatural equivalence, i.e., the inverse to ϕ(s) can be chosen naturally in s.

   How does this claim not contradict the existence of fully faithful essentially surjective morphisms of Lie groupoids or stacks that do not have an inverse (except as a bibundle)? Such a morphism of stacks would give us a pseudonatural transformation of presheaves of groupoids that on every object is a fully faithful essentially surjective morphism of groupoids. So the above claim seems to imply that it is invertible as a morphism of presheaves of groupoids.

4. • CommentRowNumber4.
   • CommentAuthorPeter Heinig
   • CommentTimeJul 22nd 2017
   • (edited Jul 22nd 2017)
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   Author: Peter Heinig
   Format: MarkdownItexDmitri Pavlov, please note that this is not claiming to answer your question in any way, I am only contextualizing/situating the question a bit, for inexperienced readers: This question calls into question a 2-categorical and equivalenceological (please forgive this neologism) analogue of the following *true* characterization from basic 1-category-theory: Let $F,G\colon\mathcal{C}\rightarrow\mathcal{D}$ be functors. Let $\Phi\colon F\Rightarrow G$ be a natural transformation. Then (each component of $\Phi$ is an isomorphism) if and only if ($F\overset{\Phi}{\rightarrow}G$ is an isomorphism within the functor category $[\mathcal{C},\mathcal{D}]$)

   Dmitri Pavlov, please note that this is not claiming to answer your question in any way, I am only contextualizing/situating the question a bit, for inexperienced readers:

   This question calls into question a 2-categorical and equivalenceological (please forgive this neologism) analogue of the following true characterization from basic 1-category-theory:

   Let $F,G\colon\mathcal{C}\rightarrow\mathcal{D}$ be functors. Let $\Phi\colon F\Rightarrow G$ be a natural transformation. Then

   (each component of $\Phi$ is an isomorphism)

   if and only if

   ($F\overset{\Phi}{\rightarrow}G$ is an isomorphism within the functor category $[\mathcal{C},\mathcal{D}]$)

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeJul 22nd 2017
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   Author: Mike Shulman
   Format: MarkdownItexI believe the answer is that an "essentially surjective" morphism of *stacks* is not actually objectwise essentially surjective; it's only essentially surjective "up to passage to covers".

   I believe the answer is that an “essentially surjective” morphism of stacks is not actually objectwise essentially surjective; it’s only essentially surjective “up to passage to covers”.

6. • CommentRowNumber6.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJul 22nd 2017
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   Author: Dmitri Pavlov
   Format: MarkdownItexRe #5: I see, so perhaps this can be interpreted as saying that (pseudo)functors (with values in groupoids, say) and pseudonatural transformations form a model category where all objects are cofibrant and fibrant, in complete analogy to the model category of Grothendieck fibrations in groupoids? A pseudofunctor can be constructed from a Grothendieck fibration with a cleavage, so perhaps the model category of pseudofunctors can be constructed as the model category of [[algebraically fibrant]] objects in Grothendieck fibrations? (Like for Kan complexes here: <https://arxiv.org/abs/1003.1342>.)

   Re #5: I see, so perhaps this can be interpreted as saying that (pseudo)functors (with values in groupoids, say) and pseudonatural transformations form a model category where all objects are cofibrant and fibrant, in complete analogy to the model category of Grothendieck fibrations in groupoids?

   A pseudofunctor can be constructed from a Grothendieck fibration with a cleavage, so perhaps the model category of pseudofunctors can be constructed as the model category of algebraically fibrant objects in Grothendieck fibrations? (Like for Kan complexes here: https://arxiv.org/abs/1003.1342.)

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeJul 23rd 2017
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   Author: Mike Shulman
   Format: MarkdownItexWell, the category of pseudofunctors and pseudonatural transformations is not complete and cocomplete as a 1-category, so it can't be a model category. I suppose it could be a "model bicategory" if that has been defined. I think the category of Grothendieck fibrations is not complete and cocomplete either unless you consider cloven fibrations and cleavage-preserving functors, in which case it's equivalent (as a 1-category already) to the category of pseudofunctors into $Cat$ and strict natural transformations between them.

   Well, the category of pseudofunctors and pseudonatural transformations is not complete and cocomplete as a 1-category, so it can’t be a model category. I suppose it could be a “model bicategory” if that has been defined.

   I think the category of Grothendieck fibrations is not complete and cocomplete either unless you consider cloven fibrations and cleavage-preserving functors, in which case it’s equivalent (as a 1-category already) to the category of pseudofunctors into $Cat$ and strict natural transformations between them.

8. • CommentRowNumber8.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJul 23rd 2017
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   Author: Dmitri Pavlov
   Format: MarkdownItexRe #7: Sorry, I meant to say that Grothendieck fibrations are fibrant objects in a model structure on all functors (the Grothendieck fibration property can be defined as a right lifting property). (And fibers can be forced to be groupoids without losing (co)completeness.)

   Re #7: Sorry, I meant to say that Grothendieck fibrations are fibrant objects in a model structure on all functors (the Grothendieck fibration property can be defined as a right lifting property). (And fibers can be forced to be groupoids without losing (co)completeness.)

9. • CommentRowNumber9.
   • CommentAuthorMike Shulman
   • CommentTimeJul 23rd 2017
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   Author: Mike Shulman
   Format: MarkdownItexWhen the fibers are groupoids, I think that's true, but when the fibers are arbitrary categories you need liftings with universal properties, which I don't think can be expressed as a fibrancy condition.

   When the fibers are groupoids, I think that’s true, but when the fibers are arbitrary categories you need liftings with universal properties, which I don’t think can be expressed as a fibrancy condition.