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1. • CommentRowNumber1.
   • CommentAuthorThomas Nikolaus
   • CommentTimeDec 16th 2010
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   Author: Thomas Nikolaus
   Format: MarkdownItexI am looking for a proof of the fact that a bifunctor between bicategories is an biequivalence if and only if it is fully faithful and essentially surjective (in the appropriate sense) that I can cite. Does anyone know a source?

   I am looking for a proof of the fact that a bifunctor between bicategories is an biequivalence if and only if it is fully faithful and essentially surjective (in the appropriate sense) that I can cite. Does anyone know a source?

2. • CommentRowNumber2.
   • CommentAuthorHarry Gindi
   • CommentTimeDec 16th 2010
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   Author: Harry Gindi
   Format: MarkdownItexBy bifunctor you mean pseudofunctor or 2-functor?

   By bifunctor you mean pseudofunctor or 2-functor?

3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeDec 16th 2010
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   Author: DavidRoberts
   Format: MarkdownItexI think Gurski has something on it. Street defines a biequivalence to be ff+eso in 'Fibrations in bicategories', and Gurski says in "An algebraic theory of tricategories" >It should now be clear that our definition of biadjoint biequivalence, in the case of the tricategory Bicat, is a fully algebraic version of the combination of biequivalence and biadjunction as given by Street in [37]. ([37] is Street's fibrations article). I don't know of a formal proof, though.

   I think Gurski has something on it. Street defines a biequivalence to be ff+eso in ’Fibrations in bicategories’, and Gurski says in “An algebraic theory of tricategories”

   It should now be clear that our definition of biadjoint biequivalence, in the case of the tricategory Bicat, is a fully algebraic version of the combination of biequivalence and biadjunction as given by Street in [37].

   ([37] is Street’s fibrations article). I don’t know of a formal proof, though.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeDec 16th 2010
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   Author: Mike Shulman
   Format: MarkdownItexI would call it folklore. It's easy to prove, by analogy with (and making use of) the corresponding fact for categories. BTW, I don't think anyone uses "bifunctor" for a functor between bicategories. The classical term was "homomorphism" (for a map that is functorial up to coherent iso) but nowadays I think the preferred thing to say is just "functor," that being the most sensible notion of functor between bicategories.

   I would call it folklore. It’s easy to prove, by analogy with (and making use of) the corresponding fact for categories.

   BTW, I don’t think anyone uses “bifunctor” for a functor between bicategories. The classical term was “homomorphism” (for a map that is functorial up to coherent iso) but nowadays I think the preferred thing to say is just “functor,” that being the most sensible notion of functor between bicategories.

5. • CommentRowNumber5.
   • CommentAuthorThomas Nikolaus
   • CommentTimeDec 16th 2010
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   Author: Thomas Nikolaus
   Format: MarkdownItexMike: Haha, I usually use 'Functor' and 'equivalence' but I thought for that post I am on the safe side if I add the prefix bi- everywhere ;) Its clear that it is relatively simple to proof that fact (given the right axiom of choice) but I wonder that no one has written it up.

   Mike: Haha, I usually use ’Functor’ and ’equivalence’ but I thought for that post I am on the safe side if I add the prefix bi- everywhere ;)

   Its clear that it is relatively simple to proof that fact (given the right axiom of choice) but I wonder that no one has written it up.

6. • CommentRowNumber6.
   • CommentAuthorzskoda
   • CommentTimeDec 16th 2010
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   Author: zskoda
   Format: MarkdownItexNobody uses "bifunctor" in this meaning just because the latter term is used in another meaning (a functor from the product category).

   Nobody uses “bifunctor” in this meaning just because the latter term is used in another meaning (a functor from the product category).

7. • CommentRowNumber7.
   • CommentAuthorTobyBartels
   • CommentTimeDec 17th 2010
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   Author: TobyBartels
   Format: MarkdownItexZoran's comment should be a genearlisation of the classical term ‘bifunction’, although right now I can't find any evidence for that term. (I *can* find ‘binary function’, however.) But yes, Thomas's bifunctors are Harry's [[pseudofunctors]]. >given the right axiom of choice Or given a suitable [[anafunctor]]-like definition (ana-2-functors).

   Zoran’s comment should be a genearlisation of the classical term ‘bifunction’, although right now I can’t find any evidence for that term. (I can find ‘binary function’, however.)

   But yes, Thomas’s bifunctors are Harry’s pseudofunctors.

   given the right axiom of choice

   Or given a suitable anafunctor-like definition (ana-2-functors).