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1. • CommentRowNumber1.
   • CommentAuthorZhen Huan
   • CommentTimeSep 28th 2021
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   Author: Zhen Huan
   Format: TextMy question is: is there a well-accepted reference for the concept, fully faithful bifunctor between bicategories? In Nikolaus and Schweigert's paper "Equivariance in higher geometry", they define \tau-prestack in Definition 2.12, page 10, via "fully faithful functor of bicategories". They explained in the definition without giving any reference that a functor of bicategories is called fully faithful, if all functors on Hom categories are equivalences of categories. I think this definition of fully faithful bifunctor is reasonable. I saw the definition of fully faithful 2-functor here on nlab: https://ncatlab.org/nlab/show/full+sub-2-category, which is defined in the same way as that in Nikolaus and Schweigert's paper. However, on this page, there is no reference as well. So my question is above at the beginning.

   My question is: is there a well-accepted reference for the concept, fully faithful bifunctor between bicategories?
   
   In Nikolaus and Schweigert's paper "Equivariance in higher geometry", they define \tau-prestack in Definition 2.12, page 10, via "fully faithful functor of bicategories". They explained in the definition without giving any reference that a functor of bicategories is called fully faithful, if all functors on Hom categories are equivalences of categories.
   I think this definition of fully faithful bifunctor is reasonable. I saw the definition of fully faithful 2-functor here on nlab: https://ncatlab.org/nlab/show/full+sub-2-category, which is defined in the same way as that in Nikolaus and Schweigert's paper. However, on this page, there is no reference as well.
   
   So my question is above at the beginning.
2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeSep 28th 2021
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   Author: Urs
   Format: MarkdownItexGood point, the literature on 2-categories is pretty bad. I just checked the references that we have at *[[2-category]]*, and none of them admits to a definition of "fully faithful" 2-functors (which does not stop some from using the term anyway). The closest is Section 7 in * [[Niles Johnson]], [[Donald Yau]], _2-Dimensional Categories_, Oxford University Press 2021 ([arXiv:2002.06055](http://arxiv.org/abs/2002.06055), [doi:10.1093/oso/9780198871378.001.0001](https://oxford.universitypressscholarship.com/view/10.1093/oso/9780198871378.001.0001/oso-9780198871378)) which I recommend as the primary source for reading up on the subject. These authors don't quite say "fully faithful as a 2-functor" but they do equivalently say (Def. 7.0.1) "essentially full on 1-cells and fully faithful on 2-cells" which is, equivalently, the expected "equivalence on all hom-categories" (by the classical theorem [here](https://ncatlab.org/nlab/show/equivalence%20of%20categories#ViaEssentiallySurjectiveAndFullyFaithful)). But no matter the terminology, the definition is certainly correct and universally accepted: The point is the theorem (7.4.1 in the above book, I have now made a note on that [here](https://ncatlab.org/nlab/show/2-functor#CharacterizationOfEquivalencesOf2Categories)) -- which holds in great generality throuhghout higher category theory, that (assuming the axiom of choice) a morphism of any type of higher categories is an equivalence precisely if, first, it is surjective on equivalence class of objects, and, second, is *fully faithful* in the sense that for any pair of objects its component on hom-objects is an equivalence of hom-objects, in whatever sense of higher categories these hom-objects themselves are. From this theorem the respective notion of full higher sub-category is implied: It must be an equivalence onto its image, hence it must satisfy the above fully faithfulness-condition, while it may violate the essential surjectivity on objects, and of course will so to the extent that it is a proper (full, higher) sub-category.

   Good point, the literature on 2-categories is pretty bad. I just checked the references that we have at 2-category, and none of them admits to a definition of “fully faithful” 2-functors (which does not stop some from using the term anyway). The closest is Section 7 in

   • Niles Johnson, Donald Yau, 2-Dimensional Categories, Oxford University Press 2021 (arXiv:2002.06055, doi:10.1093/oso/9780198871378.001.0001)

   which I recommend as the primary source for reading up on the subject. These authors don’t quite say “fully faithful as a 2-functor” but they do equivalently say (Def. 7.0.1) “essentially full on 1-cells and fully faithful on 2-cells” which is, equivalently, the expected “equivalence on all hom-categories” (by the classical theorem here).

   But no matter the terminology, the definition is certainly correct and universally accepted: The point is the theorem (7.4.1 in the above book, I have now made a note on that here) – which holds in great generality throuhghout higher category theory, that (assuming the axiom of choice) a morphism of any type of higher categories is an equivalence precisely if, first, it is surjective on equivalence class of objects, and, second, is fully faithful in the sense that for any pair of objects its component on hom-objects is an equivalence of hom-objects, in whatever sense of higher categories these hom-objects themselves are.

   From this theorem the respective notion of full higher sub-category is implied: It must be an equivalence onto its image, hence it must satisfy the above fully faithfulness-condition, while it may violate the essential surjectivity on objects, and of course will so to the extent that it is a proper (full, higher) sub-category.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeSep 28th 2021
   • (edited Sep 28th 2021)
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   Author: Urs
   Format: MarkdownItexAh, these authors here say fully explicitly what you are asking for: * [[Ofer Gabber]], [[Lorenzo Ramero]], Chapter 2 of: *Foundations for almost ring theory* ([arXiv:math/0409584](https://arxiv.org/abs/math/0409584)) See: [Def. 2.4.9](https://arxiv.org/pdf/math/0409584.pdf#page=113).

   Ah, these authors here say fully explicitly what you are asking for:

   • Ofer Gabber, Lorenzo Ramero, Chapter 2 of: Foundations for almost ring theory (arXiv:math/0409584)

   See: Def. 2.4.9.

4. • CommentRowNumber4.
   • CommentAuthorZhen Huan
   • CommentTimeSep 28th 2021
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   Author: Zhen Huan
   Format: TextMany thanks, Urs. It helps a lot. I've downloaded both.

   Many thanks, Urs. It helps a lot. I've downloaded both.