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1. • CommentRowNumber1.
   • CommentAuthorDavidRoberts
   • CommentTimeMar 16th 2011
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   Author: DavidRoberts
   Format: MarkdownItexBefore I forget, I uploaded a new version of my anafunctors paper to my page [[David Roberts]]. In particular, the finer points have been made a lot tighter. I even use technical phrases such as 'enough groupoids' and 'admits cotensors'! :) It has also been submitted for publication.

   Before I forget, I uploaded a new version of my anafunctors paper to my page David Roberts. In particular, the finer points have been made a lot tighter. I even use technical phrases such as ’enough groupoids’ and ’admits cotensors’! :) It has also been submitted for publication.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeMar 24th 2011
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   Author: Mike Shulman
   Format: MarkdownItexLooks nice! I think it would be more correct in Def. 2.21 to say that "If the strict pullback $X\times_{Y,s} Y^{\mathbf{I}} \times_{t,Y} Z$ exists, then it is a bipullback" — since the bipullback is only defined by its universal property up to equivalence, not up to isomorphism. If you always want to refer to the actual strict pullback $X\times_{Y,s} Y^{\mathbf{I}} \times_{t,Y} Z$ instead of an arbitrary bipullback (as in Cor. 4.10), then "iso-comma object" would be more correct. Also you might want to refer to Carboni-Lack-Walters for extensive categories.

   Looks nice! I think it would be more correct in Def. 2.21 to say that “If the strict pullback $X\times_{Y,s} Y^{\mathbf{I}} \times_{t,Y} Z$ exists, then it is a bipullback” — since the bipullback is only defined by its universal property up to equivalence, not up to isomorphism. If you always want to refer to the actual strict pullback $X\times_{Y,s} Y^{\mathbf{I}} \times_{t,Y} Z$ instead of an arbitrary bipullback (as in Cor. 4.10), then “iso-comma object” would be more correct.

   Also you might want to refer to Carboni-Lack-Walters for extensive categories.

3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeMar 25th 2011
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   Author: DavidRoberts
   Format: MarkdownItexThanks, Mike. You're right about the iso-comma object. Everyone in the business of using Pronk's result about bicategory localisation uses bipullback (even if by another name), but actually one only needs a 'weak bipullback' - existence but no uniqueness even at the bicategorical level. So existence of isocomma objects is certainly enough. I'll put in the reference to CLW too.

   Thanks, Mike. You’re right about the iso-comma object. Everyone in the business of using Pronk’s result about bicategory localisation uses bipullback (even if by another name), but actually one only needs a ’weak bipullback’ - existence but no uniqueness even at the bicategorical level. So existence of isocomma objects is certainly enough. I’ll put in the reference to CLW too.

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeMar 25th 2011
   • (edited Mar 25th 2011)
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   Author: zskoda
   Format: MarkdownItexDavid, did you or somebody else so far write down the details of the proof that every bicategorical reflection (that is a biadjunction with counit being an equivalence) induces a (left in my terminology) calculus of bicategorical fractions (I could work it out with a student if not, I may need this result) which leads to the equivalent reflection ? (on the other hand there is no converse in general, as in 1-categorical case there are some weak but strange assumptions needed as well)

   David, did you or somebody else so far write down the details of the proof that every bicategorical reflection (that is a biadjunction with counit being an equivalence) induces a (left in my terminology) calculus of bicategorical fractions (I could work it out with a student if not, I may need this result) which leads to the equivalent reflection ?

   (on the other hand there is no converse in general, as in 1-categorical case there are some weak but strange assumptions needed as well)

5. • CommentRowNumber5.
   • CommentAuthorDavidRoberts
   • CommentTimeMar 26th 2011
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   Author: DavidRoberts
   Format: MarkdownItexZoran, no (I didn't, and I don't know if anyone else has). There was some discussion in email between Mike and myself of how to link an example of the result you suggest to my result, namely in the case when the ambient category is regular and has enough projectives wrt regular epimorphisms (...or similar - I'd have to look up the email). Then the inclusion functor $Cat(S)_{p.o} \to Cat(S)$, where p.o means full sub-2-category on objects (=categories $X_1 \rightrightarrows X_0$) where $X_0$ is projective, is an adjoint to the localisation $Cat(S) \to Cat_{ana}(S)$, because we can give an equivalence $Cat(S)_{p.o} \simeq Cat_{ana}(S)$ under $Cat(S)$ (here $S$ is taken with the canonical singleton pretopology to define anafunctors). As an aside, anafunctors form a right category of fractions, so whatever you can glean from the above paragraph would have to be dualised. (assuming that by left calculus of fractions you mean maps are cospans) Have you seen the paper 'Bipullbacks and calculus of fractions' by Vitale (available from his website)? It has another set of axioms (stronger) than Pronks, which may help. Again, you may need to dualise the results to get left localisation.

   Zoran,

   no (I didn’t, and I don’t know if anyone else has). There was some discussion in email between Mike and myself of how to link an example of the result you suggest to my result, namely in the case when the ambient category is regular and has enough projectives wrt regular epimorphisms (…or similar - I’d have to look up the email). Then the inclusion functor $Cat(S)_{p.o} \to Cat(S)$, where p.o means full sub-2-category on objects (=categories $X_1 \rightrightarrows X_0$) where $X_0$ is projective, is an adjoint to the localisation $Cat(S) \to Cat_{ana}(S)$, because we can give an equivalence $Cat(S)_{p.o} \simeq Cat_{ana}(S)$ under $Cat(S)$ (here $S$ is taken with the canonical singleton pretopology to define anafunctors).

   As an aside, anafunctors form a right category of fractions, so whatever you can glean from the above paragraph would have to be dualised. (assuming that by left calculus of fractions you mean maps are cospans)

   Have you seen the paper ’Bipullbacks and calculus of fractions’ by Vitale (available from his website)? It has another set of axioms (stronger) than Pronks, which may help. Again, you may need to dualise the results to get left localisation.

6. • CommentRowNumber6.
   • CommentAuthorzskoda
   • CommentTimeMar 26th 2011
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   Author: zskoda
   Format: MarkdownItexBy left I mean the convention that the composition is in the Leibniz order, and in that order one generalizes the left Ore condition, the denominators are on the left. The literature is not being consistent in the conventions here. So I think I talk spans where the first (that is left, in Leibniz order) map has to be inverted.

   By left I mean the convention that the composition is in the Leibniz order, and in that order one generalizes the left Ore condition, the denominators are on the left. The literature is not being consistent in the conventions here. So I think I talk spans where the first (that is left, in Leibniz order) map has to be inverted.

7. • CommentRowNumber7.
   • CommentAuthorDavidRoberts
   • CommentTimeMar 26th 2011
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   Author: DavidRoberts
   Format: MarkdownItexAh, then we are talking about the same fractions.

   Ah, then we are talking about the same fractions.

8. • CommentRowNumber8.
   • CommentAuthorDavidRoberts
   • CommentTimeNov 16th 2012
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   Author: DavidRoberts
   Format: MarkdownItexJust so people know, I have almost completed (at long last) rewriting _Internal categories, anafunctors and localisations_ ([16 November 2012 version](http://ncatlab.org/davidroberts/files/anafunctors_2012-11-12.pdf)). In particular, if anyone feels like reading the introduction to see if it makes sense, I would be appreciative.

   Just so people know, I have almost completed (at long last) rewriting Internal categories, anafunctors and localisations (16 November 2012 version).

   In particular, if anyone feels like reading the introduction to see if it makes sense, I would be appreciative.

9. • CommentRowNumber9.
   • CommentAuthorTobyBartels
   • CommentTimeNov 16th 2012
   • (edited Nov 17th 2012)
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   Author: TobyBartels
   Format: MarkdownItexJust reading the introduction: Where you have >It is therefore *not* true if one is working with categories internal to a category $S$ which doesn’t satisfy the (external) axiom of choice. maybe add something like ‘even a category very much like the category of sets, such as a well-pointed boolean topos’, and maybe even acknowledge that people might not want to assume that the category of sets satisfies the axiom of choice (or even is boolean, for that matter, but maybe you have reason to avoid saying the word ‘constructive’). Typo in the last sentence before Theorem 1.1: ‘demands’ should be ‘demand’. More importantly in that paragraph before Theorem 1.1, it\'s unclear if it\'s part of the hypothesis or the conclusion that $p\colon U \to X_0$ belongs to $\mathbf{Cat}'(S)$. Also, I would probably just write $C$ for $\mathbf{Cat}'(S)$. For what you're doing here, there\'s no reason that $C$ must be the value of $S$ under any particular operation $\mathbf{Cat}'$, or even if it is that this operation must be derived from the pseudofunctor $\mathbf{Cat}$ in any particular way. Then $\mathbf{Cat}'_{ana}(S,J)$ becomes $C_{ana}(J)$; and indeed, the only data needed to define that are $C$ and $J$. In the paragraph introducing anafunctors, change ‘which that’ to ‘such that’ or ‘in which’. When discussing profunctors, possibly remark that Makkai named anafunctors after profunctors. (There is discussion of this in the original anafunctor paper; Makkai credits the idea to somebody else.) The word ‘papillon’ is singular; presumably you want the plural ‘papillons’. In the paragraph with Vitale, ‘one can generalises’ should be ‘one can generalise’. Also in that paragraph, ‘e.g.’ takes a comma aftewards (or if not then use `~` in TeX for the following space). In the Hilsum–Skandalis paragraph, change ‘but In fact’ to ‘but in fact’. The transition from the history to the last two paragraphs of introduction is rather abrupt. Possibly those paragraphs should go before the history, which might even have its own subsection. If I were uninterested in history, then I would miss those two paragraphs, even though they\'re important for other readers. [I have now proofread this comment.]

   Just reading the introduction:

   Where you have

   It is therefore not true if one is working with categories internal to a category $S$ which doesn’t satisfy the (external) axiom of choice.

   maybe add something like ‘even a category very much like the category of sets, such as a well-pointed boolean topos’, and maybe even acknowledge that people might not want to assume that the category of sets satisfies the axiom of choice (or even is boolean, for that matter, but maybe you have reason to avoid saying the word ‘constructive’).

   Typo in the last sentence before Theorem 1.1: ‘demands’ should be ‘demand’.

   More importantly in that paragraph before Theorem 1.1, it's unclear if it's part of the hypothesis or the conclusion that $p\colon U \to X_0$ belongs to $\mathbf{Cat}'(S)$.

   Also, I would probably just write $C$ for $\mathbf{Cat}'(S)$. For what you’re doing here, there's no reason that $C$ must be the value of $S$ under any particular operation $\mathbf{Cat}'$, or even if it is that this operation must be derived from the pseudofunctor $\mathbf{Cat}$ in any particular way. Then $\mathbf{Cat}'_{ana}(S,J)$ becomes $C_{ana}(J)$; and indeed, the only data needed to define that are $C$ and $J$.

   In the paragraph introducing anafunctors, change ‘which that’ to ‘such that’ or ‘in which’.

   When discussing profunctors, possibly remark that Makkai named anafunctors after profunctors. (There is discussion of this in the original anafunctor paper; Makkai credits the idea to somebody else.)

   The word ‘papillon’ is singular; presumably you want the plural ‘papillons’.

   In the paragraph with Vitale, ‘one can generalises’ should be ‘one can generalise’. Also in that paragraph, ‘e.g.’ takes a comma aftewards (or if not then use ~ in TeX for the following space).

   In the Hilsum–Skandalis paragraph, change ‘but In fact’ to ‘but in fact’.

   The transition from the history to the last two paragraphs of introduction is rather abrupt. Possibly those paragraphs should go before the history, which might even have its own subsection. If I were uninterested in history, then I would miss those two paragraphs, even though they're important for other readers.

   [I have now proofread this comment.]

10. • CommentRowNumber10.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 16th 2012
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    Author: DavidRoberts
    Format: MarkdownItexThanks Toby! Much appreciated.

    Thanks Toby! Much appreciated.

11. • CommentRowNumber11.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 19th 2012
    • (edited Nov 20th 2012)
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    Author: DavidRoberts
    Format: MarkdownItexOk, [new version](http://arxiv.org/abs/1101.2363) just sent back to the journal.

    Ok, new version just sent back to the journal.

12. • CommentRowNumber12.
    • CommentAuthorTobyBartels
    • CommentTimeNov 19th 2012
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    Author: TobyBartels
    Format: MarkdownItexI was worried for a moment that you would adopt *all* of my suggestions, so I\'m relieved that you didn\'t entirely. (^_^) Typo in first sentence: ‘essentually’ → ‘essentially’. New sentence in first paragraph: ‘This is true’ may be confusing, since ‘this’ mean that something earlier is *false*. Maybe ‘This applies’ or even ‘This failure applies’? Or change ‘This is true even for’ to ‘This may fail even in’ (where now ‘this’ means the thing that is false)? The ungrammatical construction ‘which that’ remains. Since you added a new Section 2, renumber the section references in the almost-last paragraph of Section 1. Your spelling is inconsistent between ‘Benabou’ and ‘Bénabou’. I\'ve still only read the introduction and history, and I didn\'t fully reread even all of that.

    I was worried for a moment that you would adopt all of my suggestions, so I'm relieved that you didn't entirely. (^_^)

    Typo in first sentence: ‘essentually’ → ‘essentially’.

    New sentence in first paragraph: ‘This is true’ may be confusing, since ‘this’ mean that something earlier is false. Maybe ‘This applies’ or even ‘This failure applies’? Or change ‘This is true even for’ to ‘This may fail even in’ (where now ‘this’ means the thing that is false)?

    The ungrammatical construction ‘which that’ remains.

    Since you added a new Section 2, renumber the section references in the almost-last paragraph of Section 1.

    Your spelling is inconsistent between ‘Benabou’ and ‘Bénabou’.

    I've still only read the introduction and history, and I didn't fully reread even all of that.

13. • CommentRowNumber13.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 19th 2012
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    Author: DavidRoberts
    Format: MarkdownItexThanks, Toby. I should have probably waited for a final proofread, but I just had to get it out the door so I can think about other stuff. I'll incorporate the corrections ready for when the referee gets back to me.

    Thanks, Toby. I should have probably waited for a final proofread, but I just had to get it out the door so I can think about other stuff. I’ll incorporate the corrections ready for when the referee gets back to me.

14. • CommentRowNumber14.
    • CommentAuthorMike Shulman
    • CommentTimeNov 28th 2012
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    Author: Mike Shulman
    Format: MarkdownItexI just got around to having a look at this. It looks good! I hope it's not too late for a few suggestions/questions. Are you committed to using a bibliography style which produces very long citations? Aside from the fact that I'm towards the end of the alphabet and therefore don't like "et.al" citations, I actually find a citation like [So and so et. al. 2010] more annoying when looking it up --- there is more to remember during the time I am flipping to the references list than with a short citation like [23] or [SS10]. I do prefer [SS10] to [23] since then it is easier to guess or remember what the reference is without looking it up every time, but I don't think the very long style has much benefit over [SS10]. I know TAC's author guidelines recommend the long style, but in my experience they don't actually care; you can use anything you want. (I mainly mention this because I'm curious whether anyone actually prefers the long style.) At the top of p4, when you say "in a finitely complete site with reflexive coequalisers" --- do you need the coequalizers to be pullback-stable? I don't know how to compose internal profunctors in general without that assumption. When you say "a pair of internal categories have equivalent externalisations if and only if they are connected by a span of internal functors which are weak equivalences" I expected to see some appearance of stack completion. Isn't this only true if you stack-complete the externalizations? Or do you mean to refer to some kind of "weak equivalence" of externalizations too? Finally, if you want something published to refer to regarding superextensive sites, there is now Example 11.12 of [my paper](http://www.tac.mta.ca/tac/volumes/27/7/27-07abs.html).

    I just got around to having a look at this. It looks good! I hope it’s not too late for a few suggestions/questions.

    Are you committed to using a bibliography style which produces very long citations? Aside from the fact that I’m towards the end of the alphabet and therefore don’t like “et.al” citations, I actually find a citation like [So and so et. al. 2010] more annoying when looking it up — there is more to remember during the time I am flipping to the references list than with a short citation like [23] or [SS10]. I do prefer [SS10] to [23] since then it is easier to guess or remember what the reference is without looking it up every time, but I don’t think the very long style has much benefit over [SS10]. I know TAC’s author guidelines recommend the long style, but in my experience they don’t actually care; you can use anything you want. (I mainly mention this because I’m curious whether anyone actually prefers the long style.)

    At the top of p4, when you say “in a finitely complete site with reflexive coequalisers” — do you need the coequalizers to be pullback-stable? I don’t know how to compose internal profunctors in general without that assumption.

    When you say “a pair of internal categories have equivalent externalisations if and only if they are connected by a span of internal functors which are weak equivalences” I expected to see some appearance of stack completion. Isn’t this only true if you stack-complete the externalizations? Or do you mean to refer to some kind of “weak equivalence” of externalizations too?

    Finally, if you want something published to refer to regarding superextensive sites, there is now Example 11.12 of my paper.

15. • CommentRowNumber15.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 28th 2012
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItexI presume that the paper will come back from the referee with at least a couple of suggestions, so I'm happy to tweak. >Are you committed to using a bibliography style which produces very long citations? No, I'm happy to change it. >do you need the coequalizers to be pullback-stable? Ah, yes, I do. I recall looking at this and thinking it looked odd as it currently stands, but I was keen to get rid of it. >Isn’t this only true if you stack-complete the externalizations? Yes, you are right. In general there is only a span of externalisations which are weak equivalences. >Finally, if you want something published to refer to regarding superextensive sites, ... Ah, good. I forgot there were in there. I'll leave the appendix in, however, because it has a few lemmas that I use.

    I presume that the paper will come back from the referee with at least a couple of suggestions, so I’m happy to tweak.

    Are you committed to using a bibliography style which produces very long citations?

    No, I’m happy to change it.

    do you need the coequalizers to be pullback-stable?

    Ah, yes, I do. I recall looking at this and thinking it looked odd as it currently stands, but I was keen to get rid of it.

    Isn’t this only true if you stack-complete the externalizations?

    Yes, you are right. In general there is only a span of externalisations which are weak equivalences.

    Finally, if you want something published to refer to regarding superextensive sites, …

    Ah, good. I forgot there were in there. I’ll leave the appendix in, however, because it has a few lemmas that I use.

16. • CommentRowNumber16.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 28th 2012
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItexSince I've been looking at this and rendering it down for so long, the result seems less special than when I first proved it; it looks almost a triviality, to my eyes. Certainly the proof is a lot less difficult than it used to be :-)

    Since I’ve been looking at this and rendering it down for so long, the result seems less special than when I first proved it; it looks almost a triviality, to my eyes. Certainly the proof is a lot less difficult than it used to be :-)

17. • CommentRowNumber17.
    • CommentAuthorMike Shulman
    • CommentTimeNov 28th 2012
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexYeah, I can see what you mean. Hope the referee doesn't feel the same way. (-:

    Yeah, I can see what you mean. Hope the referee doesn’t feel the same way. (-:

18. • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeNov 28th 2012
    • (edited Nov 28th 2012)
    • PermaLink
    Author: Urs
    Format: MarkdownItexYou probably know Feynman's famous book, where he recounts how as a physics student he made fun of his math colleagues by observing that they can only prove trivial statements. For whenever he sees them discuss on the board, in the end they will say, "ah, but now its obvious!". :-) That's what math is about...

    You probably know Feynman’s famous book, where he recounts how as a physics student he made fun of his math colleagues by observing that they can only prove trivial statements. For whenever he sees them discuss on the board, in the end they will say, “ah, but now its obvious!”. :-)

    That’s what math is about…

19. • CommentRowNumber19.
    • CommentAuthorMike Shulman
    • CommentTimeDec 23rd 2012
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItex[Woo!](http://www.tac.mta.ca/tac/volumes/26/29/26-29abs.html) Congrats!

    Woo! Congrats!

20. • CommentRowNumber20.
    • CommentAuthorDavidRoberts
    • CommentTimeDec 25th 2012
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItexThanks! (found a stray signal for my phone :-)

    Thanks! (found a stray signal for my phone :-)