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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMar 10th 2012
   • (edited Mar 10th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have worked a bit on _[[2-congruence]]_. The main addition is that I started an [Examples](http://ncatlab.org/nlab/show/2-congruence#Examples)-section, where I started writing out an explicit proof (little exercise in unwinding the definitions) of the statement: _The 2-category of 2-congruences in $Grpd$ is equivalent to that of small categories_. One should write out more. But it is getting late for me now. I should continue another day.

   I have worked a bit on 2-congruence.

   The main addition is that I started an Examples-section, where I started writing out an explicit proof (little exercise in unwinding the definitions) of the statement:

   The 2-category of 2-congruences in $Grpd$ is equivalent to that of small categories.

   One should write out more. But it is getting late for me now. I should continue another day.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMar 10th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexI am back to editing now. (There were some gaps in the Example-discussion of yesterday. Also, I realize there is something I still need to understand...)

   I am back to editing now. (There were some gaps in the Example-discussion of yesterday. Also, I realize there is something I still need to understand…)

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMar 10th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexHere is where I am stuck: I wanted to give an elementary argument that every 2-congruence in $Grpd$ is equivalent to one arising from a category $C$ as $Core(C^{\Delta[1]}) \stackrel{\to}{\to} Core(C) \times Core(C)$. What I don't see how to show from the axioms is that the internally invertible morphisms in an arbitrary 2-congruence $D$ all come from $D_0$. This must be trivial. I am probably being dumb here. I'll be doing something else now and see if this problem has gone away when I come back. :-)

   Here is where I am stuck:

   I wanted to give an elementary argument that every 2-congruence in $Grpd$ is equivalent to one arising from a category $C$ as $Core(C^{\Delta[1]}) \stackrel{\to}{\to} Core(C) \times Core(C)$.

   What I don’t see how to show from the axioms is that the internally invertible morphisms in an arbitrary 2-congruence $D$ all come from $D_0$.

   This must be trivial. I am probably being dumb here. I’ll be doing something else now and see if this problem has gone away when I come back. :-)

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeMar 12th 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexThe thing that you don't see how to show is, in fact, not true. (-: The 2-category of congruences in Gpd is equivalent to Cat only if you define its morphisms to be "anafunctors". In other words, with this approach we obtain Cat not as the category of complete Segal objects in Gpd, but by inverting some weak equivalences in the category of (incomplete) Segal objects in Gpd. 2-congruences in Gpd are a truncated version of groupoid objects in $(\infty,1)$-toposes, and the latter are (as I remarked in the other thread) not generally CSS's.

   The thing that you don’t see how to show is, in fact, not true. (-: The 2-category of congruences in Gpd is equivalent to Cat only if you define its morphisms to be “anafunctors”. In other words, with this approach we obtain Cat not as the category of complete Segal objects in Gpd, but by inverting some weak equivalences in the category of (incomplete) Segal objects in Gpd.

   2-congruences in Gpd are a truncated version of groupoid objects in $(\infty,1)$-toposes, and the latter are (as I remarked in the other thread) not generally CSS’s.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMar 12th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItex> The 2-category of congruences in Gpd is equivalent to Cat only if you define its morphisms to be "anafunctors". I was thinking: in $Grpd$ we have the axiom of choice, hence $2 Cong_s(Grpd) \simeq 2 Cong(Grpd)$. But clearly I am mixed up about something then.

   The 2-category of congruences in Gpd is equivalent to Cat only if you define its morphisms to be “anafunctors”.

   I was thinking: in $Grpd$ we have the axiom of choice, hence $2 Cong_s(Grpd) \simeq 2 Cong(Grpd)$. But clearly I am mixed up about something then.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeMar 12th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItex> in Grpd we have the axiom of choice Ah, we don't! Okay, I see :-)

   in Grpd we have the axiom of choice

   Ah, we don’t! Okay, I see :-)

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeMar 12th 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexNot in the naive sense that "every effective-epi has a section", no we don't.

   Not in the naive sense that “every effective-epi has a section”, no we don’t.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeMar 12th 2012
   • (edited Mar 12th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> no we don't. You did see my #6, right? :-) By the way, one nice thing about looking at internal categories (in the sense of complete Segal objects) in an $(n,1)$-topos for $n \geq 2$ is that we don't need -- it seems to me -- to use anafunctors anymore, i.e. that we don't need to apply an additional localization. To think of this in a pedestrian way (as I currently do): for $C$ a simplicial site, regarded as presenting an $(\infty,1)$-site, we can look at the model structure $$ [C^{op}, [\Delta^{op}, sSet]_{cSegal}]_{loc} $$ for $(\infty,1)$-category valued sheaves (where "${}_{cSegal}$" denotes localization of a global model structure at inner horns and the groupoidal interval, and where "${}_{loc}$" means localization at covering sieves of $C$). Or we can look at the model structure $$ [\Delta^{op}, [C^{op}, sSet]_{loc}]_{cSegal} $$ for complete Segal objects in the model structure presenting the $(\infty,1)$-topos over $C$. Both model structures should be manifestly the same, it seems.

   no we don’t.

   You did see my #6, right? :-)

   By the way, one nice thing about looking at internal categories (in the sense of complete Segal objects) in an $(n,1)$-topos for $n \geq 2$ is that we don’t need – it seems to me – to use anafunctors anymore, i.e. that we don’t need to apply an additional localization.

   To think of this in a pedestrian way (as I currently do): for $C$ a simplicial site, regarded as presenting an $(\infty,1)$-site, we can look at the model structure

   $$[C^{op}, [\Delta^{op}, sSet]_{cSegal}]_{loc}$$

   for $(\infty,1)$-category valued sheaves (where “${}_{cSegal}$” denotes localization of a global model structure at inner horns and the groupoidal interval, and where “${}_{loc}$” means localization at covering sieves of $C$).

   Or we can look at the model structure

   $$[\Delta^{op}, [C^{op}, sSet]_{loc}]_{cSegal}$$

   for complete Segal objects in the model structure presenting the $(\infty,1)$-topos over $C$.

   Both model structures should be manifestly the same, it seems.

9. • CommentRowNumber9.
   • CommentAuthorMike Shulman
   • CommentTimeMar 12th 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexYes, I was agreeing with your #6 (perhaps overly tersely). I also agree that adding the "completeness" condition obviates the need to use anafunctors in the $(\infty,1)$-case. There's at least one subtlety in the case $(n,1)$ for $n\lt\infty$: an arbitrary internal category in Gpd (for instance) is more like a (2,1)-category than a 1-category unless you add some "truncation" condition, and without the truncation condition the "completeness" condition is unreasonable (regarding a general (2,1)-category as an internal category in Gpd, its 2-groupoid of objects can't be represented by a 1-groupoid of objects). But probably with that condition included, you are right. Of course, that only works when internalizing in a nice place like a topos.

   Yes, I was agreeing with your #6 (perhaps overly tersely).

   I also agree that adding the “completeness” condition obviates the need to use anafunctors in the $(\infty,1)$-case. There’s at least one subtlety in the case $(n,1)$ for $n\lt\infty$: an arbitrary internal category in Gpd (for instance) is more like a (2,1)-category than a 1-category unless you add some “truncation” condition, and without the truncation condition the “completeness” condition is unreasonable (regarding a general (2,1)-category as an internal category in Gpd, its 2-groupoid of objects can’t be represented by a 1-groupoid of objects). But probably with that condition included, you are right.

   Of course, that only works when internalizing in a nice place like a topos.

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMar 12th 2012
    • PermaLink
    Author: Urs
    Format: MarkdownItexI see, thanks, I hadn't really appreciated that additional truncation issue. While I am interested in internalizing in a genuine $(\infty,1)$-topos, that's still good to keep in mind.

    I see, thanks, I hadn’t really appreciated that additional truncation issue. While I am interested in internalizing in a genuine $(\infty,1)$-topos, that’s still good to keep in mind.