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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.
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…)
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. :-)
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.
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.
in Grpd we have the axiom of choice
Ah, we don’t! Okay, I see :-)
Not in the naive sense that “every effective-epi has a section”, no we don’t.
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.
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.
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.
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