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Edited Lie groupoid a little, and new page: locally trivial category. There is an unsaturated link at the former, to Ehresmann’s notion of internal category, which is different to the default (Grothendieck’s, I believe). The difference only shows up when the ambient category doesn’t have all pullbacks (like Diff, which was Ehresmann’s pretty much default arena). It uses sketches, or something like them. There the object of composable arrows is given as part of the data. I suppose the details don’t make too much difference, but for Lie groupoids, it means that no assumption about source and target maps being submersions.
The latter page is under construction, and extends Ehresmann’s notion of locally trivial category/groupoid to more general concrete sites. I presume his theorem about transitive locally trivial groupoids and principal bundles goes through, it’s pretty well written.
I thought Ehresmann’s definition just amounted to assuming that the particular pullbacks necessary to define an internal category exist. You give the object of composable arrows as part of the data, but don’t you then require as part of the definition that it actually is a pullback of the object of arrows with itself over the object of objects? Or am I confused?
One does require that the object of composable arrows is the pullback. Up to isomorphism, one could just say the pullback exists, but to be strict, the object is given explicitly. I’m not pedantic on the issue, though. I just wanted to record that originally the submersion condition wasn’t used. I’m not sure when it was introduced.
I can imagine that in addition to the object $X_2$ of composable arrows, also the object $X_3$ of triples of composible arrows (the one coming in to express associativity of composition) is explicitely given, and also this is required to be a pullback. Then I can imagine also the higher $X_n$ are given and identified with pullbacks this way, so that in the end the datum should be that of an internal Kan complex whose horn filling maps are isomorphisms for $n\gt 1$ (this expresses the fact that the $X_n$ with $n\geq 2$ are iterated pullbacks of $X_1$ over $X_0$).
One can rephrase this by saying that a truncated simplicial object $(X_0,X_1)$ has (up to equivalence) at most one extension to a Kan complex whose horn filling maps are isomorphisms for $n\gt 1$; the condition to be satisfied in order to have this extension are precisely the axioms of internal groupoid in terms of $X_0$, $X_1$, identities, source and target maps.
This suggests (but this is just a speculation now that I’m writing) that one could define an internal (e.g. Lie) $k$-groupoid as a truncated simplicial object $(X_0,X_1,...,X_k)$ which can be extended to an internal Kan complex whose horn filling maps are isomorphisms for $n\gt k$ (this should imply that the extension is unique up to equivalence, if it exists). One should also be able to write down explicit condition in order for this extension to exist, in terms of the truncated simplicial object $(X_0,X_1,...,X_k)$ , but I can figure this gives a qite obscure combinatorics already for $n=2$
By the way, any reference for the $n=2$ case worked out?
One usually uses the internal version of the coskeleton functor. Then (modulo comment below) an internal $k$-groupoid is an internal simplicial object in the essential image of the $k(+1 or 2)$-coskeleton functor (can’t remember what degree this is meant to be) from truncated simplicial objects to simplicial objects.
Things are a bit subtle, because the Kan condition asks for a certain map of sets to be surjective, but in the internal setting (especially in the smooth setting) one needs to specify what ’surjective’ should be translated to: epimorphism? Admits local sections for some pretopology? Submersion? (in the smooth case)
created stub for submersion
Actually, at Examples at the page simplicial skeleton it tells me what index to use on the coskeleton functor: an $(n+1)$-coskeletal Kan complex is (the nerve of) an $n$-groupoid.
The term “locally trivial category” is a bit unfortunate, isn’t it? Cause these beasts in no way need to locally be like a trivial category.
At least for a groupoid the condition is more like “locally connected” in a sense.
One could say ’locally trivial internal category’. I’ve added a bit of disambiguation at the page. A locally trivial category is locally connected in just the same way as a locally trivial groupoid, but only using isomorphisms. I presume one could look at a more general version, not requiring the local sections to land in the invertible arrows, but I haven’t come across a good use for them.
David, could you remind me: are the surjective submersions precisely the regular epis in $Diff$?
cool! I see coskeleton was the one-word (and functorial) way to say what I had in mind.
Here is a possible motivation for the +2 degree suggested by David at #5: to define a 1-groupoid I need the space $X_0$ of objects, the space $X_1$ of morphisms, and the space $X_2$ to define composition of morphisms. Then I can coskeletize to obtain a simplicial object $cosk_2(X_0,X_1,X_2)$, and asking that this is a Kan complex with unique fillers for $n\gt 1$ is the same thing as asking that we have a 1-groupoid with objects given by $X_0$ and morphisms given by $X_1$. But actually, we just have to check this for $sk_3(cosk_2(X_0,X_1,X_2))$.
So it could be possible that in general one has to look at $sk_{n+2}(cosk_{n+1}(X_0,X_1,\dots,X_{n+1}))$.
started a quick section (2,1)-category of Lie groupoids but don’t have the time to do this justice right now
The difference only shows up when the ambient category doesn’t have all pullbacks
Sorry I did not read the Ehresmann aprpoach, but in general for internal categories you just need to have some particular pullbacks not all pullbacks, and this is standard.
I vaguely recall Grothendieck defined internal category/groupoid objects only when the ambient category was finitely complete. I don’t have any references to substantiate this, though. I guess I was trying to point out that the assumption of submersions is not necessary to define Lie groupoids, but is a convention, and one that was not introduced at the start.
@Urs #10 - not sure, I’m afraid.
Okay, thanks David. I’ll mention then at least azt the entry that they are regular epis.
I vaguely recall Grothendieck defined internal category/groupoid objects only when the ambient category was finitely complete.
The people write always the simplest assumption, unless there is need to use more general ones. There is no person I ever met who understand what internal category is who does not understand that one needs just those pullbacks which one needs.
Yeah, I wouldn’t make too big a deal about having all pullbacks vs having only the needed ones; I don’t think it counts as a different notion of “internal category.” However there is of course a difference in the particular case of internal categories in Diff, whether you require submersions or just that the pullbacks exist.
In that case, you may need certain condition on the rank/transversality in each interesting statement or additional construction. Condition of submersion gives maximal rank and simplifies more general interesting cases still yielding good manifold strcutures for related constructions.
Let’s just remember that we don’t need to “require” anything in fact. The general good notion of Lie groupoid is a stack on $Diff$. Some of these may be represented by internal groupoids with source-target submersion, some not. Some may even have equivalent models of either sort!
The question is not so much what to require by force, but how to nicely characterize hierarchies of “tame” objects in the $(2,1)$-topos $Sh_{(2,1)}(Diff)$ in a useful manner.
Of course that wil technically boil down to the same kind of technical discussion. But maybe helps to keep the right perspective on these questions.
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