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I edited the formatting of internal category a bit and added a link to internal infinity-groupoid
it looks like the first query box discussion there has been resolved. Maybe we can remove that box now?
I think that similar ideas for algebraic Galois group are used in Grothendieck's still unfullfilled ideas in his 1980s manuscripts and is in general known in Galois theory (covering spaces and higher Postnikov fibers have natural common generalizations in this context).
not sure if this is relevant for what you have in mind, but taking path oo-groupoids can be made into a map from a lined oo-topos to itself. For instance taking a topological oo-groupoid (might be just a topological space) to the topological oo-groupoid .
Same for smooth oo-groupoids. For the smooth model of String, one can start with the Lie 2-algebra , then form the smooth space (sheaf) which is such that its plots are precisely the flat -valued differential forms on . Then one can form the smooth oo-groupoid of that space. Finally, we can take from this simplicial sheaf degreewise the underlying concrete sheaf (=diffeological space) to get the smooth oo-groupoid . The claim is that that's a smooth version of .
This is discussed in some detail (though with slightly more antiquated tools than I have now) in section 5.2.3 here.
Concerning the categorical degree, I feel like remarking that it's only the homotopy groups (of an infinity-stack) that have intrinsic meaning, not the degrees of a truncated model.
Fur instance the goup looks like, well, a group, but it is equivalent to the 2-group that comes from the crossed module .
Similarly, the String 2-group is a -central extension of an ordinary group , hence a 2-group extension, but this is equivalent to a -central extension, which looks like a 3-group extension, but is really equivalent to the original 2-group extension.
@David: you wrote
In this context one needs to be au fait with Frechet manifolds, sadly an area where I am lacking.
I have a little facility with Frechet manifolds. Is there something here that I could help with?
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@David: <br/><br/><blockquote><br/>PI_1(X) can indeed be topologised, but composition is only continuous when X is locally nice - locally connected and semilocally simply connected.<br/></blockquote><br/><br/>This is nice: if I'm not wrong here these are exactly the conditions for X to have a universal cover; also I find it nice that one needs more and more local regularity as higher covers come into play. <br/><br/><blockquote><br/>An easy way to see that the universal cover of a topological group X is topological is that the universal cover = the homotopy fibre of X --> Pi_1(X) at a point x = the source fibre of Pi_1(X) at x. This is a subgroup of the group of arrows, as Pi_1(X) is a strict 2-group as shown by Brown-Higgins in the 70s.<br/></blockquote><br/><br/>That's exactly what I was trying to say. what i think is important to stress is that there is a subtle interplay here between topological and groupoidal aspects: from the topological perspective one ends up with an object which is clearly a simply connected cover and then has to show that this carries a natural group structure; from the groupoids point of view, the object one ends up with is clearly a group, but one now has to show that it is a simply conneced cover. And an incredible elegant way to prove both things at a time is to perform the two constructions independently, and to see that they produce the same object. here I think there should be some very general argument a priori telling me that the two constructions will lead to teh same object, but I'm still missing this argument.<br/><br/>@Urs:<br/><blockquote><br/>Concerning the categorical degree, I feel like remarking that it's only the homotopy groups (of an infinity-stack) that have intrinsic meaning, not the degrees of a truncated model.<br/></blockquote><br/><br/>edit: let me see if I correctly undersatnd this. if we start with the action groupoid of on what we are dealing with is not only a groupoid, but a groupal groupoid (since is a group and the -action is compatible with the group operation of ). so the 0-th truncation of this groupoid is a group. this 0-th truncation is evidently , so we should think of not as our basic object but rather as . right?
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Just a comment, probably no news to anyone, but just for the record:
to realize the equivalence as a homotopy equivalence (morphisms going back and forth, being weak inverses) one needs to find suitable ana-2-functors, inverting the evident morphism from one 2-groupoid to the other, but for just knowing that the two are equivalent it is sufficient to have that one morphism and checking that it is a weak equivalence.
This is true for topological infinity-groupoid realizations of the situation and even for Lie-infinity-groupoid realizations.
For that one may observe that the functor is k-surjective for all k on all stalks of Top or Diff: these are the germs of n-dimensional disks (as described by Dugger in the reference cited at topological infinity-groupoid.) So for the standard n-disk of radius r, we form the groups of group-valued maps , and and then check that the functor of ordinary 2-groupoids is k-surjective for all k for some r small enough. And it clearly is: on any disk, any U(1)-valued function (continuous or even smooth) may be lifted to an -valued function, and the nonuniquenss of the lifts are precisely given by -valued functions on the disk.
but on second thought, I think a plausible perspective could be
Yes, precisely: very different looking objects may all be equivalent to the same 2-group, (or the same oo-groupoid). The highest degree of nondegenerate cells is not an invariant under this. Instead, the main invariants are the intrinsically defined homotopy groups
I find the use of AUT(K) something which works, but which is chosen in quite an arbitrary way. maybe it's worth creating a Schreier theory entry to discuss it
Yes, I agree with all you say here. As far as I am concerned: I don't have the energy and time this particular project right now, though. Probably later sometime.
Schreier theory is included in group extension entry.
It is instructive the way Faro et al. pharse in their article, placing the role of AUT K via interpreting the Grothendieck construction. I will once write about it.
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<blockquote>The point of nonabelian cohomology is to _not_ use stable theory</blockquote><br/><br/>so that's why it's called "nonabelian" or "unstable"! I should have known, but actually it was this remark above that really made me clear this fact, thanks!<br/><br/>still, I'd like to think of going from to as a suspension. in other words, when dealing with a cohomology with degrees I'd like to have two functors raising and lowering the degrees. in the stable theory these are looping and delooping, but one could have a nice theory in any case one has a nice shift. <br/><br/>this is somehow reminiscent of "old way" triangulated categories stuff. there teh shift functor is a basic datum; then in the stable infinity-category approach the shift is no more basic but it is recovered from homotopy pullbacks and pushforwards to the zero object. but we could still think of "unstable cohomology with degrees" as something coming from a shift functor which is not defined in terms of homotopy pullbacks/pushforwards to the zero object.<br/><br/>clearly, in order to have a nice and quite general theory, one would presumibly need not only the shift but also some good functorial behaviour. and then one could check that to indeed provides an example of this structure.
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There are ways to define homotopy groups for topological groupoids
There is, by the way, something even more general: the notion of homotopy group (of an infinity-stack)
a classical example is . this suggests that the natural setting for cohomology in different degrees is a stable -category.
It is true that only if the coefficient object has arbitrary delooping that the above definition of makes sense for all . But there is a priori no requirement that such a definition needs to make sense. For general nonabelian coefficients degree cohomology is defined only up to some finite . This is a standard thing in the literature on nonabelian cohomology.
But you can always form loops to go don do lower degrees.
Am hereby moving an old Discussion-section from internal category to here
[begin forwarded discussion]
Previous versions of this entry led to the following discussions
+–{: .query} I think things are mutliply inconsistent in this entry. I do not want to change as I do not know what the intentional notation to start with was. If $p_1; s = p_2; t$ that mean that target is read at the left-hand side (composition as o, not as ;), while the diagrams before that suggest left to right composition. Then finally the diagram for groupoids has $s$ both for source and inverse, and there is only for right inverse, and one should also check convention, once it is decided above.-Zoran
You're right; I think that I caught all of the inconsistencies now. Incidentally, one needs only inverses on one side (as long as all such inverses exist), although it's probably best to put both in the definition. (For groupoids, one also needs only identities on that same side too! This proof generalises.) —Toby =–
+–{: .query}
Question: I’ve looked at the definition of category in $A$ for a while and still haven’t been able to absorb it. Could we walk through an explicit example, e.g. “This is exactly what $C_0$ is, this is exactly what $C_1$ is, this is exactly what $s,t,i$ are, and this is how it relates to the more familiar context”? For example, an algebra is a monoid in $Vect$. I’ll try to step through it myself, but it will probably need some correcting. - Eric
Eric, one example to ponder is: how is an internal category in Grp the “same” as a crossed module? As a partial hint, try to convince yourself that given a internal category, part of whose data is $(C_1, C_0, s, t)$, the group $C_1$ of arrows can be expressed as a semidirect product with $C_0$ acting on $\ker(s)$. The full details of this exercise may take some doing, but it might also be enjoyable; if you get stuck, you can look at Forrester-Barker. - Todd
Urs: I don’t know, but maybe Eric should first convince himself of what Todd may find too obvious: how the above definition of an internal category reproduces the ordinary one when one works internal to Set. Eric, is that clear? If not, let us know where you get stuck!
Tim: I have just had a go at 2-group and looked at the relationship between 2-groups and crossed modules in a little more detail, in the hope it will unbug the definition for those who have not yet ’groked’ it.
Eric: Oh thanks guys. I will try to understand how a small category is a category internal to Set first and then move on to category in Grp and the stuff Tim wrote. I’m sure this is all obvious, but don’t underestimate my ability to not understand the obvious :)
Eric: Ok. Duh. It is pretty obvious for Set EXCEPT for pullback. Pullbacks in Set are obvious, but what about other cases? Why is that important and what is an example where there are not pullbacks? In other words, is there an a example of something that is ALMOST a category in some other category except it doesn’t have pullbacks, so is not?
Tim: If I remember rightly the important case is when trying to work on ’smooth categories’, that is, general internal categories in a category of smooth manifolds. Unless you take care with the source and target maps, the pullback giving the space of composible pairs of arrows may not be a manifold. (I remember something like this being the case in Pradines work in the area.) The point is then that one works with internal categories with extra conditions on $s$ and $t$ to ensure the pullback is there when you need it.
Toby: Usually in the theory of Lie groupoids, they require $s$ and $t$ to be submersions, which guarantees that the pullback of any map along them exists. =–
[end forwarded discussion]
Reference
at internal category has a bogus pdf link which redirects to internal category! Is somebody having a correct pdf link ?
Urs 25, when you delete a discussion and archive it, then please leave the backlink to the archived version from the old place, otherwise is essentially lost. I done it this time (into references).
I have created a new entry locally internal category and listed it under related notions at internal category.
@Zoran I was not sure what you meant by the first sentence of locally internal category. I have amended it to mean what I think you meant but please check.
I added a bit more.
Very nice, Mike !
On the other hand, why saying “in the sense of the appendix of (Johnstone)” at stack semantics, rather than more correctly attributing phrase “in the sense of Penon 1974”. Is there a slight difference ? (I did not look at Penon’s article yet; strangely I can find CR Acad Paris at the partly free gallica repository till 1973 and then again from 1979, but not for 1974-1978)
I resolved the bogus pdf link which was asked about it 26 and will correct it at internal category in a minute:
I didn’t write that phrase at stack semantics. Feel free to correct it if you know the correct reference.
“in the sense of the appendix of (Johnstone)” was written by Ingo Blechschmidt.
It is not a mistake, just more people together know more about the history :) and it is good when we agree (as often mathematicians do not agree on history).
I have brushed-up the entry internal category a little. Added the remark on cartesian closure discussed in another thread to a Properties-section
I just noticed how hard it is (or was) to find the crucial discussion at 2-topos – In terms of internal categories if all one does is search the nLab for “internal category”.
So I have now added a pointer to that at internal category by way of a brief paragraph a Properties – In a topos.
Should be expanded.
added one more item to the list of examples:
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