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In a discussion from last year at the -café, John Baez and David Corfield mention (here, here, here, and here) that restricting attention to the skeleton of a groupoid is inadequate when we’re concerned with smooth groupoids. Why is the skeleton inadequate here? I’m particularly interested in the case of general relativity, since I came to this question while trying to understand general covariance in HoTT, where it seems important that we’re dealing with smooth groupoids. I’d like to be able to point to something that we want that the skeleton of can’t give us.
Urs or others will no doubt give you better information, but in FQFT we have
Proposition 1. Assuming the axiom of choice in the ambient set theory, every groupoid is equivalent to a disjoint union of delooping groupoids, example 2 – a skeleton.
Remark 2. The statement of prop. 1 becomes false as when we pass to groupoids that are equipped with geometric structure. This is the reason why for discrete geometry all Chern-Simons-type field theories (namely Dijkgraaf-Witten theory-type theories) fundamentally involve just groups (and higher groups), while for nontrivial geometry there are genuine groupoid theories, for instance the AKSZ sigma-models. But even so, Dijkgraaf-Witten theory is usefully discussed in terms of groupoid technology, in particular since the choice of equivalence in prop. 1 is not canonical.
I should add the obvious point that treating an interval just as a plain homotopy type leaves it equivalent to the point. Hence the need for all those cohesive modalities Urs tells us we need for geometry, see cohesion.
Curious that you ask this now. Just the other day somebody told me about his confusion when applying the classification result of geometrically discrete 2-groups via their skeleta to smooth 2-groupoids. Indeed, in the smooth case (and other geometric cases) that classification result does not apply.
As to say “why” not, I’d say the key is to observe that passing to a skeleton is a rather drastic operation and better would be to turn the question around and wonder why there are special situations that it works at all.
Note that if every smooth(=Lie) groupoid had a skeleton, every surjective submersion would have a section. This is false, so one just has to live with the fact skeletons don’t exist in this context, ie ignore them entirely.
I should say, in general they don’t exist. Locally trivial Lie groupoids have skeletons, but they are very special and don’t arise a lot in practice.
Thanks all for the replies.
if every smooth(=Lie) groupoid had a skeleton, every surjective submersion would have a section
My gut feeling was that this was all related to the failure of choice in Lie categories. I should spend some more time learning about these.
As to say “why” not, I’d say the key is to observe that passing to a skeleton is a rather drastic operation and better would be to turn the question around and wonder why there are special situations that it works at all.
This is essentially what I’m trying to get a better handle on. I have a sense of passing to the skeleton being drastic, but I’m trying to get a better grip on why it’s so drastic in practice. Really, I’m finding myself in conversations with people who insist on the sole importance of the skeleton, much like the discussion in the comments of the blog post I link above. So the presumption is that passing to the skeleton is the right thing to do, and I’m trying to articulate more clearly why this should make one “uncomfortable” in the case of smooth or higher groupoids. Smoothness is required for doing variational problems, but why not (a) first pass to the skeleton and then do the variational problem there, or (b) take a non-variational approach? Is passing to the skeleton drastic because it can’t be done internal to ? This would block (a), if I could convince them that one should reason internally. Working in terms of local Lagrangians is the way things are always done in some areas of physics, but GR is often presented just in terms of the EFE, without reference to the Einstein-Hilbert action. I can see some arguments that the variational formulation is helpful or even needed in some parts of GR, but I don’t know if I could really argue that variational formulation is important enough to overcome commitment to working with skeletons.
My apologies if these questions are overly naive. I’m still rather new to HoTT (I’ve really just read the book) and higher category theory, and I’m trying to see how it hooks up with what I already know.
How about the action of reflecting the real line about 0? The action groupoid has the real line as objects and non-trivial arrows between and , including the extra morphism at 0. The skeleton would be the non-negative real axis with only non-trivial morphism at 0. Now there won’t be a smooth equivalence between these two Lie groupoids.
@John Dougherty, the reason that it’s so drastic in practice is, as David R. and David C. point out, that it involves making choices which may not be made smoothly, in general. In that way it breaks the geometric structure, in general.
Do you maybe have more details on the explicit example that you and the people you are talking with have in mind? It should be straightforward to see where in these examples passing to skeleta breaks the geometry of the problem.
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