Thanks very much Noam, that’s a nice idea! As for 3-connectedness, I am not sure that I can make use of it in the way I was originally hoping, but I still think it would be a nice observation if true, probably useful for something.

(I think we’re essentially giving the same argument, basically relying on the fact that the vertex-edge-face incidences are preserved.)

I agree, that’s a good way of putting it.

]]>We have at K(n)-local stable homotopy theory:

Say that an ∞-groupoid is

strictly tameorof finite type(Hopkins-Lurie 14, def. 4.4.1) or maybe better is atruncated homotopy type with finite homotopy groupsif it has only finitely many nontrivial homotopy groups each of which is furthermore a finite group.

Rather a mouthful the ’better’ version.

And as for ’truncated homotopy type’, the use of the past participle sounds to me that it’s being to compared to what it was in some pristine untruncated form, as though we called abelian groups, abelianised groups.

]]>That is a good question! Perhaps ind-finite homotopy type? The condition without finiteness could also sensibly be called ’coskeletal’ (see Beke ’Higher Cech theory’) so perhaps ’finitely coskeletal’ or something like that for this one might work. Alternatively ’strongly homotopy finite’ would be simpler.

]]>Reflecting on Mike at #14, I wonder why they don’t work up an $\infty$-version of Enriched indexed categories.

They seem to have given a lot of thought to the indexing category having the twin properties of being orbital (making the Beck-Chevalley condition hold, at least some of the time) and being atomic (having trivial retracts). I wonder if the latter condition has something to do with Mike’s EI-inverse condition.

]]>I’ve updated Kuiper’s theorem slightly to point out that in fact most of the various topologies on $U(H)$ weaker than the norm topology agree. This is due to Espinoza-Uribe, which an earlier partial contribution by Schottenloher. So it seems that Andrew’s complaint in #6 was probably directed at $B(H)$ or $GL(H)$, since the weak operator and strong operator topologies agree on $U(H)$, but not on $GL(H)$.

I’ve also edited unitary group to add this reference, and I found that the page claimed that $U(H)$ was the maximal compact subgroup of $GL(H)$ *even in the infinite-dimensional setting* (!). So I definitely fixed that.

A note to myself to add later: $U(H)$ is a Banach Lie group in the norm topology, but not a Lie group in the strong topology; conversely, the left regular representation of a compact topological group with Haar measure is not continuous in the norm topology, but is continuous in the strong topology.

]]>Hmm. If this page is really covering two ideas, it had better not begin

The concept of a homotopy type (homotopy n-type) all of whose homotopy groups are finite groups does not have an established name. Sometimes it is called $\pi$-finiteness.

It would surely be better to say

The property of a homotopy type (homotopy n-type) that all of its homotopy groups are finite groups does not have an established name. Sometimes it is called $\pi$-finiteness.

Isn’t the second of these much more important?

By the way, is there a quick way to speak of a homotopy type with trivial homotopy groups after some point, so in the union over $n$ of $n$-types? The two references at that page have to go to the lengths of writing ’Spaces with finitely many non-trivial homotopy groups’.

]]>Apparently the answer to the question asked (over 7 years ago) in #3 above is “if it isn’t, it should have been”. Lurie has now changed the definition of $Idem$ to be the nerve of the free category containing an idempotent; see here and here. We should update the definition on the lab.

]]>I was not sure what DavidC was asking. There are two ideas here, one with only finitely many non-trivial homotopy groups and the other without that restriction. These correspond to the bit in brackets `(homotopy n-type)' and the`

homotopy type’ to the other. Looking at Lurie does not help on this point as the example referred to only looks at the ‘n-type’ bit. Both concepts are studied. Graham Ellis’s paper is a nice one for the n-type result.

I used the n-type idea in papers on the Yetter model (TQFTs), so perhaps that was one place that DavidR had seen these. They also relate to homotopy cardinality.

]]>I would think so, I’m sure I’ve seen theorems about such homotopy types, but can’t recall what or where.

]]>That is of course not the famous Giraud’s book Cohomologie non abelienne but his earlier Memoirs SMS article on descent, which is as a size of a small book as well and which contributed in its content to the later book. I will teach descent theory and nonabelian cohomology in a graduate course the next academic year.

]]>I have added a couple of clarifying lines in the idea section. I do not consider the article too long though as so many viewpoints emerge even in the most basic treatment of the topic.

]]>We have the related homotopy type with finite homotopy groups which is about

The concept of a homotopy type (homotopy n-type) all of whose homotopy groups are finite groups

Should this have the extra condition that only finitely many homotopy groups are non-trivial?

]]>I started finite ∞-group, and added that same reference to Sylow p-subgroup.

]]>I added in the (∞,1)-category of (∞,1)-modules over an E-∞ ring as an example.

Also, I pointed out that left and right closedness may be separated out.

]]>I made some very minor changes to the introduction at descent. I hesitate to do more but at present the discussion does not seem that readable to me. Can someone look at it to see what they think? The intro seems to plunge in deep very quickly and so the ‘idea’ of descent as that of gluing local information together, does not come across to me. The article is lso quite long and perhaps needs splitting up a bit.

]]>Examples are sparse at closed monoidal (infinity,1)-category. What would be good examples to add? Perhaps the stable (infinity,1)-category of spectra, although that doesn’t mention closedness.

]]>Giraud is available on Numdam.

]]>Well, I don’t have a surplus of time right now, and I’m not hugely motivated to put a lot of thought into this because internal cubical objects in sheaf 1-toposes are not a very interesting class of models for me. I suppose there might be some choicey issues regarding whether one wants the universe to classify things by actual pullbacks or only after passage to a cover. But couldn’t one also note that cubical objects in a Grothendieck topos (over a classical base $Set$) are themselves again a Grothendieck topos over the same $Set$ and apply Streicher’s construction directly?

]]>Re #28: great; I added some remarks about this to wave front set.

]]>I believe I was looking at p9 of this note by Thomas Streicher. If there was a trivial internal version for sheaf toposes, I would have expected Thomas to state it. In any case, I you have a complete argument in mind, it might be worth recording it, as it seems to be lacking from the literature at the moment. I might try to do it myself later, but I am on holiday now.

]]>What’s an example of a distribution whose wave front set is nonempty but not all of $k\neq 0$?

The Heaviside distribution on higher dimensional space has wave fronts being orthogonal to the hyperplane at which the function jumps.

More generally, the wave front set of a characteristic function of a subset $U \subset \mathbb{R}^n$ with smooth boundary $\partial U \subset \mathbb{R}^n$ (hence the distribution which integrates its argument over $U$) is the conormal bundle of $\partial U \subset \mathbb{R}^n$. (See around figure 3 in arXiv:1404.1778. Apparently this is how wave front sets are used in computer image recognition.)

Then the delta-distribution $\delta(x-y)$ regarded as a generalized function of *two* arguments has wave front set of the form $(x,(k,-k))$ (see at *Wick algebra* here). More sophisticated such examples are listed at *propagators - table*.

But how about an example on $\mathbb{R}^1$? I haven’t checked (am on vacation now…), but I suspect the reason that the Heaviside function on $\mathbb{R}^1$ has wave front at the origin pointing in both directions may be attributed to the graph of the Heaviside function having *two* kinks, one to the left and one to the right of zero. If that is right, then we should get a wave front set pointing in only one of the two directions by smoothing out one of these two kinks. So I am guessing that a distribution like $\phi \mapsto \int_0^\infty \sqrt{x} \phi(x) \, dx$ should have wave front at the origin pointing only to the left.

Right, the wave front set is what they call “conal” or “conic”. I agree that thinking of it as a subspace of the unit co-sphere bundle would be more elegant.

]]>Oh, sorry, yes, I meant $P$. I was thinking of the fact that the points of projective space are 1-dimensional subspaces, but I forgot that when you consider higher dimensional subspaces, for some reason it’s called a “Grassmanian” instead of a projective space.

I also missed that the “conic sets” are only stable under multiplication by *positive* scalars. But in that case it seems even easier: the wave front set should be a subspace of the unit sphere bundle of the cotangent bundle.

Older nLabians may or may not recall that the original URL of the nForum was nforum.mathforge.org, and that at times it was useful to have a back-up hostname for the nlab itself which was nlab.mathforge.org. As I still rent the mathforge.org domain, I’ve kept these pointing at the nforum and nlab to avoid dead links.

I’m in the process of doing a bit of housekeeping at mathforge, and in so doing I’m updating the nforum.mathforge.org and nlab.mathforge.org DNS entries so that they are *CNAME* not *A* entries. (If that means nothing to you, you probably should have stopped reading a while ago.) This is what they should have been originally: cname is like a symbolic link and means that if ncatlab.org moves then these will follow it.

For the record, once DNS hosts are updated, then they will alias as follows:

`nforum.mathforge.org`

to`nforum.ncatlab.org`

`nlab.mathforge.org`

to`ncatlab.org`

If they should be otherwise, let me know.

I’m pleased to see that the nLab/nForum are still going strong!

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