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I see this appeared:
Maybe there’s more at infinity-action to consider.
Maybe the right thread to mention this was over here. There were more suggestions there of what do with a HoTT treatment of $\infty$-groups.
Our list of references at infinity-group had been a bit thin. I have added some: here
Why isn’t
category:∞-groupoid
working at the bottom of the page?
It’s clearly due to the same issue that killed our blockquote environments.
But I suggest to everyone not to pursue this category:-labelling of entries, but instead adopt my convention, that entries get a floating-TOC to put them into their topic cluster. For two reasons:
1) It is much easier for the reader to spot in the first place. I doubt any actual user ever noticed this category-label business.
2) It is much more useful for the reader once spotted, since in the floating TOC we have control over how to organize the information about the related entries, instead of just producing a blind string of keywords
Sorry that the quote issue has not been fixed yet. I am currently working on something more major which I hope will lead to a significant speed up of the nLab, and make it significantly easier to respond to issues/requests. Maybe I’ll have something to show sometime this week or next.
If people think that the quote stuff is very urgent, I can take a look.
I think the category:people label is moderately useful, but other than that I agree that ToCs are probably more useful.
What do people here make of
Note that we have crucially used a trick to study higher groups in HoTT, namely that these can be represented by pointed, connected types?
I see the point that this feature is not available for higher monoids (at least until directed HoTT appears), but it’s surely not merely a piece of luck that there happens to be a convenient way to represent higher groups.
That “trick” has been driving much of what we discussed here over the years. In our arXiv:1207.0248 this is highlighted as theorem 2.19, citing Lurie’s lemma 7.2.2.1 in “Higher Topos Theory” and theorem 5.1.3.6 of “Higher Algebra”.
In the base $\infty$-topos $\infty Grpd$ this is a classical theorem, in its simplicial incarnation this is due to Kan, Milnor; it is also a special case of the May recognition principle.
Right, but what do you think of calling it a “trick”?
If you recall we were hereabouts before when discussing equivalence of physical theories:
Urs: This may seem like a cheap trick, but I actually think this is a useful perspective.
David: That’s the nub of it. What’s the perspective that will continue to hold that this is a cheap trick, whatever you go on to say, because of some principles which, say, would require the expression of that colimit? And conversely, can we understand your perspective to be more than just ’useful’, but getting things ’right’?
Urs: …maybe we learn from it to stick, where they exist, to elementary concepts equivalent to concepts that would need simplicial constructions.
Mike: An interesting point, that perhaps one of the things HoTT (and our current inability to deal with ∞-coherences therein) teaches us is to avoid higher homotopy coherences whenever possible. Of course, now I can hear my advisor saying “we knew that decades ago!”…
If I were writing to a generic audience the word “trick” would serve a purpose of establishing communication without irritating the audience by a perspective which, even if superior, could cause confusion with the uninitiated.
If I were writing to an audience that I expect to appreciate the foundational role of HoTT, I would try to explain that, far from being a trick, this is a phenomenon fundamental to the meaning of the whole field.
In order to unify these two perspectives, it may be worth recalling that, historically, the word “group” is a shorthand for “symmetry group”, witnessing the original idea that a group necessarily is a group of transformations of something. This original idea, which may seem naïve from the point of view of modern mathematics, finds its re-incarnation, at a higher level of insight, in the fact that $\infty$-groups are equivalently the loop space objects of pointed connected homotopy types.
I like #11.
Why did this paper not show up in my arxiv notification emails? I’m subscribed to math.AT and math.LO, and I thought cross-lists were announced to the cross-list categories; isn’t that the whole point of cross-listing?
I suppose I should finally get around to subscribing to cs.LO, since HoTT people persist in posting to there even though to my eyes there’s hardly anything computer-sciencey about what they’re doing.
Yes, #11 is the reason we used the word “trick”, but maybe in the final version we’ll reword it to indicate that it is something that maybe looks like a trick, but is indeed a fundamental fact.
Re #12, that’s indeed mysterious!
Re #4: I’ve now fixed this. Unfortunately, in so doing, any $\infty$ inside a page reference (double square brackets) became broken, which obviously affected a huge number of pages. I actually do not understand the reason that this happened, there is something very weird going on. Anyhow, I’ve now attempted to fix that problem as well, but it is possible I have missed something; please alert me if so (you will see a ? where a ∞ should be).
(Curiously, whilst there are numerous ∞Grpd occurrences, there do not appear to be any occurrences of ∞Cat! What I did was to handle all cases of “∞-“, “-∞”, “∞Grpd”, and “(∞,”. I would have thought there would be a few more, but I’ve not found any so far. It’s not the easiest thing to grep, though.)
Back to #4: just to note that if one clicks on category:∞-groupoid, there is a similar kind of issue on the All pages page that appears, but it is possible, I think, that the cause of that one is unrelated.
Thanks, Richard!
Curiously, whilst there are numerous ∞Grpd occurrences, there do not appear to be any occurrences of ∞Cat!
Incidentally, that’s for a reason. The big divide in the development of higher category theory was the understanding that the impasse of the early years dissolves if one stops insisting on a concept of $\infty$-categories right away and instead considers a filtration by concepts of $(\infty,n)$-categories. Accordingly there is an entry (∞,1)Cat, instead.
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