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David Michael Roberts (Jun 25 2021 at 00:28):I learned that everyone's favourite non-semilocally simply-connected space has a nice universal property:
https://twitter.com/HigherGeometer/status/1408025411959754754
TIL the fancy "earring space" is the one-point compactification of the countably-infinite disjoint union of copies of |R. No planar embedding required!
- theHigherGeometer (@HigherGeometer)And then a big discussion ensued. @sarahzrf pointed out two more universal properties, starting here: https://twitter.com/sarah_zrf/status/1408202936254185473
@HigherGeometer it's also the suspension of the generic convergent sequence
- free 2-sarahzrf on one generator (@sarah_zrf)I wonder how many other nice things we can say about this space?
David Michael Roberts (Jun 25 2021 at 00:29):@Andrej Bauer made a nice animation, in response
https://twitter.com/andrejbauer/status/1408119917262557184
Hey guys, I made a little animation for you that gets to the point nicely, doesn't it? https://twitter.com/HigherGeometer/status/1408025411959754754 https://twitter.com/andrejbauer/status/1408119917262557184/photo/1
- Andrej Bauer (@andrejbauer) John Baez (Jun 25 2021 at 03:35):Nice!
sarahzrf (Jul 11 2021 at 05:26):you can say quite a lot, and i've said it before!
sarahzrf (Jul 11 2021 at 05:26):im not sure whether ive said it here
sarahzrf (Jul 11 2021 at 05:27):well, its fundamental group is the terminal coalgebra of , and i strongly suspect—but have not actually proven—that π₁ in particular sends the terminal coalgebra structure on the earring space to the terminal coalgebra structure on its fundamental group
sarahzrf (Jul 11 2021 at 05:28):note that (im pretty sure, anyway) is a "morphism of categories-equipped-with-endofunctors"
sarahzrf (Jul 11 2021 at 05:29):or, well, it can be equipped as one, to be precise
sarahzrf (Jul 11 2021 at 05:30):so it induces a functor between the categories of algebras for those endofunctors—the thing i'd like to prove directly, which would furnish a novel-as-far-as-i-know computation of the earring space's fundamental group, is that this induced functor preserves the terminal object
sarahzrf (Jul 11 2021 at 05:31):here's another fun fact about the earring's fundamental group: it can be computed as a limit of the standard 1 ← F1 ← FF1 ← ... sequence, but it does not converge at ω steps—in fact, at ω steps you get the shape group!
sarahzrf (Jul 11 2021 at 05:31):rather, it converges at 2ω steps
sarahzrf (Jul 11 2021 at 05:32):this follows a pattern described in "On final coalgebras of continuous functors" by adámek, although i dont think the theorems proven there apply directly to the category of groups and the endofunctor - + Z
sarahzrf (Jul 11 2021 at 05:32):
David Michael Roberts (Jul 11 2021 at 21:56):It's that meant to be free product with Z? I guess it's obvious, but the non-standard notation threw me for a moment.
sarahzrf (Jul 12 2021 at 06:17):David Michael Roberts said:
It's that meant to be free product with Z? I guess it's obvious, but the non-standard notation threw me for a moment.
yeah, sorry!
Matteo Capucci (he/him) (Jul 13 2021 at 11:13):sarahzrf said:
here's another fun fact about the earring's fundamental group: it can be computed as a limit of the standard 1 ← F1 ← FF1 ← ... sequence, but it does not converge at ω steps—in fact, at ω steps you get the shape group!
This is so cool! What's the shape group?
Timothy Porter (Jul 13 2021 at 11:33):For the shape fundamental group look up `Shape Theory' in the nLab . The theory could also be called Cech homotopy theory. It was developed by Borsuk in the late 1960s, pushed further by SIbe Mardesic and Jack Segal, early 1970s (plus a modest addition by myself in my PhD thesis, 1972), then exploded for a few years before somewhat running out of steam. There is a categorical form initiated by Hilton and Deleanu. There are very good references in the nLab article.
It came back as terminology in Lurie's work, but usually some sort of niceness condition is applied. (I can go on for hours on the way the theory goes and some of the intuitions but will leave it at that (for the moment)!!!!!)
Ali Caglayan (Jul 14 2021 at 12:10):My favorite description of the earring space is as follows: The inclusion maps of a wedge of n circles into a wedge of n+1 circles has a retraction, take all those maps and take their (ho)limit and you get the earring space. The colimit of the original inclusion maps is the countable wedge of circles by comparison.
sarahzrf (Jul 15 2021 at 01:44):yeah, this is related to some dual statements of the stuff ive been mentioning—
sarahzrf (Jul 15 2021 at 01:44):the countable wedge of circles is [naturally equipped as] the initial algebra of the same endofunctor which the earring is [naturally equipped as] the terminal coalgebra of
sarahzrf (Jul 15 2021 at 01:44):and its fundamental group is [naturally equipped as] the initial algebra of the same endofunctor which the earring's fundamental group is [naturally equipped as] the terminal coalgebra of
sarahzrf (Jul 15 2021 at 01:46):i still need to read @Tom Leinster's paper about terminal coalgebras and self-similar spaces—i wonder if it has anything relevant to what ive been talking about w/ the earring group?
Timothy Porter (Jul 15 2021 at 05:49):There is a close relationship between (strong) shape theory and proper homotopy theory,
see the nLab. In that latter theory there are what are sometimes called the Brown-Grossman groups of a non-compact manifold (at an end). (The nLab entry gives more details, see Ed. Brown's P-functor there.) These have a shift operator, which seems to be distantly related to some of your ideas, but it puts things in a context that is more general and, whilst more geometric, also more categorical. You may find the ideas useful (or at least `fun').