Timeline for Generalized categories for "higher homotopy groupoids"
Current License: CC BY-SA 4.0
20 events
| when toggle format | what | by | license | comment | |
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| May 12, 2023 at 15:18 | history | edited | YCor | CC BY-SA 4.0 |
removed capitals from title
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
Commonmark migration
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| Apr 29, 2013 at 0:30 | vote | accept | Xander Flood | ||
| Apr 29, 2013 at 0:29 | vote | accept | Xander Flood | ||
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| Apr 29, 2013 at 0:29 | vote | accept | Xander Flood | ||
| Apr 29, 2013 at 0:29 | |||||
| Apr 28, 2013 at 14:44 | comment | added | Fernando Muro | @Xander: as I said, you can modify the construction in order to turn it into an honest category, with a tensor product, in dimensions above the fundamental groupoid, braided in the metastable dimension, and symmetric from there on. It's done in several references, e.g. in the work of the Granada group. | |
| Apr 28, 2013 at 3:35 | comment | added | Xander Flood | @Fernando: I'm quite aware that it is not a category, and in fact question (2) above was whether this thing (which is certainly not a category) is equivalent to a (higher) category in the sense of encoding the same data. | |
| Apr 25, 2013 at 9:02 | comment | added | Fernando Muro | @Xander: your description fails to give a category anyway, if you want a (higher) category, you can't define objetcs to be homotopy classes of maps. | |
| Apr 23, 2013 at 14:15 | answer | added | Ronnie Brown | timeline score: 18 | |
| Apr 23, 2013 at 6:30 | comment | added | Xander Flood | @S. Carnahan: Here's my reasoning behind the $T^k$ comment, although I don't doubt that it is flawed: if we take inductively that $k-1$-morphisms are maps from $T^{k-1}$ (as is certainly the case when $k=1$) then a homotopy of such a map to itself is a map $h:T^{k-1}\times I$, where $h(-,0)=h(-,1)$, which of course factors two a map from $T^k$. Am I misunderstanding the context? | |
| Apr 23, 2013 at 6:25 | comment | added | Xander Flood | @David and Fernando: I actually would've preferred that it not be skeletal for the reason you described, but the description I gave of composition would fail otherwise. Basically, since n-morphisms at $f$ and at $g$ compose to give something at $f\star g$, and since that composition of $(n-1)$-endomorpisms is only associative up to homotopy, we need to consider everything up to homotopy in order to get associativity. Also, I'll take a look at globular n-groupoids to see if that clears things up! | |
| Apr 23, 2013 at 6:02 | answer | added | Tim Porter | timeline score: 5 | |
| Apr 23, 2013 at 5:39 | comment | added | David Roberts♦ | It looks as though Xander is aiming for a totally skeletal higher groupoid, which is not a natural thing to consider. | |
| Apr 23, 2013 at 5:20 | comment | added | Fernando Muro | The fundamental groupoid $\Pi_1$ encodes $\pi_0$ and $\pi_1$. There are severa equivalent definitions in the literature of higher homotopy groupoids $\Pi_{n-1}$ encoding $\pi_n$ and $\pi_{n}$. I guess you mean object set when you write hom-set, but mind that you cannot reduce it to a skeletong if you want to make it strictly functorial. You can do it of couse up to equivalence, in the same way as you can reduce the set of objects of $\Pi_1(X)$ to be bijective with $\Pi_0(X)$, but this new $\Pi_1(X)$ would only be a pseudo-functor. | |
| Apr 23, 2013 at 5:12 | comment | added | David Roberts♦ | Xander, look up globular n-groupoids, in conjunction with the names I gave you, if you have no idea what a Kan complex is, as they take a more geometric approach. | |
| Apr 23, 2013 at 3:45 | comment | added | S. Carnahan♦ | I don't understand how you get $T^k$ from the notion of a map from a $k-1$-morphism (i.e., a map whose source is a $k-1$-simplex) to itself. | |
| Apr 23, 2013 at 1:42 | comment | added | Xander Flood | The n-lab article on the fundamental infinity-groupoid is pretty dense. Does anyone happen to know of another reference? (I don't know what exactly a Kan complex is, or how they correspond to infinity groupoids.) The obvious way of building of the category would be for $k$-morphisms to be just homotopies of $(k-1)$-morphisms. In that case though, it seems that $k$-endomorphisms are maps from a torus $T^k = (S^1)^k$, so that shouldn't be the right construction. | |
| Apr 23, 2013 at 0:28 | comment | added | David Roberts♦ | "groupesque" -> n-groupoid. Grothendieck wrote about this to Larry Breen in 1975, and again to Quillen in 1983. These ideas have been written up by Maltsiniotis and by Ara. However, there are versions due to Trimble, Batanin, ... | |
| Apr 22, 2013 at 23:11 | comment | added | Ryan Budney | Have you done any searching for these terms on Google? ncatlab.org/nlab/show/fundamental+infinity-groupoid | |
| Apr 22, 2013 at 23:08 | history | asked | Xander Flood | CC BY-SA 3.0 |