Timeline for On the paradox that $n$-coskeletal simplicial sets model all homotopy types
Current License: CC BY-SA 4.0
11 events
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| Sep 18, 2018 at 11:43 | answer | added | S. Douteau | timeline score: 2 | |
| Jun 18, 2018 at 12:55 | comment | added | Andrea Gagna | The nerve of a poset is a Kan complex iff it is trivial. If you take a fibrant replacement of it (i.e. you consider it as an object of the $\infty$-cosmos of $\infty$-groupoids), so that you can correctly define its homotopy groups, this will not be 1-connected in general. | |
| Jun 12, 2018 at 22:43 | comment | added | Tim Campion | @DenisNardin You're right, that doesn't make any sense. So the whole "salvaged paradox" thing falls apart because there isn't really a colimit to be approaching -- $Sd^\infty$ doesn't exist as a colimit (though it does as a limit, it's hard to see how to map out of this). Nevertheless, I'm having trouble dispelling the sense that there should be an argument to be had somewhere in this neighborhood. Is there something to say about hashing out homotopy groups of coskeletal simplicial sets in terms of lifting properties against skeletal things... | |
| Jun 12, 2018 at 22:12 | history | rollback | Tim Campion |
Rollback to Revision 2
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| Jun 12, 2018 at 22:07 | history | edited | Tim Campion | CC BY-SA 4.0 |
deleted 48 characters in body
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| Jun 12, 2018 at 21:59 | comment | added | Tim Campion | @SamGunningham Exactly :) | |
| Jun 12, 2018 at 21:25 | comment | added | Denis Nardin | I don't understand the map in (1.). What is the map $K\to Sd K$ that you're using? (the map I know goes the other way...) | |
| Jun 12, 2018 at 19:23 | comment | added | Sam Gunningham | Wait, scratch that - the statement about truncation is only true for Kan complexes... | |
| Jun 12, 2018 at 19:22 | comment | added | Sam Gunningham | Is your claim really false? The $n$-coskeleton is a model for the $(n-1)$-truncation of the homotopy type (see here: mathoverflow.net/questions/243164/…). So an $n$-connected $n$-coskeletal simplicial set will have all homotopy groups zero, right? | |
| Jun 12, 2018 at 19:14 | history | edited | Tim Campion | CC BY-SA 4.0 |
edited title
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| Jun 12, 2018 at 19:05 | history | asked | Tim Campion | CC BY-SA 4.0 |