Timeline for Kan condition for bar construction
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Apr 12, 2018 at 16:10 | answer | added | geodude | timeline score: 2 | |
| Apr 5, 2018 at 19:19 | comment | added | Arnaud D. | This paper gives a definition of the Kan condition for simplicial objects in any regular category, and proves that every simplicial object is Kan if and only if the category is Mal'cev, i.e. if and only every reflexive relation is an equivalence relation. This generalises Fernando Muro's observation about the functor factorising through the category of groups. | |
| Apr 5, 2018 at 16:26 | comment | added | Chris Schommer-Pries | If C=Sets and T is the monoid for pointed sets, and $A=(A,a_0)$, then the Bar construction, as a simplicial set, is the union of $A \times \Delta[0]$ (i.e. const. simp. set on A) with $\Delta[1]$ along $\Delta[0]$ identified as $a_0 \in A$. It is not Kan, but it is a quasicategory. | |
| Apr 5, 2018 at 16:22 | history | edited | geodude | CC BY-SA 3.0 |
added 70 characters in body
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| Apr 5, 2018 at 15:35 | comment | added | Najib Idrissi | @geodude A simplicial group is always a Kan complex. | |
| Apr 5, 2018 at 15:33 | comment | added | geodude | @FernandoMuro Why is it yes if there is an "underlying group"? That sounds interesting. | |
| Apr 5, 2018 at 15:30 | comment | added | geodude | @FernandoMuro I see. You are right, the nLab page talks briefly about a possible internalization, but it is unclear how it would be in the general case. So let's assume we are in a concrete category. | |
| Apr 5, 2018 at 15:27 | comment | added | Fernando Muro | I don't know, maybe the answer depends on the properties of the 'underlying set' functor. It's definitely yes if this functor factors through the category of groups. Actually, I cannot make sense of your question unless you take underlying sets, since I don't know what the Kan condition is for simplicial sets in an arbitrary category. | |
| Apr 5, 2018 at 15:22 | comment | added | geodude | @FernandoMuro I could. What would that imply? | |
| Apr 5, 2018 at 15:22 | comment | added | Fernando Muro | Are you assuming that the objects of your category have underlying sets? | |
| Apr 5, 2018 at 9:11 | vote | accept | geodude | ||
| Apr 5, 2018 at 14:55 | |||||
| Apr 5, 2018 at 8:05 | history | asked | geodude | CC BY-SA 3.0 |