Timeline for Kan Complexes, proof of extension of a map to a product
Current License: CC BY-SA 4.0
8 events
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| Dec 20, 2021 at 22:04 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Nov 20, 2021 at 21:34 | answer | added | Doron Grossman-Naples | timeline score: 1 | |
| Nov 16, 2021 at 21:23 | comment | added | Raphaël Kalfon | Yes that's what I thought, thank you. It's a bit less confusing now, however I cannot figure out a rigorous proof for this statement. I could for example extend the map restricted to elements of the form $(*,x)$ where $*$ are degenerates of $\Delta[n]$ obtained from a vertex and $x$ are points in $\Lambda^k[m]$. But I have no way to know whether or not this extension extends to all of $\Delta[n] \times \Delta[m]$. I would be very grateful for a detailed proof. More particularly, to what precise maps does he apply the extension conditions ? | |
| Nov 16, 2021 at 20:03 | comment | added | Gregory Arone | I looked up Curtis' paper, and he does seem to have it backwards (lemma 1.14 in the version published in Advances in Mathematics). Must be just a typo. $A$ and $B$ should be switched. | |
| Nov 16, 2021 at 19:53 | comment | added | Gregory Arone | Furthermore, the point is that since $\Lambda^k[m]\hookrightarrow \Delta[m]$ is an anodyne extension, so is $\Delta[n]\times \Lambda^k[m]\hookrightarrow \Delta[n]\times \Delta[m]$. In fact, you could replace $\Delta[n]$ with any simplicial set. See, for example, this nlab page ncatlab.org/nlab/show/anodyne+morphism, in particular Proposition 3.1 | |
| Nov 16, 2021 at 19:49 | comment | added | Andreas Blass | As far as I can see, the statement "$\Delta[n] \times \Lambda^k[m]$ can be obtained from $\Delta[n] \times \Delta[m]$ by successively adjoining a simplex and one of its faces" has the two complexes interchanged; it should instead say that "$\Delta[n] \times \Delta^k[m]$ can be obtained from $\Delta[n] \times \Lambda[m]$ by successively adjoining a simplex and one of its faces." | |
| S Nov 16, 2021 at 19:36 | review | First questions | |||
| Nov 16, 2021 at 19:59 | |||||
| S Nov 16, 2021 at 19:36 | history | asked | Raphaël Kalfon | CC BY-SA 4.0 |