Timeline for A more natural proof of Dold-Kan?
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7 events
| when toggle format | what | by | license | comment | |
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| Oct 2 at 22:59 | comment | added | Andy | @Gaussler As you say it has to be true if we want an isomorphism. Here is one bijection (idk if it works with the differentials): fix $n$ and consider $C(\mathbb{Z} \Delta[n])$. This is a big complex. In it consider the subcomplex supported on simplices whose first vertex is $0$. $k$-simplices starting at $0$ are in bijection with $[n] \twoheadrightarrow [k]$ (via taking the section $[k] \to [n]$ that takes the first preimage. Then it also easy to see any choice of images of these in the codomain lifts uniquely to a chain map (given a simplex add the vertex 0 and consider the differential...) | |
| Jan 7, 2017 at 14:40 | comment | added | Gaussler | @ChrisBrav Can you perhaps shed some light on one (or both) of your two ways of identifying $\operatorname{Ch}_{\ge0}(C(\mathbb{Z}\Delta[n]),C)$ and $\bigoplus_{[n]\twoheadrightarrow[k]} C_k$? I currently don’t quite see a natural approach to this. | |
| Feb 11, 2015 at 14:43 | answer | added | Charles Rezk | timeline score: 27 | |
| Feb 10, 2015 at 22:54 | comment | added | Fernando Muro | I think that the levelwise direct sum decomposition of simplicial abelian groups is a very nice computation based in the observation that a simplicial object contains lots of split monos and epis. Therefore, I don't think it's pulled out of a hat. Sometimes computations are really necessary. Not everything is 'categorical nonsense' (I hate this unfair expression, but I don't know a better way of putting my ideas into words right now). | |
| Feb 10, 2015 at 21:38 | answer | added | Karol Szumiło | timeline score: 9 | |
| Feb 7, 2015 at 14:47 | history | edited | user42325 | CC BY-SA 3.0 |
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| Feb 7, 2015 at 14:42 | history | asked | user42325 | CC BY-SA 3.0 |