Timeline for Are the Alexander-Whitney and Eilenberg-Zilber maps homotopy inverse in arbitrary abelian categories?
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Jun 29, 2018 at 10:53 | comment | added | Richard Hepworth | @Charles Thanks very much for sketching this answer! | |
| Jun 28, 2018 at 21:47 | comment | added | Charles Rezk | It would be nice if somebody with time on their hands would produce an explicit formula for $H_k$, I doubt it is very difficult. | |
| Jun 28, 2018 at 21:46 | comment | added | Charles Rezk | (The terms of the normalized complex are natural summands of these objects, split off by idempotents which can be written as linear combinations of simplicial operators, so there is a similar statement for natural transformations $N(A\otimes B)_k\to N(A\otimes B)_{k+1}$.) So there is an explicit formula for abelian groups, but which clearly is meaingful in a general context, and very likely proves what you want ... | |
| Jun 28, 2018 at 21:44 | comment | added | Charles Rezk | When $\mathcal{A}$ is abelian groups, this condition means that $H_k$ must have a formula of the form: $a\otimes b\mapsto \sum_{f,g} m_{f,g}\; (af)\otimes (ag)$, where $f,g\colon [k+1]\to [k]$ are simplicial operators (which I think of as acting on simplices from the right), and $m_{f,g}$ is an integer. This is because you can classify natural transformations $A_k\otimes B_k\to A_{k+1}\otimes B_{k+1}$, and they all have this form .... | |
| Jun 28, 2018 at 21:41 | comment | added | Charles Rezk | This is surely true assuming the correct hypotheses on $\mathcal{A}$, which I suspect are: additive category with retracts equipped with biadditive monoidal structure. The key point to keep in mind is that the proofs that produce a chain homotopy $EZ\circ AW\sim Id$ actually show the existence of a natural chain homotopy, i.e., a collection of transformations $H_k\colon N(A\otimes B)_k\to N(A\otimes B)_{k+1}$ which are natural in $(A,B)$ ... | |
| Jun 28, 2018 at 14:51 | review | First posts | |||
| Jun 28, 2018 at 15:15 | |||||
| Jun 28, 2018 at 14:50 | history | asked | Richard Hepworth | CC BY-SA 4.0 |