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Apr 12, 2021 at 3:36 comment added Tyler Lawson I think the issue is with homotopies. Eg the standard simplicial model for BG has the properties that you describe. But a homotopy is allowed to be nontrivial on a 1-simplex $p \times \Delta^1$ so long as p is not the basepoint.
Apr 12, 2021 at 0:30 comment added Tim Campion Er -- the same proof would seem to show that $f$ is a map of simplicial abelian groups, assuming that $A$ is a simplicial abelian group, which is false. Probably there is something up with the claim that that diagram commutes...
Apr 12, 2021 at 0:21 history edited Tim Campion CC BY-SA 4.0
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Apr 12, 2021 at 0:12 history answered Tim Campion CC BY-SA 4.0