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What maps of simplicial sets exist between

  • the image under the Dold-Kan correspondence of a chain complex shifted up in degree

  • and the image under the right adjoint to simplicial looping of the DK-image of the unshifted complex

?

Here is the same question in detail:

Write

$$ (G \dashv \bar W) : sGrp \stackrel{\leftarrow}{\underset{\bar W}{\to}} sSet_0 \hookrightarrow sSet $$

for the adjunction between simplicial groups and reduced simplicial sets whose left adjoint is the simplicial loop group functor (as for instance in Goerss-Jardine, chapter V);

and write

$$ Ch_\bullet^+ \overset{\Xi}{\to} sAbGrp \hookrightarrow sGrp \overset{U}{\to} sSet $$

for the Dold-Kan correspondence, where in both cases I care about the images as simplicial sets.

Then for $V \in Ch_\bullet^+$ a chain complex and $V[1]$ (or $V[-1]$ if you prefer) its shift up in degree (its delooping as a chain complex) the two simplicial sets

$$ U \Xi (V[1]) $$

and

$$ \bar W (\Xi V) $$

should have the same homotopy type. What nice natural maps of simplicial sets do we have between them?

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There's an explicit natural isomorphism between the two functors.

Rick Jardine says as much, but for the image of the functors in the category of chain complexes (i.e. after applying the normalization). You can find this in Goerss, Jardine Remark III.5.6, or in greater depth in section 4.6 of Jardine's book on Generalized Etale Cohomology.

The combinatorics for the isomorphism in simplicial abelian groups means that the isomorphism takes a little longer to state, but I could send you a pdf with everything written out if this would be useful.

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