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Given a (co/contravariant) functor $F$ from the simplicial category $\Delta$ to an abelian category $A$, we can form its Cech complex (or "alternating face map complex" on the nLab), i.e. $CF^n=F([n])$, and $\partial^n$ is $\sum_{i=0}^n(-1)^iF(\delta^n_i)$, where $\delta^n_i$ is the face map.

By the Dold-Kan correspondence, we can then reconstruct a (co)simplicial object $\Gamma CF$ which has this $CF^\bullet$ as its Moore complex. In particular, its Cech complex is a direct sum of $CF^\bullet$ and a nullhomotopic complex, and is homotopy equivalent to $CF^\bullet$.

So my question is: can we say anything more direct about relationship between our $F$ and $\Gamma CF$ than "they have homotopy-equivalent Cech complexes"? More generally, what relation does homotopy equivalence impose on (co)simplicial objects?

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  • $\begingroup$ If you have an open cover you can get a simplicial object. $\endgroup$
    – Zhen Lin
    Aug 17, 2015 at 7:48
  • $\begingroup$ How? I see how you can get the face maps from the projections, but I don't see what the degeneracy maps would be. $\endgroup$ Aug 17, 2015 at 8:37
  • $\begingroup$ The simplicial identities force them to be (generalised) diagonal embeddings. Use the universal property of projections. $\endgroup$
    – Zhen Lin
    Aug 17, 2015 at 10:41
  • $\begingroup$ Oh, I see. I was having an indexing issue that made me think there should be more degeneracy maps than diagonals. $\endgroup$ Aug 17, 2015 at 16:14

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The answer, if anybody is still wondering, is simplicial homotopy.

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