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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeFeb 24th 2011
   • (edited Feb 24th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI think I can see the following, but am wondering if this is discussed in the literature: for $A$ a cochain dg-algebra in non-negative degree let $\Xi A \in Alg^\Delta$ be the cosimplicial algebra obtained as its image under dual Dold-Kan. Then regarded as an object in $[\Delta^{op}, Alg^{op}] \hookrightarrow [\Delta^{op}, (Alg^\Delta)^{op}]$ it should always be Reedy cofibrant, for the projective model structure on $Alg^\Delta$ where the fibrations are degreewise surjections: because the morphism out of the latching object in degree $n$ is dually a projection out of the algebra $(\Xi A)_n$ discarding some top degree generators, hence is an epi, hence a fibration in $Alg^\Delta_{proj}$ hence a cofibration $(Alg^\Delta_{proj})^{op}$. This statement should be of general relevance and is probably discussed somewhere. Does anyone have anything?

   I think I can see the following, but am wondering if this is discussed in the literature:

   for $A$ a cochain dg-algebra in non-negative degree let $\Xi A \in Alg^\Delta$ be the cosimplicial algebra obtained as its image under dual Dold-Kan. Then regarded as an object in $[\Delta^{op}, Alg^{op}] \hookrightarrow [\Delta^{op}, (Alg^\Delta)^{op}]$ it should always be Reedy cofibrant, for the projective model structure on $Alg^\Delta$ where the fibrations are degreewise surjections:

   because the morphism out of the latching object in degree $n$ is dually a projection out of the algebra $(\Xi A)_n$ discarding some top degree generators, hence is an epi, hence a fibration in $Alg^\Delta_{proj}$ hence a cofibration $(Alg^\Delta_{proj})^{op}$.

   This statement should be of general relevance and is probably discussed somewhere. Does anyone have anything?