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Let $... \to G_2 \to G_1$ an inverse system of abelian groups with inverse limit $G$, let $n \geq 2$ and $F$ a field. The induced inverse system $$... \to C_*(K(G_2,n);F) \to C_*(K(G_1,n);F) \ (*)$$ on chains with F-coeffients seen as inverse system of conilpotent coassociative dg-coalgebras over $F$ has inverse limit $C_*(K(G,n);F).$

Is there a left induced model structure on the category of conilpotent coassociative dg-coalgebras over $F$ such that $C_*(K(G,n);F)$ is the homotopy inverse limit of the diagram $(*)$?

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  • $\begingroup$ The first statement sounds odd without an explicit choice of a model for the $K(G,n)$'s - can you say a bit more ? $\endgroup$
    – Maxime Ramzi
    Commented May 20, 2022 at 7:13

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