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Is it possible to complete commutative dg associative (conilpotent) coalgebras over $\mathbb{Q}$ in a way so that when we complete the symmetric coalgebra $Sym(V)$ it becomes completed with respect to the word length?

A similar question is if it is possible to complete coalgebras over a Koszul operad $\mathcal P$ so that the bar construction of an algebra over $\mathcal P^!$ becomes completed with respect to the word length.

Thanks!

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    $\begingroup$ Can you say a bit more about what you mean by "completed with respect to the word length"? Depending on what you mean, I might have an answer to your question. $\endgroup$
    – David White
    Commented Apr 4, 2018 at 22:34
  • $\begingroup$ @DavidWhite: If $\{x_i\}$ is a collection where $x_i\in Sym^i(V)$, then I want that the infinite sum $\sum_ix_i$ to be well-defined. In the case for the bar construction, I want something similar, but now in the category of coalgebras over $\mathcal P^!$. $\endgroup$
    – Bashar Saleh
    Commented Apr 5, 2018 at 14:43
  • $\begingroup$ It looks like if you want to take products instead of direct sums, am I right? What obstruction do you find? $\endgroup$
    – Fernando Muro
    Commented Apr 5, 2018 at 15:19
  • $\begingroup$ @FernandoMuro: Yes, is there any construction on coalgebras so that when I apply it on say $Sym(V)$ it becomes $\prod Sym^i(V)$? $\endgroup$
    – Bashar Saleh
    Commented Apr 5, 2018 at 15:45

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