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I was wondering if the category of $L_\infty$ algebras is complete and in particular I am looking for an explicit construction of the pullback for

$\require{AMScd}$ \begin{CD} @. B\\ \phantom V @VV b V\\ A @>>a> C \end{CD} in the case that $A, B, C$ are Lie-n-algebras, i.e. concentrated in only finitely many degrees.

I cannot find any reference suited to my basic knowledge of category theory. Any help is appreciated.

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  • $\begingroup$ It depends on the morphisms you take. With "strict" ones, yes, with $\infty$-morphisms I don't know for sure, but in those cases one is often more interested in the homotopy colimit, which does exist. $\endgroup$
    – Fernando Muro
    Commented Nov 10, 2020 at 12:41
  • $\begingroup$ Hi @FernandoMuro and thanks! In my case, I'm not considering strict $L_\infty$-morphisms. By the way, how can I argue the existence of homotopy limits? Does it follow from the fact that the category of $ L_\infity$ algebras is an algebra over an operad? (sorry for the naive questions, but I'm a beginner in these things) $\endgroup$
    – A.Miti
    Commented Nov 11, 2020 at 13:16
  • $\begingroup$ you're right, your guess is correct. $\endgroup$
    – Fernando Muro
    Commented Nov 12, 2020 at 0:35

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