Complete filtered L∞-algebra

Let $(V,Q)$ be an L∞-algebra in the shifted convention of Shifted coalgebra definition of an L∞-algebra, with $W=V[1]$ and corestrictions $q_n:S^n(W)\to W$. A complete filtered L∞-algebra is this data together with a descending filtration by graded subspaces

$F^{1}W\supseteq F^{2}W\supseteq F^{3}W\supseteq \cdots$

such that both of the following conditions hold:

1. For every fixed degree $j$, the natural map from $W^j$ to the compatible homogeneous cosets $(w_r+(F^{r}W{)}^j{)}_{r\ge 1},w_r\in W^j,w_{r+1}-w_r\in (F^{r}W{)}^j,$ is a bijection. Thus the filtration is separated and complete in each degree: . All graded inverse limits here and in the filtered Maurer–Cartan constructions are taken degreewise. The underlying graded space remains ${⨁}_j{W}^j$; completeness does not assert convergence of sums of terms in unboundedly varying degrees.
2. For every $n\ge 1$ and positive integers $r_1,\dots ,r_n$, $q_n(F^{r_1}W,\dots ,F^{r_n}W)\subseteq F^{r_1+\cdots +r_n}W\mathrm{.}$

This definition imposes an algebraic filtration topology only; it makes no analytic convergence assertion. The next theorem proves that the Maurer–Cartan series converges for each $w\in F^1{W}^0$.