Filtered L∞-morphism

Let $W=V[1]$ and $Z=U[1]$ be the shifted spaces of two complete filtered L∞-algebras over the same characteristic-zero field, in the convention of Complete filtered L∞-algebra. Write $Q,R$ for their degree-one square-zero coderivations and $q_n,r_n$ for their corestrictions. A filtered L∞-morphism from $W$ to $Z$ consists of degree-zero graded symmetric maps $f_n:S^{n}W\to Z$, $n\ge 1$, with

$f_n(F^{i_1}W,\dots ,F^{i_n}W)\subseteq F^{i_1+\cdots +i_n}Z(i_j\ge 1),$

whose associated coaugmented coalgebra map $F:k\oplus S^c(W)\to k\oplus S^c(Z)$ satisfies $FQ=RF$, after extending $Q,R$ by $Q(1)=R(1)=0$. There is no constant Taylor coefficient: $F(1)=1$. Explicitly, for homogeneous inputs,

$F(x_1\cdots x_n)={\sum }_{s=1}^n\frac{1}{s!}{\sum }_{(B_1,\dots ,B_s)}ϵ(B_1,\dots ,B_s;x)f_{\mathrm{\mid }{B}_1\mathrm{\mid }}(x_{B_1})\cdots f_{\mathrm{\mid }{B}_s\mathrm{\mid }}(x_{B_s})\mathrm{.}$

The inner sum runs over ordered partitions of $\{1,\dots ,n\}$ into nonempty blocks. Inside each block retain the original input order; $ϵ$ is the Koszul sign of the permutation which lists those blocks in the indicated order. The factor $1\mathrm{/}s!$ removes ordering of the graded symmetric output blocks. The length-one projection of $F$ on $S^{n}W$ is $f_n$.

The composite of $f:W\to Z$ and $g:Z\to T$ has associated map $GF$; its Taylor coefficients are the length-one projections of $GF$. The identity has $f_1=\mathrm{id}$ and $f_n=0$ for $n>1$. These definitions do not require $F^{1}W=W$. The complete positive-filtration subalgebra $P=F^{1}W$ is the space on which completed exponentials and Maurer–Cartan sums will be used; it need not be an ideal in $W$.