complexes.

This observation motivates the definition of ${{\bf Ger}}_{\infty}^+$ quasi-isomorphism

\[dfn:Ger-plus-q-iso\] A [*${{\bf Ger}}_{\infty}^+$ quasi-morphism*]{} from $(V,A)$ to $(V',A')$ is a morphism of ${{\bf S}}$-coalgebras $${\hat{T}}: {{\bf S}}(V,A) \to {{\bf S}}(V',A')$$ for which the restrictions (\[T-1-V\]), (\[T-1-A\]) are quasi-isomorphisms of cochain complexes.

Algebras over the operad ${{\bf Ger}}^{+}_{\infty}$ {#sect:Ger-plus-alg}
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The following theorem gives us an alternative description[^3] of algebras over the operad ${{\bf Ger}}^{+}_{\infty}$.

\[thm:Ger-plus-alg\] A ${{\bf Ger}}^{+}_{\infty}$-algebra structure on the pair of cochain complexes $(V,A)$ is a triple:

-   A ${{\bf Ger}}_{\infty}$-structure on $V$,

-   An $A_{\infty}$-structure on $A$, and

-   An $L_{\infty}$-morphism from $V$ to the Hochschild cochain complex ${C^{\bullet}}(A,A)$ of the $A_{\infty}$-algebra $A$.

A ${{\bf Ger}}^{+}_{\infty}$-algebra structure on $(V,A)$ is a degree $1$ codifferential ${\hat{Q}}$ on the cofree coalgebra ${{\bf S}}(V,A)$. This differential splits into the ${{\mathfrak{c}}}$-component ${\hat{Q}}^{{{\mathfrak{c}}}}$ $$\label{hQ-mc}
{\hat{Q}}^{{{\mathfrak{c}}}} : {{\bf Ger}}^{\vee}(V) \to {{\bf Ger}}^{\vee}(V)$$ and the ${{\mathfrak{o}}}$-component $$\label{hQ-ma}
{\hat{Q}}^{{{\mathfrak{o}}}} : {{\bf S}}^{{{\mathfrak{o}}}}(V,A) \to {{\bf S}}^{{{\mathfrak{o}}}}(V,A)\,.$$

Since ${{\bf S}}(V,A)$ is cofree, the codifferential ${\hat{Q}}$ is uniquely determined by its composition $$\label{Q}
Q = p \circ {\hat{Q}}: {{\bf S}}(V,A) \to V \oplus A$$ with the projection $$p : {{\bf S}}(V,A) \to V \oplus A\,.$$ The composition $Q$ also splits into the ${{\mathfrak{c}}}$-component $$\label{Q-mc}
Q^{{{\mathfrak{c}}}} : {{\bf 