C}}}^{\vee})\,.$$ In other words, an OCHA algebra structure on a pair of graded vector spaces $(V,A)$ is a degree $1$ codifferential on the free ${{\mathcal{O C}}}^{\vee}$- coalgebra ${{\mathcal{O C}}}^{\vee}(V,A)$.

Comparing Definitions \[dfn:S-coalg\] and \[dfn:OC-vee\] it is not hard to see that ${{\mathcal{O C}}}^{\vee}$ is a sub-cooperad of ${{\bf S}}$. Hence ${{\mathcal{O C}}}_{\infty}$ is a sub DG operad of ${{\bf Ger}}^+_{\infty}$: $$\label{OC-inf-in-Ger-plus}
{{\mathcal{O C}}}_{\infty} \hookrightarrow {{\bf Ger}}^+_{\infty}\,.$$ Furthermore, Theorem \[thm:Ger-plus-alg\] is a generalization of the following statement from [@OCHAStasheff1]

\[thm:OCHA\] An ${{\mathcal{O C}}}_{\infty}$-algebra structure on the pair of cochain complexes $(V,A)$ is a triple:

-   A ${{\Lambda}}{{\bf Lie}}_{\infty}$-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 $A$.

\[OCHA-Cbu-A\] For every $A_{\infty}$-algebra $A$ the pair $({C^{\bullet}}(A,A), A)$ carries a tautological OCHA structure: the ${{\Lambda}}{{\bf Lie}}_{\infty}$-structure on ${C^{\bullet}}(A,A)$ is simply the ${{\Lambda}}{{\bf Lie}}$-algebra structure given by the Hochschild differential (\[Hoch\]) and the Gerstenhaber bracket (\[Gerst\]); the $A_{\infty}$-structure on $A$ is the given one; finally, the $L_{\infty}$-morphism from ${C^{\bullet}}(A,A)$ to ${C^{\bullet}}(A,A)$ is the identity map.

Homotopies of ${{\bf Ger}}^+_{\infty}$-algebra structures
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Let $(V,A)$ be a pair of cochain complexes. Let ${\hat{Q}}$ and ${\hat{Q}}'$ be ${{\bf Ger}}^+_{\infty}$-algebra structures on $(V,A)$.

\[dfn:homotopies\] 