Given a cosimplicial commutative algebra $A^\bullet$ over a field of characteristic zero, there are two ways of producing an $A\_\infty$-structure on its realization $|A^\bullet| := \int^\Delta C^\*(\Delta^\bullet)\otimes A^\bullet$, where $C^\*(-)$ is the simplicial cochain complex, i.e. $|A^\bullet|$ is the cochain complex with $|A^\bullet|^n = A^n$ and differential the alternating sum of coface maps:


* The Alexander-Whitney map endows the realization functor with a lax monoidal structure, so that it sends the monoid $A^\bullet$ in cosimplicial groups to a monoid in cochain complexes, i.e. a differential graded algebra. Explicitly, the product of $a\_p\in A^p$ and $a\_q\in A^q$ is (up to sign) $f\_{p,q}(a\_p)b\_{p,q}(a\_q)$, where $f\_{p,q}: A^p\to A^{p+q}$ and $b\_{p,q}:A^q\to A^{p+q}$ are the "front and back" cofaces. Note that this does not use the commutativity of $A^\bullet$, and the product is not commutative in general. Call this (dga, and thus in particular) $A\_\infty$-algebra $|A^\bullet|\_{CG}$.
* The coend $\Omega^\*(A^\bullet):=\int^\Delta \Omega\_P^\*(\Delta^\bullet)\otimes A^\bullet$, where $\Omega^\*\_P(\Delta^n) = k[t\_0,\dots,t\_n,\mathrm dt\_0,\dots,\mathrm dt\_n]/(t\_0+\dots+t\_n-1,\mathrm dt\_0 + \dots + \mathrm dt\_n)$ carries a natural (commutative) dga structure. As explained by [Cheng and Getzler](https://arxiv.org/abs/math/0610912), there is a simplicial retraction $\Omega\_P^\*(\Delta^\bullet) \rightleftarrows C^\*(\Delta^\bullet)$, giving rise to a retraction $\Omega^\*(A^\bullet)\rightleftarrows|A^\bullet|$ along which this structure can be transfered to an $A\_\infty$-structure on $|A\_\bullet|$ (actually, even a $C\_\infty$-structure). The operations are given by sums over trees. Call this $A\_\infty$-algebra $|A^\bullet|\_{CG}$.


Obviously these two structures are quite different: For instance, the $2$-ary operation of the first is associative, but not commutative, whereas that of the second is commutative, but not associate.


There is a special case where I know that these $A\_\infty$ algebras are equivalent: If $M$ is a smooth manifold and $A^\bullet = \operatorname{Sing}^\bullet(M)$ is the cosimplicial commutative algebra of functions on smooth simplices, there is a zigzag
$$
|A^\bullet|\_{CG}\rightarrow \Omega^\*(A^\bullet)\rightarrow \int^\Delta \Omega^\*(\Delta^\bullet)\otimes A^\bullet \leftarrow \Omega^\*(M)\rightarrow |A^\bullet|\_{AW}
$$
where the maps are, in order:


* the canonical $A\_\infty$-morphism produced by the homotopy transfer
* the (simplex-wise) inclusion of polynomial forms into smooth forms
* the map $\omega\mapsto (\sigma\mapsto \sigma^\*\omega)$
* an explicit $A\_\infty$-isomorphism obtained by Chen's iterated integrals, see for instance [here](https://mathoverflow.net/questions/10311/infinity-de-rham-quasi-isomorphism).



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> Is there a natural $A\_\infty$-isomorphism between $|A^\bullet|\_{AW}$ and $|A^\bullet|\_{CG}$? Can it be made reasonably explicit? If there is, does it extend the above construction for $A^\bullet = \operatorname{Sing}^\bullet(M)$, or is there at least a natural (and explicit?) $A\_\infty$-homotopy between them?
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Are these two natural $A_\infty$-structures on the realization of a cosimplicial commutative algebra isomorphic?
$\DeclareMathOperator{\Sing}{Sing}$It turns out that the answer is yes, essentially for formal reasons; in case someone finds this question, let me sketch the argument.




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Taking $X = \Delta^n = \{(t\_0,\dots,t\_n)\in \mathbb R^{n+1}\mid \sum\_i t\_i = 1\}$ the algebraic $n$-simplex, there are algebra maps $\Omega\_{P}^\*(\Delta^n)\to \Omega^\*(X)$ and $\Sing^\*(X)\to C^\*(\Delta^n)$, and we obtain a pair of composable $A\_\infty$-morphisms $(C^\*(\Delta^n))\_{CG}\rightsquigarrow \Omega^\*(\Delta^n)\rightsquigarrow (C^\*(\Delta^n))\_{AW}$.
The iterated integral $A\_\infty$-morphism $\Omega^\*(X)\rightsquigarrow \Sing^\*(X)$ is (strictly) functorial with respect to smooth maps $X\to Y$; this means that the above construction defines a simplicial object in the category of triples of $A\_\infty$-algebras together with a pair of composable $A\_\infty$-morphisms, where morphisms are strict morphisms of $A\_\infty$-algebras. Call this category $\operatorname{Alg}\_\infty^{[2]}$. Note that the underlying morphism of chain complexes from $(C^\*(\Delta^\bullet))\_{CG}$ to $(C^\*(\Delta^\bullet))\_{AW}$ is the identity.


Given a cosimplicial commutative algebra $A^\bullet$, we can tensor this diagram with $A^\bullet$ to get a functor $\Delta\times\Delta^{op}\to \operatorname{Alg}\_\infty^{[2]}$, whose end defines an object in $\operatorname{Alg}\_\infty^{[2]}$. Since $\operatorname{Alg}\_\infty^{[2]}$ is the category of algebras over a colored operad, limits (and hence also ends) are calculated on underlying objects; this object is therefore given by $|A^\bullet|\_{CG}\rightsquigarrow \int^{\Delta}A^\bullet\otimes \Omega\_P^\*(\Delta^\bullet)\rightsquigarrow |A^\bullet|\_{AW}$, and the underlying chain map of the composition is the identity.


For $A^\bullet = \Sing^\bullet(X)$, the resulting morphism is the iterated integral map; this again follows from its strict functoriality with respect to smooth maps.


