# Are these two natural $A_\infty$-structures on the realization of a cosimplicial commutative algebra isomorphic?

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, 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.

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?

$$\DeclareMathOperator{\Sing}{Sing}$$It turns out that the answer is yes, essentially for formal reasons; in case someone finds this question, let me sketch the argument.
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 $$\operatorname{Alg}_\infty^{}$$. 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 $$\Delta\times\Delta^{op}\to \operatorname{Alg}_\infty^{}$$, whose end defines an object in $$\operatorname{Alg}_\infty^{}$$. Since $$\operatorname{Alg}_\infty^{}$$ 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.