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?


1 Answer 1


$\DeclareMathOperator{\Sing}{Sing}$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 $\operatorname{Alg}_\infty^{[2]}$. 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.


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