Let $C^{\bullet}(A,M)$ be the Hochschild cochain complex of a DG-algebra $A$ with coefficients in a DG-bimodule $M$. Let $\zeta \in C^0(A,M)$ be a cocycle. Let $a \in A$ be a Maurer-Cartan element, $d(a)+a^2 = 0$. We can write $\zeta = \zeta_0 + \zeta_1 + \ldots + \zeta_n$, where $\zeta_i \in \operatorname{Hom}_k(A^{\otimes i},M)$. In $M$, we can twist the differential by $a$, setting $d'(m) = d(m)\pm am \pm ma$. Then the following holds:

$\zeta(a): = \zeta_0 + \zeta_1(a) +\zeta_2(a \otimes a) + \ldots + \zeta_n(a^{\otimes n})$ is closed in $(M,d')$.

The statement can be checked by an explicit computation; however, the operation of applying $\zeta$ to a MC-element $a$ remains somewhat mysterious to me. For instance, it is using something of the form $\exp(a) = \sum_{i=0}^\infty a^{\otimes i}$...

My question is extremely poorly formulated, but how should one understand this operation? Some analogies, maybe, or reinterpretations, or examples...

The only related thing I know is the following: if $a$ is a MC-element in $A$, then $\exp(a)$ gives the structure of DG-comodule over $\operatorname{Bar}(A)$ on $k$.



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