# Applying a Hochschild cocycle to a Maurer-Cartan element: how one should think of this?

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 $$k$$.