Why Lie integration is called "integration"?

Question

According to nlab on Lie integration, The Lie integration of $\mathfrak{a}$ is essentially the simplicial object whose k-cells are the d-paths in $\mathfrak{a}$, where $\mathfrak{a}$ is a $\infty$-Lie Algebroid. A $k$-path is an element \begin{equation} \Sigma \in \text{Hom}_{\infty \text{LieAlgd}}(T\Delta^k,\mathfrak{a}) \end{equation} From which one can deduce a morphism of dg-algebras: \begin{equation} \Sigma^\ast \in \text{Hom}_{\text{dgAlg}}(\text{CE}(\mathfrak{a}),\Omega^\bullet(\Delta^k)) \end{equation} where $\text{CE}(\mathfrak{a})$ is the Chevalley-Eilenberg algebra of $\mathfrak{a}$, and $\Omega^\bullet$ is the de Rham complex. Why is it called integration? What it have to do with the usual notion of integration of differential forms?

Answer 1

The functor that implements the equivalence between simply connected Lie groups and Lie algebras sends a simply connected Lie group to its tangent space at the origin equipped with the appropriate Lie bracket, and a morphism of simply connected Lie groups to its derivative at the origin.

Thus, the inverse functor reconstructs morphisms of Lie groups from their derivatives at the origin, so is referred to as the integration functor.

More generally, Lie integration explains how to reconstruct a smooth function X→G, where X is a connected smooth manifold and G is a Lie group, from its value at some point of X and its derivative (a differential 1-form on X valued in the Lie algebra of G). A necessary and sufficient condition for this is that the Maurer–Cartan equation holds. In this case, the original function can be uniquely reconstructed from the given data, and the process is naturally called integration, since it generalizes the usual integration on a line.