Naturality properties and comparison results for topological and infinitesimal embedded jump loci

We use augmented commutative differential graded algebra (ACDGA) models to study $G$-representation varieties of fundamental groups $\pi=\pi_1(M)$ and their embedded cohomology jump loci, around the trivial representation 1. When the space $M$ admits a finite family of maps, uniformly modeled by ACDGA morphisms, and certain finiteness and connectivity assumptions are satisfied, the germs at 1 of ${\rm Hom} (\pi,G)$ and of the embedded jump loci can be described in terms of their infinitesimal counterparts, naturally with respect to the given families. This approach leads to fairly explicit answers when $M$ is either a compact K\"ahler manifold, the complement of a central complex hyperplane arrangement, or the total space of a principal bundle with formal base space, provided the Lie algebra of the linear algebraic group $G$ is a non-abelian subalgebra of $\mathfrak{sl}_2(\mathbb{C})$.