Decorated Local Systems and Character Varieties

The focus of this paper is the study of the moduli space of representations of fundamental groupoids of surfaces $\Sigma$ with boundaries with values in $G:=GL_n(\mathbb C)$. In absence of marked points on the boundary, this moduli space is realized in many equivalent ways: as the moduli space of linear local systems on $\Sigma$, as the moduli space of representations of the fundamental groupoid $\Pi_1 (\Sigma)$, as the space of monodromy data and as character variety. By adding marked points to the boundary of $\Sigma$ in order to capture irregular singularities, the Betti moduli space has been generalized in several ways by many authors. Although it is clear that these different approaches describe essentially the same spaces of mathematical objects, exactly how they fit together has not yet been established. Motivated by the broader programme of establishing an explicit and conceptually coherent relationship between the existing approaches to the study of the decorated Betti moduli space, in this paper, we develop a categorical framework that allows for a systematic definition of the \dfn{decorated Betti moduli spaces} space, in the presence of higher order poles, designed to specialize to the different points of view encountered in the literature.