Decorated Betti moduli space

The focus of this talk le is the study of moduli spaces of representations of fundamental groups 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 $\mathfrak{Loc}_G (\Sigma)$ of linear local systems on $\Sigma$, as the moduli space $\mathfrak{Rep}_G \big( \Pi_1 (\Sigma) \big)$ of representations of the fundamental groupoid $\Pi_1 (\Sigma)$, as the moduli space $\mathfrak{Rep}_G \big( \pi_1 (\Sigma, p) \big)$ of monodromy data and as character variety $\mathfrak{X}_G (\Sigma)$. We call all these moduli spaces. Their equivalence can be stated as follows:

\begin{equation} \text{$\mathfrak{Loc}_G (\Sigma)$} \simeq\text{$\mathfrak{Rep}_G \big( \Pi_1 (\Sigma) \big)$} \simeq\text{$\mathfrak{Rep}_G \big( \pi_1 (\Sigma, p) \big)$} \simeq \text{$\mathfrak{X}_G \Sigma)$} \end{equation}

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 as summarized below. 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. In this talk, I will develop a categorical framework that allows for a clear and systematic definition of the \textbf{decorated Betti moduli spaces} designed to specialize to the different points of view encountered in the literature.