Ramification of a G-local system

Definition

Fix a Betti, de Rham, or étale category of local systems, and let $U=X\setminus D$ be a punctured smooth curve. A $G$-local system on $U$ is ramified at a marked point $x\in D$ when it does not extend across $x$ as an unramified object in that category.

In the Betti category over $\mathbb C$, extension across $x$ is equivalent to trivial local monodromy. For an algebraic flat connection, extension means extension with no pole. A regular-singular connection may still have nontrivial monodromy, while an irregular-singular connection requires formal type and Stokes data in addition to ordinary monodromy.

In the étale category, ramification is the nontrivial action of inertia. Tame ramification is trivial on wild inertia; wild ramification is not.