nLab
monodromy

Idea

Monodromy is the name for the action of the the homotopy groups of a space $X$ on fibers of covering spaces or locally constant ∞-stacks on $X$.

Details

Let $H$ be an (∞,1)-topos and $X\in H$ an object. At least in nice situations the locally constant ∞-stacks on $X$, represented by morphisms $X\to \mathrm{LConst}\mathrm{Core}(\mathrm{\infty }\mathrm{Grpd})$ are equivalently encoded by the adjunct morphism $\Pi (X)\to \mathrm{\infty }\mathrm{Grpd}$ out of the bare path ∞-groupoid. This morphism exhibits the monodromy of the locally constant ∞-stack.

Specifically, the restriction $B{\Omega }_x\Pi (X)\hookrightarrow \Pi (X)\to \mathrm{\infty }\mathrm{Grpd}$ to the delooping $B{\Omega }_x\Pi (X)$ of the loop space object ${\Omega }_x\Pi (X)$ at a chosen baspoint $x:*\to X$ is the monodromy action of loops based at $x\in X$ on the fiber of the locally constant $\mathrm{\infty }$-stack over $x$.

• Monodromy trasnformation, at Springer eom