nLab
monodromy

Contents

Idea

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

Details

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

Specifically, the restriction BΩ xΠ(X)Π(X)Grpd\mathbf{B}\Omega_x \Pi(X) \hookrightarrow \Pi(X) \to \infty Grpd to the delooping BΩ xΠ(X)\mathbf{B}\Omega_x \Pi(X) of the loop space object Ω xΠ(X)\Omega_x \Pi(X) at a chosen baspoint x:*Xx : {*} \to X is the monodromy action of loops based at xXx \in X on the fiber of the locally constant \infty-stack over xx.

  • Monodromy trasnformation, at Springer eom

Revised on September 8, 2010 17:59:39 by Zoran Škoda (161.53.130.104)