Hopf algebroids and rigid tensor categories associated with dynamical Yang–Baxter maps

We construct Hopf algebroids by means of dynamical Yang–Baxter maps, set-theoretic solutions to a version of the quantum dynamical Yang–Baxter equation. This Hopf algebroid with suitable properties implies two rigid tensor categories: one is a category consisting of finite-dimensional dynamical representations; the other is a category consisting of L-operators on finite-dimensional spaces. They are isomorphic to each other.