From NLS type matrix refactorisation problems to set-theoretical solutions of the 2- and 3-simplex equations

We present a method for constructing hierarchies of solutions to $n$-simplex equations by variating the spectral parameter in their Lax representation. We use this method to derive new solutions to the set-theoretical 2- and 3-simplex equations which are related to the Adler map and Nonlinear Schrödinger (NLS) type equations. Moreover, we prove that some of the derived Yang–Baxter maps are completely integrable.