  The 4-simplex equation is a higher-dimensional analogue of Zamolodchikov's
tetrahedron equation and the Yang--Baxter equation which are two of the most
fundamental equations of mathematical physics. In this paper, we introduce a
method for constructing 4-simplex maps, namely solutions to the set-theoretical
4-simplex equation, using Lax matrix refactorisation problems. Employing this
method, we construct 4-simplex maps which at a certain limit give tetrahedron
maps classified by Kashaev, Korepanov and Sergeev. Moreover, we construct a
Kadomtsev--Petviashvili type of 4-simplex map. Finally, we introduce a method
for constructing 4-simplex maps which can be restricted on level sets to
parametric 4-simplex maps using Darboux transformations of integrable PDEs. We
construct a nonlinear Schr\"odinger type parametric 4-simplex map which is the
first parametric 4-simplex map in the literature.
