Modeling objects from algebraic geometry in the language of polyhedral combinatorics has been a success story as it paves the way to algorithmic and experimental methods. The purpose of this project is to investigate how more complicated varieties or generalisations can be combinatorialised, too. Examples include moduli spaces and quotient varieties. Dealing with abstract algebraic varieties in terms of systems of affine charts is an obvious choice – yet the combinatorial and algorithmic ramifications are extremely intricate, e.g. due to the sheer size of the relevant objects. The goal of this project is to develop general infrastructure for dealing with piecewise-linear analogs of algebraic varieties which can be described by finitely many polyhedra and a combinatorial or algebraic gluing procedure.