This is a report on joint work with Nathan Broomhead and David Ploog. Tilting theory is of central importance in representation theory, and central to tilting theory are the notions of tilting module and tilting object, which are used to characterise derived equivalences. The combinatorics of tilting objects is very rich, and in the 2000s, it was realised the combinatorics could be used to categorify cluster algebras, for example with the simplicial complex of cluster-tilting objects modelling the cluster complex. Another natural generalisation of tilting object is that of a silting object. One may think of tilting objects identifying "hearts" inside a derived category which have that derived category as their derived category. Silting objects identify "hearts" inside a derived category which are algebraic in nature. The combinatorics of silting objects is also very rich, and we discuss a CW complex arising from them in the case of algebras with discrete derived category. We will also discuss how this CW complex is related to Bridgeland's stability manifold.