Homotopy colimits – comparison lemmas for combinatorial applications

We provide a “toolkit” of basic lemmas for the comparison of homotopy types of homotopy colimits of diagrams of spaces over small categories. We show how this toolkit can be used in quite different fields of applications. We demonstrate this with respect to Björner's “Generalized Homotopy Complementation Formula” [5], the topology of toric varieties, the study of homotopy types of arrangements of subspaces, the analysis of homotopy types of subgroup complexes.