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Two of the most useful tools in topological combinatorics are the nerve lemmaand discrete Morse theory. In this note we introduce a theorem thatinterpolates between them and allows decompositions of complexes intonon-contractible pieces as long as discrete Morse theory ensures that theybehave well enough. The proof is based on diagrams of spaces, but that theoryis not needed for the formulation or applications of the theorem.

A nerve lemma for gluing together incoherent discrete Morse functions