  Motivated by applications in Topological Data Analysis, we consider
decompositions of a simplicial complex induced by a cover of its vertices. We
study how the homotopy type of such decompositions approximates the homotopy of
the simplicial complex itself. The difference between the simplicial complex
and such an approximation is quantitatively measured by means of the so called
obstruction complexes. Our general machinery is then specialized to clique
complexes, Vietoris-Rips complexes and Vietoris-Rips complexes of metric
gluings. For the latter we give metric conditions which allow to recover the
first and zero-th homology of the gluing from the respective homologies of the
components.
