  If two compact quantum metric spaces are close in the metric sense, then how
similar are they, as noncommutative spaces? In the classical realm of
Riemannian geometry, informally, if two manifolds are close in the
Gromov-Hausdorff distance, and belong to a class of manifolds with bounded
curvature and diameter, then the spectra of their Laplacian or Dirac operators
are also close under many scenari. Of particular interest is the case where a
sequence of manifolds converge for the Gromov-Hausdorff distance to a manifold
of lower dimension, and the question of the continuity, in some sense, of the
spectra of geometrically relevant operators. In this paper, we initiate the
study of the continuity of spectra and other properties of metric spectral
triples under collapse in the noncommutative realm. As a first step in this
study, we work with collapse for the spectral propinquity, an analogue of the
Gromov-Hausdorff distance for spectral triples introduced by the second author,
i.e. a form of metric for differential structures. Inspired by results from
collapse in Riemannian geometry, we begin with the study of spectral triples
which decompose, in some sense, in a vertical and a horizontal direction, and
we collapse these spectral triples along the vertical direction. We obtain
convergence results, and by the work of the second author, we conclude
continuity results for the spectra of the Dirac operators of these spectral
triples. Examples include collapse of product of spectral triples with one
Abelian factor, $U(1)$ principal bundles over Riemannian spin manifolds, and
noncommutative principal bundles, including C*-crossed-products and other
noncommutative bundles.
