  The Reeb space, which generalizes the notion of a Reeb graph, is one of the
few tools in topological data analysis and visualization suitable for the study
of multivariate scientific datasets. First introduced by Edelsbrunner et al.,
it compresses the components of the level sets of a multivariate mapping and
obtains a summary representation of their relationships. A related construction
called mapper, and a special case of the mapper construction called the Joint
Contour Net have been shown to be effective in visual analytics. Mapper and JCN
are intuitively regarded as discrete approximations of the Reeb space, however
without formal proofs or approximation guarantees. An open question has been
proposed by Dey et al. as to whether the mapper construction converges to the
Reeb space in the limit.
  In this paper, we are interested in developing the theoretical understanding
of the relationship between the Reeb space and its discrete approximations to
support its use in practical data analysis. Using tools from category theory,
we formally prove the convergence between the Reeb space and mapper in terms of
an interleaving distance between their categorical representations. Given a
sequence of refined discretizations, we prove that these approximations
converge to the Reeb space in the interleaving distance; this also helps to
quantify the approximation quality of the discretization at a fixed resolution.
