  Summarizing topological information from datasets and maps defined on them is
a central theme in topological data analysis. \textsf{Mapper}, a tool for such
summarization, takes as input both a possibly high dimensional dataset and a
map defined on the data, and produces a summary of the data by using a cover of
the codomain of the map. This cover, via a pullback operation to the domain,
produces a simplicial complex connecting the data points.
  The resulting view of the data through a cover of the codomain offers
flexibility in analyzing the data. However, it offers only a view at a fixed
scale at which the cover is constructed. Inspired by the concept, we explore a
notion of a tower of covers which induces a tower of simplicial complexes
connected by simplicial maps, which we call {\em multiscale mapper}. We study
the resulting structure, its stability, and design practical algorithms to
compute its associated persistence diagrams efficiently. Specifically, when the
domain is a simplicial complex and the map is a real-valued piecewise-linear
function, the algorithm can compute the exact persistence diagram only from the
1-skeleton of the input complex. For general maps, we present a combinatorial
version of the algorithm that acts only on \emph{vertex sets} connected by the
1-skeleton graph, and this algorithm approximates the exact persistence diagram
thanks to a stability result that we show to hold. We also relate the
multiscale mapper with the \v{C}ech complexes arising from a natural pullback
pseudometric defined on the input domain.
