  In recent work, generalized persistence modules have proved useful in
distinguishing noise from the legitimate topological features of a data set.
Algebraically, generalized persistence modules can be viewed as representations
for the poset algebra. The interplay between various metrics on persistence
modules has been of wide interest, most notably, the isometry theorem of Bauer
and Lesnick for (one-dimensional) persistence modules. The interleaving metric
of Bubenik, de Silva and Scott endows the collection of representations of a
poset with values in any category with the structure of a metric space. This
metric makes sense for any poset, and has the advantage that post-composition
by any functor is a contraction. In this paper, we prove an isometry theorem
using this interleaving metric on a full subcategory of generalized persistence
modules for a large class of posets.
