  The interleaving distance, although originally developed for persistent
homology, has been generalized to measure the distance between functors modeled
on many posets or even small categories. Existing theories require that such a
poset have a superlinear family of translations or a similar structure.
However, many posets of interest to topological data analysis, such as zig-zag
posets and the face relation poset of a cell-complex, do not admit interesting
translations, and consequently don't admit a nice theory of interleavings. In
this paper we show how one can side-step this limitation by providing a general
theory where one maps to a poset that does admit interesting translations, such
as the lattice of down sets, and then defines interleavings relative to this
map. Part of our theory includes a rigorous notion of discretization or
"pixelization" of poset modules, which in turn we use for interleaving
inference. We provide an approximation condition that in the setting of
lattices gives rise to two possible pixelizations, both of which are guaranteed
to be close in the interleaving distance. Finally, we conclude by considering
interleaving inference for cosheaves over a metric space and give an explicit
description of interleavings over a grid structure on Euclidean space.
