  In Schiebinger et al. (2017), the authors use optimal transport of measures
on empirical distributions arising from biological experiments to relate the
single cell RNA sequencing profiles for induced pluripotent stem cells
differentiating. But such algorithms could be arbitrarily applied to any
datasets from any collection of experiments. We consider here a natural
question that arises: in a manner consistent with conventionally accepted
assumptions about biology, in which cases can the results of two experiments be
mapped to each other in this manner? The answer to this question is of
fundamental practical importance in developing algorithms that use this method
for analysing and integrating complex datasets collected as part of the Human
Cell Atlas. Here, we develop a formulation of biology in terms of sheaves of
$C^*(X)$-modules for a smooth manifold $X$ equipped with certain structures,
that enables this question to be formally answered, leading to formal
statements about experimental inference and phenotypic identifiability. These
structures capture a perspective on biology that is consistent with a standard,
widely accepted biological perspective and is mathematically intuitive. Our
methods provide a framework in which to design complex experiments and the
algorithms to analyse them in a way that their conclusions can be believed.
