The aim of this article is to describe a new perspective on functoriality of
persistent homology and explain its intrinsic symmetry that is often
overlooked. A data set for us is a finite collection of functions, called
measurements, with a finite domain. Such a data set might contain internal
symmetries which are effectively captured by the action of a set of the domain
endomorphisms. Different choices of the set of endomorphisms encode different
symmetries of the data set. We describe various category structures on such
enriched data sets and prove some of their properties such as decompositions
and morphism formations. We also describe a data structure, based on coloured
directed graphs, which is convenient to encode the mentioned enrichment. We
show that persistent homology preserves only some aspects of these landscapes
of enriched data sets however not all. In other words persistent homology is
not a functor on the entire category of enriched data sets. Nevertheless we
show that persistent homology is functorial locally. We use the concept of
equivariant operators to capture some of the information missed by persistent
homology
