  In this paper, we consider the problem of determining when two tensor
networks are equivalent under a heterogeneous change of basis. In particular,
to a string diagram in a certain monoidal category (which we call tensor
diagrams), we formulate an associated abelian category of representations. Each
representation corresponds to a tensor network on that diagram. We then
classify which tensor diagrams give rise to categories that are finite, tame,
or wild in the traditional sense of representation theory. For those tensor
diagrams of finite and tame type, we classify the indecomposable
representations. Our main result is that a tensor diagram is wild if and only
if it contains a vertex of degree at least three. Otherwise, it is of tame or
finite type.
