  The content of this paper can be roughly organized into a three-level
hierarchy of generality. At the first, most general level, we introduce a new
language which allows us to express various categorical structures in a
systematic and explicit manner in terms of so-called 2-schemes. Although
2-schemes can formalize categorical structures such as symmetric monoidal
categories, they are not limited to this, and can be used to define structures
with no categorical analogue. Most categorical structures come with an
effective graphical calculus such as string diagrams for symmetric monoidal
categories, and the same is true more generally for interesting 2-schemes. In
this work, we focus on one particular non-categorical 2-scheme, whose instances
we refer to as tensor types. At the second level of the hierarchy, we work out
different flavors of this 2-scheme in detail. The effective graphical calculus
of tensor types is that of tensor networks or Penrose diagrams, that is, string
diagrams without a flow of time. As such, tensor types are similar to compact
closed categories, though there are various small but potentially important
differences. Also, the two definitions use completely different mechanisms
despite both being examples of 2-schemes. At the third level of the hierarchy,
we provide a long list of different families of concrete tensor types, in a way
which makes them accessible to concrete computations, motivated by their
potential use in physics. Different tensor types describe different types of
physical models, such as classical or quantum physics, deterministic or
statistical physics, many-body or single-body physics, or matter with or
without symmetries or fermions.
