Control theory uses "signal-flow diagrams" to describe processes where
real-valued functions of time are added, multiplied by scalars, differentiated
and integrated, duplicated and deleted. These diagrams can be seen as string
diagrams for the symmetric monoidal category FinVect_k of finite-dimensional
vector spaces over the field of rational functions k = R(s), where the variable
s acts as differentiation and the monoidal structure is direct sum rather than
the usual tensor product of vector spaces. For any field k we give a
presentation of FinVect_k in terms of the generators used in signal flow
diagrams. A broader class of signal-flow diagrams also includes "caps" and
"cups" to model feedback. We show these diagrams can be seen as string diagrams
for the symmetric monoidal category FinRel_k, where objects are still
finite-dimensional vector spaces but the morphisms are linear relations. We
also give a presentation for FinRel_k. The relations say, among other things,
that the 1-dimensional vector space k has two special commutative
dagger-Frobenius structures, such that the multiplication and unit of either
one and the comultiplication and counit of the other fit together to form a
bimonoid. This sort of structure, but with tensor product replacing direct sum,
is familiar from the "ZX-calculus" obeyed by a finite-dimensional Hilbert space
with two mutually unbiased bases.Comment: 42 pages LaTe
