Categories in Control
John C. Baez, Jason Erbele · Theory and applications of categories · 2015
Control theory uses 'signal-flow diagrams' to describe processes where realvalued 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 signalflow 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 †-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.