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 PROP FinRel_k, the strict version of the category of finite-dimensional vector spaces over the field of rational functions k = R(s) and linear relations, where the variable s acts as differentiation and the monoidal structure is direct sum rather than the usual tensor product of vector spaces. Control processes are also described by controllability and observability—whether the input can drive the process to any state, and whether any state can be determined from later outputs. For any field k we give a presentation of FinRel_k in terms of generators of the free PROP of signal-flow diagrams together with the equations that give FinRel_k its structure. The `cap' and `cup' generators, missing when the morphisms are linear maps, make it possible to model feedback. 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. In order to address controllability and observability, we construct the PROP Stateful_k and relate it back to the PROP of signal-flow diagrams. This provides a way to graphically express controllability and observability for linear time-invariant processes.

Category theory is known as a language of mathematics. The basic concepts of the language are systematized in a fibrant double category, a two-dimensional structure also known as a bicategory equipped with proarrows.

We give a new definition of the structure: a bifibrant double category is a "two-sided bifibration'' from a category to itself, with a weak composition and identity. We construct the three-dimensional category of bifibrant double categories.

A category forms a bifibrant double category, by forming the union of the arrow double category with its opposite; we call this the weave double category. Then a two-sided bifibration or matrix category is a span of categories which forms a bimodule of weave double categories. These form a three-dimensional category: the objects are categories; the three kinds of morphism are functor, profunctor, and matrix category; the three kinds of square are transformation, matrix functor, and matrix profunctor, and the cubes are matrix transformations. This structure is a "bifibrant triple category without interchange'', which we call a metalogic.

A bifibrant double category is then a pseudomonad in the metalogic of matrix categories; this defines the objects of a three-dimensional construction: a double functor is a morphism of pseudomonads, a vertical profunctor is a ``vertical monad'' between pseudomonads, and a horizontal profunctor is a bimodule of pseudomonads; a vertical transformation is a morphism of vertical monads, a horizontal transformation is a morphism of bimodules, and a double profunctor is a bimodule of vertical monads. A double transformation is a transformation of vertical bimodules. These form the metalogic of bifibrant double categories.

The concept of a system has proliferated through natural

and social sciences. While myriad theories of systems

exist, there is no mathematical general theory of

systems. In this thesis, we take a first step towards

formulating such a theory. Our focus is on developing a

syntax for compositional systems equipped with a rewriting

theory. We pull from category theory and linguistics to

accomplish this. The basic syntactical unit is a

structured cospan and rewriting is introduced via the

double pushout method. Two versions of rewriting are

proposed: one that tracks intermediate steps and another

disregards them. Benefits and drawbacks of both versions

are discussed. We apply our results to the decomposition

of closed systems, obtaining a structurally inductive

viewpoint of rewriting such systems.