Decorated Cospans

Let $\mathcal C$ be a category with finite colimits, writing its coproduct $+$, and let $(\mathcal D, \otimes)$ be a braided monoidal category. We describe a method of producing a symmetric monoidal category from a lax braided monoidal functor $F: (\mathcal C,+) \to (\mathcal D, \otimes)$, and of producing a strong monoidal functor between such categories from a monoidal natural transformation between such functors. The objects of these categories, our so-called `decorated cospan categories', are simply the objects of $\mathcal C$, while the morphisms are pairs comprising a cospan $X \rightarrow N \leftarrow Y$ in $\mathcal C$ together with an element $1 \to FN$ in $\mathcal D$. Moreover, decorated cospan categories are multigraph categories---each object is equipped with a special commutative Frobenius monoid---and their functors preserve this structure.Comment: 25 page

Universal Constructions for (Co)Relations: categories, monoidal categories, and props

Calculi of string diagrams are increasingly used to present the syntax and algebraic structure of various families of circuits, including signal flow graphs, electrical circuits and quantum processes. In many such approaches, the semantic interpretation for diagrams is given in terms of relations or corelations (generalised equivalence relations) of some kind. In this paper we show how semantic categories of both relations and corelations can be characterised as colimits of simpler categories. This modular perspective is important as it simplifies the task of giving a complete axiomatisation for semantic equivalence of string diagrams. Moreover, our general result unifies various theorems that are independently found in literature and are relevant for program semantics, quantum computation and control theory.Comment: 22 pages + 3 page appendix, extended version of arXiv:1703.0824

Additive monotones for resource theories of parallel-combinable processes with discarding

A partitioned process theory, as defined by Coecke, Fritz, and Spekkens, is a symmetric monoidal category together with an all-object-including symmetric monoidal subcategory. We think of the morphisms of this category as processes, and the morphisms of the subcategory as those processes that are freely executable. Via a construction we refer to as parallel-combinable processes with discarding, we obtain from this data a partially ordered monoid on the set of processes, with f > g if one can use the free processes to construct g from f. The structure of this partial order can then be probed using additive monotones: order-preserving monoid homomorphisms with values in the real numbers under addition. We first characterise these additive monotones in terms of the corresponding partitioned process theory. Given enough monotones, we might hope to be able to reconstruct the order on the monoid. If so, we say that we have a complete family of monotones. In general, however, when we require our monotones to be additive monotones, such families do not exist or are hard to compute. We show the existence of complete families of additive monotones for various partitioned process theories based on the category of finite sets, in order to shed light on the way such families can be constructed.Comment: In Proceedings QPL 2015, arXiv:1511.0118

The Algebra of Open and Interconnected Systems

Herein we develop category-theoretic tools for understanding network-style diagrammatic languages. The archetypal network-style diagrammatic language is that of electric circuits; other examples include signal flow graphs, Markov processes, automata, Petri nets, chemical reaction networks, and so on. The key feature is that the language is comprised of a number of components with multiple (input/output) terminals, each possibly labelled with some type, that may then be connected together along these terminals to form a larger network. The components form hyperedges between labelled vertices, and so a diagram in this language forms a hypergraph. We formalise the compositional structure by introducing the notion of a hypergraph category. Network-style diagrammatic languages and their semantics thus form hypergraph categories, and semantic interpretation gives a hypergraph functor. The first part of this thesis develops the theory of hypergraph categories. In particular, we introduce the tools of decorated cospans and corelations. Decorated cospans allow straightforward construction of hypergraph categories from diagrammatic languages: the inputs, outputs, and their composition are modelled by the cospans, while the 'decorations' specify the components themselves. Not all hypergraph categories can be constructed, however, through decorated cospans. Decorated corelations are a more powerful version that permits construction of all hypergraph categories and hypergraph functors. These are often useful for constructing the semantic categories of diagrammatic languages and functors from diagrams to the semantics. To illustrate these principles, the second part of this thesis details applications to linear time-invariant dynamical systems and passive linear networks.Comment: 230 pages. University of Oxford DPhil Thesi