7 research outputs found
Calculating Colimits Compositionally
We show how finite limits and colimits can be calculated compositionally
using the algebras of spans and cospans, and give as an application a proof of
the Kleene Theorem on regular languages
Calculating Colimits Compositionally
We show how finite limits and colimits can be calculated
compositionally using the algebras of spans and cospans, and give as an application a proof of the Kleene Theorem on regular languages
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
Calculating colimits compositionally
3noneROSEBRUGH R; SABADINI N; R. WALTERSRosebrugh, R; Sabadini, Nicoletta; Walters, ROBERT FRANK CARSLA