Algebraic transformation of unary partial algebras II: Single-pushout approach

AbstractThe single-pushout approach to graph transformation is extended to the algebraic transformation of partial many-sorted unary algebras. Such a generalization has been motivated by the need to model the transformation of structures which are richer and more complex than graphs and hypergraphs.The main result presented in this article is an algebraic characterization of the single-pushout transformation in the categories of all conformisms, all closed quomorphisms, and all closed-domain closed quomorphisms of unary partial algebras over a given signature, together with a corresponding operational characterization that may serve as a basis for implementation.Moreover, all three categories are shown to satisfy all of the HLR (high-level replacement) conditions for parallelism, taking as occurrences the total morphisms in each category. Another important result presented in this article is the definition of HLR conditions for amalgamation, which are also satisfied by the categories of partial homomorphisms considered here, taking again the corresponding total morphisms as occurrences

Open Graphs and Monoidal Theories

String diagrams are a powerful tool for reasoning about physical processes, logic circuits, tensor networks, and many other compositional structures. The distinguishing feature of these diagrams is that edges need not be connected to vertices at both ends, and these unconnected ends can be interpreted as the inputs and outputs of a diagram. In this paper, we give a concrete construction for string diagrams using a special kind of typed graph called an open-graph. While the category of open-graphs is not itself adhesive, we introduce the notion of a selective adhesive functor, and show that such a functor embeds the category of open-graphs into the ambient adhesive category of typed graphs. Using this functor, the category of open-graphs inherits "enough adhesivity" from the category of typed graphs to perform double-pushout (DPO) graph rewriting. A salient feature of our theory is that it ensures rewrite systems are "type-safe" in the sense that rewriting respects the inputs and outputs. This formalism lets us safely encode the interesting structure of a computational model, such as evaluation dynamics, with succinct, explicit rewrite rules, while the graphical representation absorbs many of the tedious details. Although topological formalisms exist for string diagrams, our construction is discreet, finitary, and enjoys decidable algorithms for composition and rewriting. We also show how open-graphs can be parametrised by graphical signatures, similar to the monoidal signatures of Joyal and Street, which define types for vertices in the diagrammatic language and constraints on how they can be connected. Using typed open-graphs, we can construct free symmetric monoidal categories, PROPs, and more general monoidal theories. Thus open-graphs give us a handle for mechanised reasoning in monoidal categories.Comment: 31 pages, currently technical report, submitted to MSCS, waiting review

Category theory : definitions and examples

Category theory was invented as an abstract language for describing certain structures and constructions which repeatedly occur in many branches of mathematics, such as topology, algebra, and logic. In recent years, it has found several applications in computer science, e.g., algebraic specification, type theory, and programming language semantics. In this paper, we collect definitions and examples of the basic concepts in category theory: categories, functors, natural transformations, universal properties, limits, and adjoints

Diagram spaces, diagram spectra, and spectra of units

This article compares the infinite loop spaces associated to symmetric spectra, orthogonal spectra, and EKMM S-modules. Each of these categories of structured spectra has a corresponding category of structured spaces that receives the infinite loop space functor \Omega^\infty. We prove that these models for spaces are Quillen equivalent and that the infinite loop space functors \Omega^\infty agree. This comparison is then used to show that two different constructions of the spectrum of units gl_1 R of a commutative ring spectrum R agree.Comment: 62 pages. The definition of the functor \mathbb{Q} is changed. Sections 8, 9, 17 and 18 contain revisions and/or new materia