Results 1  10
of
21
Bicategories of Processes
 JOURNAL OF PURE AND APPLIED ALGEBRA
, 1997
"... The suspensionloop construction is used to define a process in a symmetric monoidal category. The algebra of such processes is that of symmetric monoidal bicategories. Processes in categories with products and in categories with sums are studied in detail, and in both cases the resulting bicate ..."
Abstract

Cited by 42 (14 self)
 Add to MetaCart
The suspensionloop construction is used to define a process in a symmetric monoidal category. The algebra of such processes is that of symmetric monoidal bicategories. Processes in categories with products and in categories with sums are studied in detail, and in both cases the resulting bicategories of processes are equipped with operations called feedback. Appropriate versions of traced monoidal properties are verified for feedback, and a normal form theorem for expressions of processes is proved. Connections with existing theories of circuit design and computation are established via structure preserving homomorphisms.
Fibrations of Graphs
 DISCRETE MATH
, 1996
"... A fibration of graphs is a morphism that is a local isomorphism of inneighbourhoods, much in the same way a covering projection is a local isomorphism of neighbourhoods. This paper develops systematically the theory of graph fibrations, emphasizing in particular those results that recently found ..."
Abstract

Cited by 25 (6 self)
 Add to MetaCart
A fibration of graphs is a morphism that is a local isomorphism of inneighbourhoods, much in the same way a covering projection is a local isomorphism of neighbourhoods. This paper develops systematically the theory of graph fibrations, emphasizing in particular those results that recently found application in the theory of distributed systems.
First Order Linear Logic in Symmetric Monoidal Closed Categories
, 1991
"... There has recently been considerable interest in the development of `logical frameworks ' which can represent many of the logics arising in computer science in a uniform way. Within the Edinburgh LF project, this concept is split into two components; the first being a general proof theoretic encodin ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
There has recently been considerable interest in the development of `logical frameworks ' which can represent many of the logics arising in computer science in a uniform way. Within the Edinburgh LF project, this concept is split into two components; the first being a general proof theoretic encoding of logics, and the second a uniform treatment of their model theory. This thesis forms a case study for the work on model theory. The models of many first and higher order logics can be represented as fibred or indexed categories with certain extra structure, and this has been suggested as a general paradigm. The aim of the thesis is to test the strength and flexibility of this paradigm by studying the specific case of Girard's linear logic. It should be noted that the exact form of this logic in the first order case is not entirely certain, and the system treated here is significantly different to that considered by Girard.
Maps II: Chasing Diagrams in Categorical Proof Theory
, 1996
"... In categorical proof theory, propositions and proofs are presented as objects and arrows in a category. It thus embodies the strong constructivist paradigms of propositionsastypes and proofsasconstructions, which lie in the foundation of computational logic. Moreover, in the categorical setting, ..."
Abstract

Cited by 7 (4 self)
 Add to MetaCart
In categorical proof theory, propositions and proofs are presented as objects and arrows in a category. It thus embodies the strong constructivist paradigms of propositionsastypes and proofsasconstructions, which lie in the foundation of computational logic. Moreover, in the categorical setting, a third paradigm arises, not available elsewhere: logicaloperationsasadjunctions. It offers an answer to the notorious question of the equality of proofs. So we chase diagrams in algebra of proofs. On the basis of these ideas, the present paper investigates proof theory of regular logic: the f; 9gfragment of the first order logic with equality. The corresponding categorical structure is regular fibration. The examples include stable factorisations, sites, triposes. Regular logic is exactly what is needed to talk about maps, as total and singlevalued relations. However, when enriched with proofsasarrows, this familiar concept must be supplied with an additional conversion rule, conn...
Dataflow Networks are Fibrations
 In Category Theory and Computer Science
, 1991
"... Dataflow networks are a paradigm for concurrent computation in which a collection of concurrently and asynchronously executing processes communicate by sending messages over FIFO message channels. In a previous paper, we showed that dataflow networks could be represented as certain spans in a catego ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
Dataflow networks are a paradigm for concurrent computation in which a collection of concurrently and asynchronously executing processes communicate by sending messages over FIFO message channels. In a previous paper, we showed that dataflow networks could be represented as certain spans in a category of automata, or more abstractly, in a category of domains, and we identified some universal properties of various operations for building networks from components. Not all spans corresponded to dataflow processes, and we raised the question of what might be an appropriate categorical characterization of those spans that are "dataflowlike. " In this paper, we answer this question by obtaining a characterization of the dataflowlike spans as split right fibrations, either in a 2category of automata or a 2category of domains. This characterization makes use of the theory of fibrations in a 2category developed by Street. In that theory, the split right fibrations are the algebras of a cert...
An Australian conspectus of higher categories

, 2004
"... Much Australian work on categories is part of, or relevant to, the development of higher categories and their theory. In this note, I hope to describe some of the origins and achievements of our efforts that they might perchance serve as a guide to the development of aspects of higherdimensional wo ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
Much Australian work on categories is part of, or relevant to, the development of higher categories and their theory. In this note, I hope to describe some of the origins and achievements of our efforts that they might perchance serve as a guide to the development of aspects of higherdimensional work. I trust that the somewhat autobiographical style will add interest rather than be a distraction. For so long I have felt rather apologetic when describing how categories might be helpful to other mathematicians; I have often felt even worse when mentioning enriched and higher categories to category theorists. This is not to say that I have doubted the value of our work, rather that I have felt slowed down by the continual pressure to defend it. At last, at this meeting, I feel justified in speaking freely amongst motivated researchers who know the need for the subject is well established. Australian Category Theory has its roots in homology theory: more precisely, in the treatment of the cohomology ring and the Künneth formulas in the book by Hilton and Wylie [HW]. The first edition of the book had a mistake concerning the cohomology ring of a product. The Künneth formulas arise from splittings of the natural short exact sequences
Proofs Without Syntax
 Annals of Mathematics
"... [M]athematicians care no more for logic than logicians for mathematics. Augustus de Morgan, 1868 Proofs are traditionally syntactic, inductively generated objects. This paper presents an abstract mathematical formulation of propositional calculus (propositional logic) in which proofs are combinatori ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
[M]athematicians care no more for logic than logicians for mathematics. Augustus de Morgan, 1868 Proofs are traditionally syntactic, inductively generated objects. This paper presents an abstract mathematical formulation of propositional calculus (propositional logic) in which proofs are combinatorial (graphtheoretic), rather than syntactic. It defines a combinatorial proof of a proposition φ as a graph homomorphism h: C → G(φ), where G(φ) is a graph associated with φ and C is a coloured graph. The main theorem is soundness and completeness: φ is true if and only if there exists a combinatorial proof h: C → G(φ). 1.
Model structures for homotopy of internal categories
 Theory Appl. Categ
"... CatC of internal categories and functors in a given finitely complete categoryC. Several nonequivalent notions of internal equivalence exist; to capture these notions, the model structures are defined relative to a given Grothendieck topology on C. Under mild conditions on C, the regular epimorphis ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
CatC of internal categories and functors in a given finitely complete categoryC. Several nonequivalent notions of internal equivalence exist; to capture these notions, the model structures are defined relative to a given Grothendieck topology on C. Under mild conditions on C, the regular epimorphism topology determines a modelstructure where we is the class of weak equivalences of internal categories (in the sense of Bunge and Par'e). For a Grothendieck topos C we get a structure that, thoughdifferent from Joyal and Tierney's, has an equivalent homotopy category. In case C is semiabelian, these weak equivalences turn out to be homology isomorphisms, and themodel structure on CatC induces a notion of homotopy of internal crossed modules. Incase C is the category
Weak Bisimulation and Open Maps (Extended Abstract)
, 1999
"... A systematic treatment of weak bisimulation and observational congruence on presheaf models is presented. The theory is developed with respect to a "hiding" functor from a category of paths to observable paths. Via a view of processes as bundles , we are able to account for weak morphisms (rough ..."
Abstract

Cited by 4 (4 self)
 Add to MetaCart
A systematic treatment of weak bisimulation and observational congruence on presheaf models is presented. The theory is developed with respect to a "hiding" functor from a category of paths to observable paths. Via a view of processes as bundles , we are able to account for weak morphisms (roughly only required to preserve observable paths) and to derive a saturation monad (on the category of presheaves over the category of paths). Weak morphisms may be encoded as strong ones via the Kleisli construction associated to the saturation monad. A general