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15
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 ..."
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Cited by 24 (6 self)
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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.
Exact Completions and Toposes
 University of Edinburgh
, 2000
"... Toposes and quasitoposes have been shown to be useful in mathematics, logic and computer science. Because of this, it is important to understand the di#erent ways in which they can be constructed. Realizability toposes and presheaf toposes are two important classes of toposes. All of the former and ..."
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Cited by 13 (4 self)
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Toposes and quasitoposes have been shown to be useful in mathematics, logic and computer science. Because of this, it is important to understand the di#erent ways in which they can be constructed. Realizability toposes and presheaf toposes are two important classes of toposes. All of the former and many of the latter arise by adding "good " quotients of equivalence relations to a simple category with finite limits. This construction is called the exact completion of the original category. Exact completions are not always toposes and it was not known, not even in the realizability and presheaf cases, when or why toposes arise in this way. Exact completions can be obtained as the composition of two related constructions. The first one assigns to a category with finite limits, the "best " regular category (called its regular completion) that embeds it. The second assigns to
Mathematical models of computational and combinatorial structures. Invited address for Foundations
 of Software Science and Computation Structures (FOSSACS 2005
, 2005
"... Abstract. The general aim of this talk is to advocate a combinatorial perspective, together with its methods, in the investigation and study of models of computation structures. This, of course, should be taken in conjunction with the wellestablished views and methods stemming from algebra, category ..."
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Cited by 9 (3 self)
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Abstract. The general aim of this talk is to advocate a combinatorial perspective, together with its methods, in the investigation and study of models of computation structures. This, of course, should be taken in conjunction with the wellestablished views and methods stemming from algebra, category theory, domain theory, logic, type theory, etc. In support of this proposal I will show how such an approach leads to interesting connections between various areas of computer science and mathematics; concentrating on one such example in some detail. Specifically, I will consider the line of my research involving denotational models of the pi calculus and algebraic theories with variablebinding operators, indicating how the abstract mathematical structure underlying these models fits with that of Joyal’s combinatorial species of structures. This analysis suggests both the unification and generalisation of models, and in the latter vein I will introduce generalised species of structures and their calculus. These generalised species encompass and generalise various of the notions of species used in combinatorics. Furthermore, they have a rich mathematical structure (akin to models of Girard’s linear logic) that can be described purely within Lawvere’s generalised logic. Indeed, I will present and treat the cartesian closed structure, the linear structure, the differential structure, etc. of generalised species axiomatically in this mathematical framework. As an upshot, I will observe that the setting allows for interpretations of computational calculi (like the lambda calculus, both typed and untyped; the recently introduced differential lambda calculus of Ehrhard and Regnier; etc.) that can be directly seen as translations into a more basic elementary calculus of interacting agents that compute by communicating and operating upon structured data.
Unique Factorisation Lifting Functors and Categories of LinearlyControlled Processes
 Mathematical Structures in Computer Science
, 1999
"... We consider processes consisting of a category of states varying over a control category as prescribed by a unique factorisation lifting functor. After a brief analysis of the structure of general processes in this setting, we restrict attention to linearlycontrolled ones. To this end, we introduce ..."
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We consider processes consisting of a category of states varying over a control category as prescribed by a unique factorisation lifting functor. After a brief analysis of the structure of general processes in this setting, we restrict attention to linearlycontrolled ones. To this end, we introduce and study a notion of pathlinearisable category in which any two paths of morphisms with equal composites can be linearised (or interleaved) in a canonical fashion. Our main result is that categories of linearlycontrolled processes (viz., processes controlled by pathlinearisable categories) are sheaf models. Introduction This work is an investigation into the mathematical structure of processes. The processes to be considered embody a notion of state space varying according to a control. This we formalise as a category of states (and their interrelations) Xequipped with a control functor X C f . There are different ways in which the control category C may be required to control t...
Higher order symmetry of graphs
 In Lecture given at the September Meeting of the Irish Mathematical Society, available on the Author's web
, 1994
"... It advertises the joining of two themes: groups and symmetry; and categorical methods and analogues of set theory. Groups are expected to be associated with symmetry. Klein’s famous Erlanger Programme asserted that the study of a geometry was the study of the group of automorphisms of that geometry. ..."
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Cited by 6 (3 self)
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It advertises the joining of two themes: groups and symmetry; and categorical methods and analogues of set theory. Groups are expected to be associated with symmetry. Klein’s famous Erlanger Programme asserted that the study of a geometry was the study of the group of automorphisms of that geometry. The structure of group alone may not give all the expression one needs of the intuitive idea of symmetry. One often needs structured groups (for example topological, Lie, algebraic, order,...). Here we consider groups with the additional structure of directed graph, which we abbreviate to graph. This type of structure appears in [18, 14]. We shall associate with a graph A a group AUT(A) which is also a graph. The vertices of AUT(A) are the automorphism of the graph A and the edges between automorphisms give an expression of “adjacency ” of automorphisms. The vertices of this graph form a group, and so also do the edges. The automorphisms of A adjacent to the identity will be called the inner automorphisms of the graph A. One aspect of the problem is to describe these inner automorphisms in terms of the internal structure of the graph A. The second theme is that of regarding the usual category of sets and mappings as but one environment for doing mathematics, and one which may be replaced by others. We use the word “environment ” here rather than “foundation”, because the former word implies a more relativistic approach.
A homotopical algebra of graphs related to zeta series
 Homology, Homotopy and its Applications 10 (2008), 1–13. MR2506131 (2010f:18010). [C95] Crans, Sjoerd E. Quillen
"... Abstract: The purpose of this paper is to develop a homotopical algebra for graphs, relevant to zeta series and spectra of finite graphs. More precisely, we define a Quillen model structure in a category of graphs (directed and possibly infinite, with loops and multiple arcs allowed). The weak equiv ..."
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Abstract: The purpose of this paper is to develop a homotopical algebra for graphs, relevant to zeta series and spectra of finite graphs. More precisely, we define a Quillen model structure in a category of graphs (directed and possibly infinite, with loops and multiple arcs allowed). The weak equivalences for this model structure are the Acyclics (graph morphisms which preserve cycles). The cofibrations and fibrations for the model are determined from the class of Whiskerings (graph morphisms produced by grafting trees). Our model structure seems to fit well with the importance of acyclic directed graphs in many applications. In addition to the weak factorization systems which form this model structure, we also describe two FreydKelly factorization systems based on Folding, Injecting, and Covering graph morphisms. 0. Introduction. In this paper we develop a notion of homotopy within graphs, and demonstrate its relevance to the study of zeta series and spectrum of a finite graph. We will work throughout with a particular category of graphs, described in Section 1 below. Our graphs will be directed and possibly infinite, with loops and multiple arcs allowed.
ALGEBRAIC CATEGORIES WHOSE PROJECTIVES ARE EXPLICITLY FREE
"... Abstract. Let M = (M, m, u) be a monad and let (MX, m) be the free Malgebra on the object X. Consider an Malgebra (A, a), a retraction r: (MX, m) → (A, a) and a section t: (A, a) → (MX, m) of r. The retract (A, a) is not free in general. We observe that for many monads with a ‘combinatorial flav ..."
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Abstract. Let M = (M, m, u) be a monad and let (MX, m) be the free Malgebra on the object X. Consider an Malgebra (A, a), a retraction r: (MX, m) → (A, a) and a section t: (A, a) → (MX, m) of r. The retract (A, a) is not free in general. We observe that for many monads with a ‘combinatorial flavor ’ such a retract is not only a free algebra (MA0, m), but it is also the case that the object A0 of generators is determined in a canonical way by the section t. We give a precise form of this property, prove a characterization, and discuss examples from combinatorics, universal algebra, convexity and topos theory. 1.
2007a), Components, Complements and the Reflection Formula, Theory and
"... ABSTRACT. Some basic features of the simultaneous inclusion of discrete fibrations and discrete opfibrations in categories over a base category X are considered. In particular, we illustrate the formulas (↓P)x = ten(x/X, P) ; (P↓)x = hom(X/x, P) ..."
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ABSTRACT. Some basic features of the simultaneous inclusion of discrete fibrations and discrete opfibrations in categories over a base category X are considered. In particular, we illustrate the formulas (↓P)x = ten(x/X, P) ; (P↓)x = hom(X/x, P)
Variations on Realizability: Realizing the Propositional Axiom of Choice
 Math. Structures Comput. Sci
, 2000
"... Introduction 1.1 Historical background Early investigators of realizability were interested in metamathematical questions. In keeping with the traditions of the time they concentrated on interpretations of one formal system in another. They considered an ad hoc collection of increasingly ingenious ..."
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Introduction 1.1 Historical background Early investigators of realizability were interested in metamathematical questions. In keeping with the traditions of the time they concentrated on interpretations of one formal system in another. They considered an ad hoc collection of increasingly ingenious interpretations mainly to establish consistency, independence and conservativity results. van Oosten's contribution to the Workshop (see van Oosten [56] and the extended account van Oosten [57]) gave inter alia an account of these concerns from a modern perspective. (One should also draw attention to realizability used to provide interpretations of Brouwer's theory of Choice Sequences. An early approach is in Kleene Vesley [28]; for modern work in the area consult Moschovakis [35], [36], [37].) In the early days of categorical logic one considered realizability as providing models for constructive mathematics; while the metamathematics could be retrieved by `coding' the mod
2007), Components, Complements and Reflection Formulas, preprint
"... Abstract. We illustrate the formula (↓p)x = Γ!(x/p), which gives the reflection ↓p of a category p: P → X over X in discrete fibrations. One of its proofs is based on a “complement operator ” which takes a discrete fibration A to the functor ¬A, right adjoint to Γ!(A × −) : Cat/X → Set and valued in ..."
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Cited by 1 (1 self)
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Abstract. We illustrate the formula (↓p)x = Γ!(x/p), which gives the reflection ↓p of a category p: P → X over X in discrete fibrations. One of its proofs is based on a “complement operator ” which takes a discrete fibration A to the functor ¬A, right adjoint to Γ!(A × −) : Cat/X → Set and valued in discrete opfibrations. Some consequences and applications are presented. 1.