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30
Presheaf Models for Concurrency
, 1999
"... In this dissertation we investigate presheaf models for concurrent computation. Our aim is to provide a systematic treatment of bisimulation for a wide range of concurrent process calculi. Bisimilarity is defined abstractly in terms of open maps as in the work of Joyal, Nielsen and Winskel. Their wo ..."
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Cited by 49 (19 self)
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In this dissertation we investigate presheaf models for concurrent computation. Our aim is to provide a systematic treatment of bisimulation for a wide range of concurrent process calculi. Bisimilarity is defined abstractly in terms of open maps as in the work of Joyal, Nielsen and Winskel. Their work inspired this thesis by suggesting that presheaf categories could provide abstract models for concurrency with a builtin notion of bisimulation. We show how
Cofibrantly generated natural weak factorisation systems
, 2007
"... There is an “algebraisation ” of the notion of weak factorisation system (w.f.s.) known as a natural weak factorisation system. In it, the two classes of maps of a w.f.s. are replaced by two categories of mapswithstructure, where the extra structure on a map now encodes a choice of liftings with r ..."
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There is an “algebraisation ” of the notion of weak factorisation system (w.f.s.) known as a natural weak factorisation system. In it, the two classes of maps of a w.f.s. are replaced by two categories of mapswithstructure, where the extra structure on a map now encodes a choice of liftings with respect to the other class. This extra structure has pleasant consequences: for example, a natural w.f.s. on C induces a canonical natural w.f.s. structure on any functor category [A, C]. In this paper, we define cofibrantly generated natural weak factorisation systems by analogy with cofibrantly generated w.f.s.’s. We then construct them by a method which is reminiscent of Quillen’s small object argument but produces factorisations which are much smaller and easier to handle, and show that the resultant natural w.f.s. is, in a suitable sense, freely generated by its generating cofibrations. Finally, we show that the two categories of mapswithstructure for a natural w.f.s. are closed under all the constructions we would expect of them: (co)limits, pushouts / pullbacks, transfinite composition, and so on. 1
The Coalgebraic Dual Of Birkhoff's Variety Theorem
, 2000
"... We prove an abstract dual of Birkho's variety theorem for categories E of coalgebras, given suitable assumptions on the underlying category E and suitable : E ## E . We also discuss covarieties closed under bisimulations and show that they are denable by a trivial kind of coequation { nam ..."
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We prove an abstract dual of Birkho's variety theorem for categories E of coalgebras, given suitable assumptions on the underlying category E and suitable : E ## E . We also discuss covarieties closed under bisimulations and show that they are denable by a trivial kind of coequation { namely, over one "color". We end with an example of a covariety which is not closed under bisimulations. This research is part of the Logic of Types and Computation project at Carnegie Mellon University under the direction of Dana Scott.
Factorization systems and fibrations: Toward a fibred Birkhoff variety theorem
, 2002
"... It is wellknown that a factorization system on a category (with sufficient pullbacks) gives rise to a fibration. This paper characterizes the fibrations that arise in such a way, by making precise the logical structure that is given by factorization systems. The underlying motivation is to obtain g ..."
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Cited by 9 (0 self)
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It is wellknown that a factorization system on a category (with sufficient pullbacks) gives rise to a fibration. This paper characterizes the fibrations that arise in such a way, by making precise the logical structure that is given by factorization systems. The underlying motivation is to obtain general Birkho results in a fibred setting.
Modal Predicates and Coequations
, 2002
"... We show how coalgebras can be presented by operations and equations. We discuss the basic properties of this presentation and compare it with the usual approach. ..."
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Cited by 5 (2 self)
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We show how coalgebras can be presented by operations and equations. We discuss the basic properties of this presentation and compare it with the usual approach.
Tensor products of finitely cocomplete and abelian categories
 Journal of Algebra
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Completeness for the coalgebraic cover modality
 Logical Methods in Computer Science
"... We study the finitary version of the coalgebraic logic introduced by L. Moss. The syntax of this logic, which is introduced uniformly with respect to a coalgebraic type functor T: Set → Set, extends that of classical propositional logic with the socalled coalgebraic cover modality ∇T. The semantics ..."
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Cited by 3 (1 self)
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We study the finitary version of the coalgebraic logic introduced by L. Moss. The syntax of this logic, which is introduced uniformly with respect to a coalgebraic type functor T: Set → Set, extends that of classical propositional logic with the socalled coalgebraic cover modality ∇T. The semantics of ∇T is defined in terms of a categorically defined relation lifting operation T. As the main contributions of our paper we introduce a derivation system M, and prove that M provides a sound and complete axiomatization for the collection of coalgebraically valid inequalities. Our soundness and completeness proof is algebraic, and we employ Pattinson’s stratification method, showing that our derivation system can be stratified in ω many layers, corresponding to the modal depth of the formulas involved. In the proof of our main result we identify some new concepts and obtain some auxiliary results of independent interest. We survey properties of the notion T of relation lifting, induced by an arbitrary but fixed set functor T. We introduce a category Pres of Boolean algebra presentations, and establish an adjunction between Pres and the category BA of Boolean algebras. Given the fact that our derivation system M involves only formulas of depth one, it can be encoded as a functor
A Categorical Theory of Patches
"... Abstract—When working with distant collaborators on the same files, one often uses a distributed version control system, which is a program tracking the history of files and helps importing modifications brought by others as patches. The implementation of such a system requires to handle lots of sit ..."
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Abstract—When working with distant collaborators on the same files, one often uses a distributed version control system, which is a program tracking the history of files and helps importing modifications brought by others as patches. The implementation of such a system requires to handle lots of situations depending on the operations users have performed on files, and it is thus difficult to ensure that all the corner cases have been correctly addressed. Here, instead of verifying the implementation of such a system, we adopt a “converse” approach: we introduce a theoretical model, which is defined abstractly by the universal property that it should satisfy, and work out a concrete description of it. We begin by defining a category of files and patches, where the operation of merging the effect of two coinitial patches is defined by pushout. Since the category is not closed under pushouts (two patches can