Results 1  10
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12
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 45 (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
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 { namely, ..."
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Cited by 11 (0 self)
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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.
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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Cited by 9 (0 self)
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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
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 4 (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.
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 4 (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.
Homotopy theory of C ∗algebras
, 2008
"... In this work we construct from ground up a homotopy theory of C ∗algebras. This is achieved in parallel with the development of classical homotopy theory by first introducing an unstable model structure and second a stable model structure. The theory makes use of a full fledged import of homotopy t ..."
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In this work we construct from ground up a homotopy theory of C ∗algebras. This is achieved in parallel with the development of classical homotopy theory by first introducing an unstable model structure and second a stable model structure. The theory makes use of a full fledged import of homotopy theoretic techniques into the subject of C ∗algebras. The spaces in C ∗homotopy theory are certain hybrids of functors represented by C ∗algebras and spaces studied in classical homotopy theory. In particular, we employ both the topological circle and the C ∗algebra circle of complexvalued continuous functions on the real numbers which vanish at infinity. By using the inner workings of the theory, we may stabilize the spaces by forming spectra and bispectra with respect to either one of these circles or their tensor product. These stabilized spaces or spectra are the objects of study in stable C ∗homotopy theory. The stable homotopy category of C ∗algebras gives rise to invariants such as stable homotopy groups and bigraded cohomology and homology theories. We
NONCOMMUTATIVE CORRESPONDENCE CATEGORIES, SIMPLICIAL SETS AND PRO C ∗ALGEBRAS
, 906
"... Abstract. We construct an additive functor from the category of separable C ∗algebras with morphisms enriched over Kasparov’s KK0groups to the noncommutative correspondence category NCC K dg, whose objects are small DG categories and morphisms are given by the equivalence classes of some DG bimodu ..."
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Abstract. We construct an additive functor from the category of separable C ∗algebras with morphisms enriched over Kasparov’s KK0groups to the noncommutative correspondence category NCC K dg, whose objects are small DG categories and morphisms are given by the equivalence classes of some DG bimodules up to a certain Ktheoretic identification. Motivated by a construction of Cuntz we associate a pro C ∗algebra to any simplicial set, which is functorial with respect to proper maps of simplicial sets and those of pro C ∗algebras. This construction respects homotopy between proper maps after enforcing matrix stability on the category of pro C ∗algebras. The first result can be used to deduce derived Morita equivalence between DG categories of topological bundles associated to separable C ∗algebras up to a Ktheoretic identification from the knowledge of KKequivalence between the C ∗algebras. The second construction gives an indication that one can possibly develop a noncommutative proper homotopy theory in the context of topological algebras, e.g., pro C ∗algebras.
Small Objects in Categories of Algebras
, 1999
"... This Diplom thesis [J98a] was written in the research group "Categorical Methods in Algebra and Topology (KatMAT)" at the University of Bremen in 1998. In the setting of categorical universal algebra/ categorical model theory it examines algebraic and categorical smallness conditions on objects in c ..."
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This Diplom thesis [J98a] was written in the research group "Categorical Methods in Algebra and Topology (KatMAT)" at the University of Bremen in 1998. In the setting of categorical universal algebra/ categorical model theory it examines algebraic and categorical smallness conditions on objects in categories of algebras and the relationships between them as considered e. g. in [GU71, BH76, AR94]. Parts of this thesis are being published in [J98b].