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Complete Cuboidal Sets in Axiomatic Domain Theory (Extended Abstract)
 In Proceedings of 12th Annual Symposium on Logic in Computer Science
, 1997
"... ) Marcelo Fiore !mf@dcs.ed.ac.uk? Gordon Plotkin y !gdp@dcs.ed.ac.uk? John Power !ajp@dcs.ed.ac.uk? Department of Computer Science Laboratory for Foundations of Computer Science University of Edinburgh, The King's Buildings Edinburgh EH9 3JZ, Scotland Abstract We study the enrichment of ..."
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) Marcelo Fiore !mf@dcs.ed.ac.uk? Gordon Plotkin y !gdp@dcs.ed.ac.uk? John Power !ajp@dcs.ed.ac.uk? Department of Computer Science Laboratory for Foundations of Computer Science University of Edinburgh, The King's Buildings Edinburgh EH9 3JZ, Scotland Abstract We study the enrichment of models of axiomatic domain theory. To this end, we introduce a new and broader notion of domain, viz. that of complete cuboidal set, that complies with the axiomatic requirements. We show that the category of complete cuboidal sets provides a general notion of enrichment for a wide class of axiomatic domaintheoretic structures. Introduction The aim of Axiomatic Domain Theory (ADT) is to provide a conceptual understanding of why domains are adequate as mathematical models of computation. (For a discussion see [12, x Axiomatic Domain Theory ].) The approach taken is to axiomatise the structure needed on a category so that its objects can be considered as domains, and its maps as continuous...
Weak Factorization Systems and Topological Functors
 Appl. Categ. Struct
, 2000
"... Weak factorization systems, important in homotopy theory, are related to injective objects in comma{categories. Our main result is that full functors and topological functors form a weak factorization system in the category of small categories, and that this is not cobrantly generated. We also prese ..."
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Cited by 16 (6 self)
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Weak factorization systems, important in homotopy theory, are related to injective objects in comma{categories. Our main result is that full functors and topological functors form a weak factorization system in the category of small categories, and that this is not cobrantly generated. We also present a weak factorization system on the category of posets which is not cobrantly generated. No such weak factorization systems were known until recently. This answers an open problem posed by M. Hovey.
Using Limits of Parchments to Systematically Construct Institutions of Partial Algebras
 Recent Trends in Data Type Specifications. 11th Workshop on Specification of Abstract Data Types, volume 1130 of Lecture Notes in Computer Science
, 1996
"... this paper, so we leave them out here. Thus we can apply the idea of combining things via colimits to institutions themselves, with the special point that we have to take limits here instead of colimits. Taking limits in CAT results in categories of "amalgamated objects", i. e. we put signatures an ..."
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Cited by 15 (5 self)
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this paper, so we leave them out here. Thus we can apply the idea of combining things via colimits to institutions themselves, with the special point that we have to take limits here instead of colimits. Taking limits in CAT results in categories of "amalgamated objects", i. e. we put signatures and models together at the level of single objects. In contrast to this, sentences are combined with colimits in Set (due to the contravariant direction of the sentence component). That is, sets of sentences are combined. To show how this works, we introduce some wellknown institutions and morphisms between them.
Extending Development Graphs With Hiding
, 2001
"... Development graphs are a tool for dealing with structured specifications in a formal program development in order to ease the management of change and reusing proofs. In this work, we extend development graphs with hiding (e.g. hidden operations). Hiding is a particularly difficult to realize operat ..."
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Cited by 15 (10 self)
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Development graphs are a tool for dealing with structured specifications in a formal program development in order to ease the management of change and reusing proofs. In this work, we extend development graphs with hiding (e.g. hidden operations). Hiding is a particularly difficult to realize operation, since it does not admit such a good decomposition of the involved specifications as other structuring operations do. We develop both a semantics and proof rules for development graphs with hiding. The rules are proven to be sound, and also complete relative to an oracle for conservative extensions. We also show that an absolute complete set of rules cannot exist. The whole framework is developed in a way independent of the underlying logical system (and thus also does not prescribe the nature of the parts of a specification that may be hidden).
LeftDetermined Model Categories and Universal Homotopy Theories
"... We say that a model category is leftdetermined if the weak equivalences are generated (in a sense specified below) by the cofibrations. While the model category of simplicial sets is not leftdetermined, we show that its nonoriented variant, the category of symmetric simplicial sets (in the sense ..."
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Cited by 13 (2 self)
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We say that a model category is leftdetermined if the weak equivalences are generated (in a sense specified below) by the cofibrations. While the model category of simplicial sets is not leftdetermined, we show that its nonoriented variant, the category of symmetric simplicial sets (in the sense of Lawvere and Grandis) carries a natural leftdetermined model category structure. This is used to give another and, as we believe simpler, proof of a recent result of D. Dugger about universal homotopy theories. 1
The strength of Mac Lane set theory
 ANNALS OF PURE AND APPLIED LOGIC
, 2001
"... SAUNDERS MAC LANE has drawn attention many times, particularly in his book Mathematics: Form and Function, to the system ZBQC of set theory of which the axioms are Extensionality, Null Set, Pairing, ..."
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Cited by 12 (1 self)
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SAUNDERS MAC LANE has drawn attention many times, particularly in his book Mathematics: Form and Function, to the system ZBQC of set theory of which the axioms are Extensionality, Null Set, Pairing,
Higher fundamental functors for simplicial sets, Cahiers Topologie Géom
 Diff. Catég
"... Abstract. An intrinsic, combinatorial homotopy theory has been developed in [G3] for simplicial complexes; these form the cartesian closed subcategory of simple presheaves in!Smp, the topos of symmetric simplicial sets, or presheaves on the category!å of finite, positive cardinals. We show here how ..."
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Cited by 11 (8 self)
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Abstract. An intrinsic, combinatorial homotopy theory has been developed in [G3] for simplicial complexes; these form the cartesian closed subcategory of simple presheaves in!Smp, the topos of symmetric simplicial sets, or presheaves on the category!å of finite, positive cardinals. We show here how this homotopy theory can be extended to the topos itself,!Smp. As a crucial advantage, the fundamental groupoid Π1:!Smp = Gpd is left adjoint to a natural functor M1: Gpd =!Smp, the symmetric nerve of a groupoid, and preserves all colimits – a strong van Kampen property. Similar results hold in all higher dimensions. Analogously, a notion of (nonreversible) directed homotopy can be developed in the ordinary simplicial topos Smp, with applications to image analysis as in [G3]. We have now a homotopy ncategory functor ↑Πn: Smp = nCat, left adjoint to a nerve Nn = nCat(↑Πn(∆[n]), –). This construction can be applied to various presheaf categories; the basic requirements seem to be: finite products of representables are finitely presentable and there is a representable 'standard interval'.
A Classification of Accessible Categories
 Max Kelly volume, J. Pure Appl. Alg
, 1996
"... For a suitable collection D of small categories, we define the Daccessible categories, generalizing the #accessible categories of Lair, Makkai, and Pare; here the ..."
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For a suitable collection D of small categories, we define the Daccessible categories, generalizing the #accessible categories of Lair, Makkai, and Pare; here the