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Solving Recursive Domain Equations with Enriched Categories
, 1994
"... Both preorders and metric spaces have been used at various times as a foundation for the solution of recursive domain equations in the area of denotational semantics. In both cases the central theorem states that a `converging' sequence of `complete' domains/spaces with `continuous' ..."
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Both preorders and metric spaces have been used at various times as a foundation for the solution of recursive domain equations in the area of denotational semantics. In both cases the central theorem states that a `converging' sequence of `complete' domains/spaces with `continuous' retraction pairs between them has a limit in the category of complete domains/spaces with retraction pairs as morphisms. The preorder version was discovered first by Scott in 1969, and is referred to as Scott's inverse limit theorem. The metric version was mainly developed by de Bakker and Zucker and refined and generalized by America and Rutten. The theorem in both its versions provides the main tool for solving recursive domain equations. The proofs of the two versions of the theorem look astonishingly similar, but until now the preconditions for the preorder and the metric versions have seemed to be fundamentally different. In this thesis we establish a more general theory of domains based on the noti...
Isomorphisms and Splitting of Idempotents in Semicategories
"... denoted by (f; g) 7! fg, subject to the condition that d(fg) = dg, c(fg) = cf , and df = cg whenever fg is defined (these structures are slightly more general than C. Ehresmann's multiplicative graphs, cf. [5]). The nodes of A, i.e. the elements of d[A] = c[A], are called objects or identities ..."
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denoted by (f; g) 7! fg, subject to the condition that d(fg) = dg, c(fg) = cf , and df = cg whenever fg is defined (these structures are slightly more general than C. Ehresmann's multiplicative graphs, cf. [5]). The nodes of A, i.e. the elements of d[A] = c[A], are called objects or identities (notation: Ident(A)) and are denoted by capital letters A, B etc. or in the form id A , id B etc.; the elements of A are called morphisms. A functor between composition graphs is a graph morphism F such that FfFg is defined and equal to F (fg) whenever fg is defined. A composition graph is called identitive if the terms fdf and cff are equal to f whenever they are defined, and strongly identitive if, moreover, these terms are always defined. Note that each identitive composition graph A can be made strongly identitive by defining additional compositions; the arising composition graph is called the strongly identitive modification of A. A composite fg is called pseudodefined in A if it is define
CATEGORICAL DATASPECIFICATIONS
, 1995
"... We introduce MDsketches, which are a particular kind of Finite Sum sketches. Two interesting results about MDsketches are proved. First, we show that, given two MDsketches, it is algorithmically decidable whether their model categories are equivalent. Next we show that dataspecifications, as use ..."
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We introduce MDsketches, which are a particular kind of Finite Sum sketches. Two interesting results about MDsketches are proved. First, we show that, given two MDsketches, it is algorithmically decidable whether their model categories are equivalent. Next we show that dataspecifications, as used in databasedesign and software engineering, can be translated to MDsketches. As a corollary, we obtain that equivalence of dataspecifications is decidable.
What is a differential partial combinatory algebra?
, 2011
"... In this thesis we combine Turing categories with Cartesian left additive restriction categories and again with differential restriction categories. The result of the first combination is a new structure which models nondeterministic computation. The result of the second combination is a structure wh ..."
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In this thesis we combine Turing categories with Cartesian left additive restriction categories and again with differential restriction categories. The result of the first combination is a new structure which models nondeterministic computation. The result of the second combination is a structure which models the notion of linear resource consumption. We also study the structural background required to understand what new features Turing structure should have in light of addition and differentiation – most crucial to this development is the way in which idempotents split. For the combination of Turing categories with Cartesian left additive restriction categories we will also provide a model.
TIGHTLY BOUNDED COMPLETIONS
"... Abstract. By a ‘completion ’ on a 2category K we mean here an idempotent pseudomonad on K. We are particularly interested in pseudomonads that arise from KZdoctrines. Motivated by a question of Lawvere, we compare the Cauchy completion [23], defined in the setting of VCat for V a symmetric monoida ..."
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Abstract. By a ‘completion ’ on a 2category K we mean here an idempotent pseudomonad on K. We are particularly interested in pseudomonads that arise from KZdoctrines. Motivated by a question of Lawvere, we compare the Cauchy completion [23], defined in the setting of VCat for V a symmetric monoidal closed category, with the Grothendieck completion [7], defined in the setting of SIndexed Cat for S a topos. To this end we introduce a unified setting (‘indexed enriched category theory’) in which to formulate and study certain properties of KZdoctrines. We find that, whereas all of the KZdoctrines that are relevant to this discussion (Karoubi, Cauchy, Stack, Grothendieck) may be regarded as ‘bounded’, only the Cauchy and the Grothendieck completions are ‘tightly bounded ’ – two notions that we introduce and study in this paper. Tightly bounded KZdoctrines are shown to be idempotent. We also show, in a different approach to answering the motivating question, that the Cauchy completion (defined using ‘distributors ’ [2]) and the Grothendieck completion (defined using ‘generalized functors’ [21]) are actually equivalent constructions1.
A note on Cauchy completeness for preorders
"... In this paper, we study the notion of Cauchycomplete preorder in a regular category, following work in [CS86], introducing the logic of a regular category. We give a different, stronger characterization than in loc.cit. for those preorders. Using this, we provide a new construction of the Cauchyco ..."
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In this paper, we study the notion of Cauchycomplete preorder in a regular category, following work in [CS86], introducing the logic of a regular category. We give a different, stronger characterization than in loc.cit. for those preorders. Using this, we provide a new construction of the Cauchycompletion in a exact category. AMS Classification: 18A40, 03G30.