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Variations on Algebra: monadicity and generalisations of equational theories
 Formal Aspects of Computing
, 2001
"... this paper the author was partially supported by an SERC/EPSRC Advanced Research Fellowship, EPSRC Research grant GR/L54639, and EU Working Group APPSEM ..."
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this paper the author was partially supported by an SERC/EPSRC Advanced Research Fellowship, EPSRC Research grant GR/L54639, and EU Working Group APPSEM
Enriched Lawvere Theories
"... We define the notion of enriched Lawvere theory, for enrichment over a monoidal biclosed category V that is locally finitely presentable as a closed category. We prove that the category of enriched Lawvere theories is equivalent to the category of finitary monads on V. Morever, the Vcategory of mod ..."
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We define the notion of enriched Lawvere theory, for enrichment over a monoidal biclosed category V that is locally finitely presentable as a closed category. We prove that the category of enriched Lawvere theories is equivalent to the category of finitary monads on V. Morever, the Vcategory of models of a Lawvere Vtheory is equivalent to the Vcategory of algebras for the corresponding Vmonad. This all extends routinely to local presentability with respect to any regular cardinal. We finally consider the special case where V is Cat, and explain how the correspondence extends to pseudo maps of algebras.
Lax Naturality Through Enrichment
, 1995
"... We develop the relationship between algebraic structure and monads enriched over the monoidal biclosed category LocOrd l of small locally ordered categories, with closed structure given by Lax(A; B). We state the theorem, give a series of examples, and incorporate an account of sketches and cont ..."
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We develop the relationship between algebraic structure and monads enriched over the monoidal biclosed category LocOrd l of small locally ordered categories, with closed structure given by Lax(A; B). We state the theorem, give a series of examples, and incorporate an account of sketches and contravariance into the theory. This was motivated by C.A.R. Hoare's use of category theoretic structures to model data refinement. 1 Introduction In 1987, C.A.R. Hoare wrote a draft paper, "Data refinement in a categorical setting" [10] in which he used category theory to provide an abstract formalism for his development of data refinement over the previous twenty years [9]. The notion of data refinement is central to the programming method called stepwise refinement proposed by Wirth [19], and gave rise to work on abstract data types such as the IOTA programming system developed by Nakajima, Honda and Nakahara [16]. As Hoare said in [10], there was evidently a unified body of category theo...
Data Refinement and Algebraic Structure
, 1996
"... We recall Hoare's formulation of data refinement in terms of upward, downward and total simulations between locally ordered functors from the structured locally ordered category generated by a programming language with an abstract data type to a semantic locally ordered category: we use a simple imp ..."
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We recall Hoare's formulation of data refinement in terms of upward, downward and total simulations between locally ordered functors from the structured locally ordered category generated by a programming language with an abstract data type to a semantic locally ordered category: we use a simple imperative language with a data type for stacks as leading example. We give a unified category theoretic account of the sort of structures on a category that allow upward simulation to extend from ground types and ground programs to all types and programs of the language. This answers a question of Hoare about the category theory underlying his constructions. It involves a careful study of algebraic structure on the category of small locally ordered categories, and a new definition of sketch of such structure. This is accompanied by a range of detailed examples. We extend that analysis to total simulations for modelling constructors of mixed variance such as higher order types. 1 Introduction ...
A General Completeness Result in Refinement
 in: Proceedings of the 14th International Workshop on Algebraic Development Techniques, no. 1827 in Lecture Notes in Computer Science
, 1999
"... . In a paper in 1986, Hoare, He and Sanders proposed a formulation of refinement for a system equivalent to the #calculus using a relation based semantics. To give a proof method to show that one program is a refinement of another, they introduced downward simulation and upward simulation, but the ..."
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. In a paper in 1986, Hoare, He and Sanders proposed a formulation of refinement for a system equivalent to the #calculus using a relation based semantics. To give a proof method to show that one program is a refinement of another, they introduced downward simulation and upward simulation, but the proof method based upon either of them is not complete with respect to their notion of refinement, so they claimed "joint" completeness based upon both notions of simulation with respect to their notion of refinement. We give a new definition of refinement in terms of structure respecting lax transformations, and show that the proof method based upon downward simulation is complete with respect to this notion of refinement. Although our theory works for the #calculus, we present the result for the calculus to make the presentation simpler. We use results in enriched category theory to show this, and the central notion here is that of algebraic structure on locally ordered categories, not o...
Sketches
 JOURNAL OF PURE AND APPLIED ALGEBRA
, 1999
"... We generalise the notion of sketch. For any locally nitely presentable category, one can speak of algebraic structure on the category, or equivalently, a finitary monad on it. For any such finitary monad, we de ne the notions of sketch and strict model and prove that any sketch has a generic stric ..."
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We generalise the notion of sketch. For any locally nitely presentable category, one can speak of algebraic structure on the category, or equivalently, a finitary monad on it. For any such finitary monad, we de ne the notions of sketch and strict model and prove that any sketch has a generic strict model on it. This is all done with enrichment in any monoidal biclosed