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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
Categorical structures enriched in a quantaloid: Categories, distributions and functors
 Theory Appl. Categ
"... We study the different guises of the projective objects in Cocont(Q): they are the “completely distributive ” cocomplete Qcategories (the left adjoint to the Yoneda embedding admits a further left adjoint); equivalently, they are the “totally continuous ” cocomplete Qcategories (every object is th ..."
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We study the different guises of the projective objects in Cocont(Q): they are the “completely distributive ” cocomplete Qcategories (the left adjoint to the Yoneda embedding admits a further left adjoint); equivalently, they are the “totally continuous ” cocomplete Qcategories (every object is the supremum of the presheaf of objects “totally below ” it); and also are they the Qcategories of regular presheaves on a regular Qsemicategory. As a particular case, the Qcategories of presheaves on a Qcategory are precisely the “totally algebraic” cocomplete Qcategories (every object is the supremum of the “totally compact” objects below it). We think that these results should be part of a yettobeunderstood “quantaloidenriched domain theory”. 1
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.
A Note on Actions of a Monoidal Category
, 2001
"... An action : V A! A of a monoidal category V on a category A corresponds to a strong monoidal functor F : V ! [A; A] into the monoidal category of endofunctors of A. In many practical cases, the ordinary functor f : V ! [A; A] underlying the monoidal F has a right adjoint g; and when this is so, ..."
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An action : V A! A of a monoidal category V on a category A corresponds to a strong monoidal functor F : V ! [A; A] into the monoidal category of endofunctors of A. In many practical cases, the ordinary functor f : V ! [A; A] underlying the monoidal F has a right adjoint g; and when this is so, F itself has a right adjoint G as a monoidal functorso that, passing to the categories of monoids (also called \algebras") in V and in [A; A], we have an adjunction MonF a MonG between the category MonV of monoids in V and the category Mon[A; A] = MndA of monads on A. We give sucient conditions for the existence of the right adjoint g, which involve the existence of right adjoints for the functors X { and { A, and make A (at least when V is symmetric and closed) into a tensored and cotensored Vcategory A. We give explicit formulae, as large ends, for the right adjoints g and MonG, and also for some related right adjoints, when they exist; as well as another explicit expression for MonG as a large limit, which uses a new representation of any monad as a (large) limit of monads of two special kinds, and an analogous result for general endofunctors.
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...
Linearlyused state in models of callbyvalue
"... Abstract. We investigate the phenomenon that every monad is a linear state monad. We do this by studying a fullycomplete statepassing translation from an impure callbyvalue language to a new linear type theory: the enriched callbyvalue calculus. The results are not specific to store, but can b ..."
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Abstract. We investigate the phenomenon that every monad is a linear state monad. We do this by studying a fullycomplete statepassing translation from an impure callbyvalue language to a new linear type theory: the enriched callbyvalue calculus. The results are not specific to store, but can be applied to any computational effect expressible using algebraic operations, even to effects that are not usually thought of as stateful. There is a bijective correspondence between generic effects in the source language and state access operations in the enriched callbyvalue calculus. From the perspective of categorical models, the enriched callbyvalue calculus suggests a refinement of the traditional Kleisli models of effectful callbyvalue languages. The new models can be understood as enriched adjunctions. 1
Towards `dynamic domains': totally continuous cocomplete Qcategories, Theoret
 Comput. Sci
, 2007
"... Abstract. It is common practice in both theoretical computer science and theoretical physics to describe the (static) logic of a system by means of a complete lattice. When formalizing the dynamics of such a system, the updates of that system organize themselves quite naturally in a quantale, or mor ..."
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Abstract. It is common practice in both theoretical computer science and theoretical physics to describe the (static) logic of a system by means of a complete lattice. When formalizing the dynamics of such a system, the updates of that system organize themselves quite naturally in a quantale, or more generally, a quantaloid. In fact, we are lead to consider cocomplete quantaloidenriched categories as fundamental mathematical structure for a dynamic logic common to both computer science and physics. Here we explain the theory of totally continuous cocomplete categories as generalization of the wellknown theory of totally continuous suplattices. That is to say, we undertake some first steps towards a theory of “dynamic domains”.
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 ...
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