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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...
SProc Categorically
 in: Proceedings CONCUR'94 (SpringerVerlag
, 1994
"... . We provide a systematic reconstruction of Abramsky's category SProc of synchronous processes [Abr93]: SProc is isomorphic to a span category on a category of traces. The significance of the work is twofold: It shows that the original presentation of SProc in mixed formulations is unneces ..."
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. We provide a systematic reconstruction of Abramsky's category SProc of synchronous processes [Abr93]: SProc is isomorphic to a span category on a category of traces. The significance of the work is twofold: It shows that the original presentation of SProc in mixed formulations is unnecessary  a simple categorical description exists. Furthermore, the techniques employed in the reconstruction suggest a general method of obtaining process categories with structure similar to SProc. In particular, the method of obtaining bisimulation equivalence in our setting, which represents an extension of the work of Joyal, Nielsen and Winskel [JNW93], has natural application in many settings. 1 Introduction In [Abr93], Abramsky proposed a new paradigm for the semantics of computation, interaction categories, where the following substitutions are made: Denotational semantics Categories Interaction categories Domains objects Interface specifications Continuous functions maps Commun...
2Categorical Specification of Partial Algebras
 Recent Trends in Data Type Specification, Proc. 9th Workshop on Specification of Abstract Data Types, Caldes de Malavella
, 1992
"... The purpose of this paper is to present a short survey of possible results of an application of general concepts from categorical algebra to the specification of partial algebras with conditional existence equations. The general concept, which models theories (= formulas and equivalence classes of t ..."
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The purpose of this paper is to present a short survey of possible results of an application of general concepts from categorical algebra to the specification of partial algebras with conditional existence equations. The general concept, which models theories (= formulas and equivalence classes of terms) as categories, is extended to 2categories, such that rewriting between terms can be made explicit. To make clear the benefits of such an approach the results are presented in the usual terminology of algebraic specifications. 1 Introduction The aim of this paper is to use concepts from categorical algebra to obtain a clear and extendable description of partial algebras and their specification. For the more general case, categorical algebra provides such a clear and extendable methodology, which could briefly be described as follows. ffl A theory is a category with a certain structure, called the syntactic category. (E.g. an equational theory of total many sorted operations is a cat...
GraphBased Logic snd Sketches II: FiniteProduct Categories and Equational Logic
, 1997
"... ion. In this case the equational rule of inference reads Given a set of typed variables and x 2 Vbl[S] n V , e = V e 0 e = V [fxg e 0 We define ø : = Type[e] = Type[e 0 ] and oe : = Type[x], and f : P ! ø : = Arr [e; V; ø ] f 0 : P ! ø : = Arr \Theta e 0 ; V; ø g : Q ! ø : = Arr [e; ..."
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ion. In this case the equational rule of inference reads Given a set of typed variables and x 2 Vbl[S] n V , e = V e 0 e = V [fxg e 0 We define ø : = Type[e] = Type[e 0 ] and oe : = Type[x], and f : P ! ø : = Arr [e; V; ø ] f 0 : P ! ø : = Arr \Theta e 0 ; V; ø g : Q ! ø : = Arr [e; V [ fxg; ø ] g 0 : Q ! ø : = Arr \Theta e 0 ; V [ fxg; ø Using Lemma 4.4.1, we may choose a map h : Y (V [ fxg) ! V such that g = f ffi h and g 0 = f 0 ffi h. Thus coded as arrows the rule reads f = f 0 f ffi h = f 0 ffi h GRAPHBASED LOGIC AND SKETCHES II 23 5.6. Substitutivity. Given a set V of typed variables, x 2 V , and expressions u and u 0 for which Type[x] = Type[u] = Type[u 0 ] and Type[e] = Type[e 0 ] = ø , e = V e 0 u =W u 0 e[x / u] = V nfxg[W e 0 [x / u 0 ] We already have the representations f : = Arr [e; V; ø ] = Sep[e] Par[e] Dia [e; V; ø ] f 0 : = Arr \Theta e 0 ; V; ø = Sep[e 0 ] Par[e 0 ] Dia[e 0 ; V; ø ] g : = Arr [u...
GRAPHBASED LOGIC AND SKETCHES II: FINITEPRODUCT CATEGORIES AND EQUATIONAL LOGIC
, 1996
"... Abstract. It is shown that the proof theory for sketches and forms provided in [Bagchi and Wells, 1996] is strong enough to produce all the theorems of the entailment system for multisorted equational logic provided in [Goguen and Meseguer, 1982]. 1. ..."
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Abstract. It is shown that the proof theory for sketches and forms provided in [Bagchi and Wells, 1996] is strong enough to produce all the theorems of the entailment system for multisorted equational logic provided in [Goguen and Meseguer, 1982]. 1.