Categorical Data � Specifications

We introduce MD sketches which are a particular kind of Finite Sum sketches Two interesting results about MD sketches are proved First we show that given two MD sketches it is algorithmically decidable whether their model categories are equivalent Next we show that data speci cations as used in database design and software engineering can be translated to MD sketches As a corollary we obtain that equivalence of data speci cations is decidable Introduction This paper is about the correspondence between two kinds of speci cation mechanisms sketches and data speci cations Sketches were invented by Charles Ehresmann in the late sixties for the purpose of specifying algebraic structures Since then they have been studied intensively An outline of the theory of sketches with a large number of references can be found in It is well known that the categories of models of sketches in Set are exactly the accessible categories An accessible category is complete i it is the model category of a Limit sketch i it is a locally presentable category Locally presentable categories have the pleasant property that it is possible to de ne a canonical Limit sketch with the given category as model category All these results can be found in For other accessible categories it is not possible to de ne a canonical sketch with the given category as model category In a number of recent papers various equivalence preserving transformations between sketches are studied These transformations preserve up to equivalence the model category in Set but do not necessarily preserve model categories in other categories than Set Hence it is clear that model categories in Set do not determine the sketch in the same way as this is the case for Limit sketches In this paper we are interested in algorithmic decidability of the equivalence problem and we will prove that the equivalence problem is decidable for the class of MD sketches Semantic data speci cations like Chen s Entity Relationship diagrams on the other hand have been used for many years in the early stages of database design However most speci cation systems used in practice are of an informal nature and have little or no mathematical foundation This makes it impossible to prove interesting results about them formalized these speci cation systems using categorical language and proved The rst author is a Research Assistant of the Belgian Fund for Scienti c Research Received by the editors June Published on November Mathematics Subject Classi cation A C P