Data types in distributive categories

Walters, R. F. C.

doi:10.1017/s0004972700003506

Cited by 18 publications

(5 citation statements)

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“…It has been observed by several authors in different ways (Walters 1989;Kasangian and Vigna 1991;Johnson and Walters 1992;Kasangian and Labella 1992;Johnstone 1992) that many set-valued functors of importance in computer science (such as the freemonoid and free-semigroup functors, and the functor that assigns to a set X the set of -labelled trees) are not representable but are 'very nearly so', in that 'the only reason' why they are not representable is their failure to preserve the terminal object -more formally, they become representable when extended to the free product-completion of their domains. It has also been observed, notably by R. F. C. Walters (Walters 1989a;Walters 1989b;Walters 1992) and by S. H. Schanuel (cf.…”

Section: Introductionmentioning

confidence: 87%

Connected limits, familial representability and Artin glueing

Carboni

Johnstone

1995

Math. Struct. Comp. Sci.

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We consider the following two properties of a functor F from a presheaf topos to the category of sets: (a) F preserves connected limits, and (b) the Artin glueing of F is again a presheaf topos. We show that these two properties are in fact equivalent. In the process, we develop a general technique for associating categorical properties of a category obtained by Artin glueing with preservation properties of the functor along which the glueing takes place. We also give a syntactic characterization of those monads on Set whose functor parts have the above properties, and whose units and multiplications are cartesian natural transformations.

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Section: Introductionmentioning

confidence: 87%

Connected limits, familial representability and Artin glueing

Carboni

Johnstone

1995

Math. Struct. Comp. Sci.

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“…Similarly, although flow-charts have been discussed entirely in terms of sums, in practice the state space of an edge of a flow chart is a product not explicit in the multigraph. A richer theory, hinted at in [6] and to be discussed in future papers, will make explicit both sums and products and the relation between them.…”

Section: Discussionmentioning

confidence: 99%

The duality between flow charts and circuits

Kasangian

Walters

1990

Bull. Austral. Math. Soc.

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This paper contains a precise description of the duality between the formal evolutions of flow charts and of circuits. In addition, it contains a new description of the free category-with-products on a multigraph as a familially representable construction. INTRODUCTIONIf A is a set of elementary actions then the elements of the free monoid A* may be thought of as formal evolutions of the set of actions. This can be generalised. Given a graph G of elementary actions the arrows in the free category -with appropriate structure -on G may again be thought of as the formal evolutions of the elementary actions, taking into account the specificity of the graph. This paper is concerned in particular with the formal evolution of both flow charts and of digital circuits. The appropriate notion of graph in both cases is multigraph. The formal evolutions of a flow chart are arrows in the free category-with-sums on a multigraph; the formal evolutions of a circuit are arrows in the free category-withproducts on a multigraph.As a result there is a precise categorical duality between the evolution of flow charts and of circuits. Roughly speaking, this is the duality between terms and trees.In addition to this duality theorem, we give a new description of the free categorywith-products on a multigraph. Diers [2] introduced the notion of a locally representable functor, that is, a functor represented by a family of objects rather than one object. Johnson and Walters, calling the notion instead familially representable functor, found further examples (see [4], [5]). A simple example is the free-monoid functor. The family of objects in this case is the natural numbers, and the free monoid on A may be thought of as {n -+ i ; n a natural number}. In section 5 we indicate that the free categorywith-products on a multigraph is similarly familially representable -an idea suggested by the well-known description in terms of families of the free category-with-products on a category.

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“…w-complete posets with least elements. The category Predom is distributive [17], i.e. it has finite products and coproducts, with the natural isomorphism…”

Section: Gz-interpretationsmentioning

confidence: 99%