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On Bunched Typing
, 2002
"... We study a typing scheme derived from a semantic situation where a single category possesses several closed structures, corresponding to dierent varieties of function type. In this scheme typing contexts are trees built from two (or more) binary combining operations, or in short, bunches. Bunched ..."
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Cited by 33 (2 self)
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We study a typing scheme derived from a semantic situation where a single category possesses several closed structures, corresponding to dierent varieties of function type. In this scheme typing contexts are trees built from two (or more) binary combining operations, or in short, bunches. Bunched typing and its logical counterpart, bunched implications, have arisen in joint work of the author and David Pym. The present paper gives a basic account of the type system, and then focusses on concrete models that illustrate how it may be understood in terms of resource access and sharing. The most
Modelling Conditional Rewriting Logic in Structured Categories
, 1996
"... We reformulate and generalize the functorial model of Meseguer's conditional full rewriting logic by using inserter, a weighted limit in 2categories. Indeed 2categories are categories enriched in Cat. Therefore this method also can be extended to sesquicategories and other enriched categories, wi ..."
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Cited by 3 (0 self)
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We reformulate and generalize the functorial model of Meseguer's conditional full rewriting logic by using inserter, a weighted limit in 2categories. Indeed 2categories are categories enriched in Cat. Therefore this method also can be extended to sesquicategories and other enriched categories, with which we can model various aspects of rewritings and strategies. 1 Introduction J. Meseguer introduced his conditional rewriting logic in [12,13], and gave a functorial semantics for it in [13] and a 2algebraic theory semantics in [12]. But his treatments of conditionals are not fully 2categorical because he essentially uses subequalizers. As Lambek pointed out in his original paper [11], a subequalizer is not a 2categorical limit in the sense that this does not require the 2dimensional universality. In 2category (or enriched category) theory, the weighted limit was proposed as the more suitable notion than that of 2limit [7,1]. Our first approach is to use a weighted limit in 2ca...
Sketches: Outline with References
 Dept. of Computer Science, Katholieke Universiteit Leuven
, 1994
"... This document is an outline of the theory of sketches with pointers to the literature. An extensive bibliography is given. Some coverage is given to related areas such as algebraic theories, categorial model theory and categorial logic as well. An appendix beginning on page 11 provides definitions o ..."
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Cited by 2 (0 self)
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This document is an outline of the theory of sketches with pointers to the literature. An extensive bibliography is given. Some coverage is given to related areas such as algebraic theories, categorial model theory and categorial logic as well. An appendix beginning on page 11 provides definitions of some of the less standard terms used in the paper, but the reader is expected to be familiar with the basic ideas of category theory. A rough machine generated index begins on page 21. I would have liked to explain the main ideas of all the papers referred to herein, but I am not familiar enough with some of them to do that. It seemed more useful to be inclusive, even if many papers were mentioned without comment. One consequence of this is that the discussions in this document often go into more detail about the papers published in North America than about those published elsewhere. The DVI file for this article is available by anonymous FTP from ftp.cwru.edu in the directory
Free Products of Higher Operad Algebras
, 909
"... Abstract. One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product of 2categories. In this paper we continue the developments of [3] and [2] by understanding the natural generalisations of Gray’s little brother, ..."
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Cited by 2 (2 self)
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Abstract. One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product of 2categories. In this paper we continue the developments of [3] and [2] by understanding the natural generalisations of Gray’s little brother, the funny tensor product of categories. In fact we exhibit for any higher categorical structure definable by an noperad in the sense of Batanin [1], an analogous tensor product which forms a symmetric monoidal closed structure
An Algebraic Foundation for Graphbased Diagrams in Computing
"... We develop an algebraic foundation for some of the graphbased structures underlying a variety of popular diagrammatic notations for the specication, modelling and programming of computing systems. Using hypergraphs and higraphs as leading examples, a locally ordered category Graph(C) of graphs in a ..."
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Cited by 1 (0 self)
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We develop an algebraic foundation for some of the graphbased structures underlying a variety of popular diagrammatic notations for the specication, modelling and programming of computing systems. Using hypergraphs and higraphs as leading examples, a locally ordered category Graph(C) of graphs in a locally ordered category C is dened and endowed with symmetric monoidal closed structure. Two other operations on higraphs and variants, selected for relevance to computing applications, are generalised in this setting. 1
Bireflectivity
, 1996
"... Motivated by a model for syntactic control of interference, we introduce a general categorical concept of bireflectivity. Bireflective subcategories of a category A are subcategories with left and right adjoint equal, subject to a coherence condition. We characterize them in terms of splitidempoten ..."
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Motivated by a model for syntactic control of interference, we introduce a general categorical concept of bireflectivity. Bireflective subcategories of a category A are subcategories with left and right adjoint equal, subject to a coherence condition. We characterize them in terms of splitidempotent natural transformations on id A . In the special case that A is a presheaf category, we characterize them in terms of the domain, and prove that any bireflective subcategory of A is itself a presheaf category. We define diagonal structure on a symmetric monoidal category which is still more general than asking the tensor product to be the categorical product. We then obtain a bireflective subcategory of [C op ; Set] and deduce results relating its finite product structure with the monoidal structure of [C op ; Set] determined by that of C. We also investigate the closed structure. Finally, for completeness, we give results on bireflective subcategories in Rel(A), the category of relati...
Additive closed symmetric monoidal structures on Rmodules
"... In this paper, we classify additive closed symmetric monoidal structures on the category of left Rmodules by using Watts ’ theorem. An additive closed symmetric monoidal structure is equivalent to an Rmodule ΛA,B equipped with two commuting right Rmodule structures represented by the symbols A an ..."
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In this paper, we classify additive closed symmetric monoidal structures on the category of left Rmodules by using Watts ’ theorem. An additive closed symmetric monoidal structure is equivalent to an Rmodule ΛA,B equipped with two commuting right Rmodule structures represented by the symbols A and B, an Rmodule K to serve as the unit, and certain isomorphisms. We use this result to look at simple cases. We find rings R for which there are no additive closed symmetric monoidal structures on Rmodules, for which there is exactly one (up to isomorphism), for which there are exactly seven, and for which there are a proper class of isomorphism classes of such structures. We also prove some general structual results; for example, we prove that the unit K must always be a finitely generated Rmodule. Key words: symmetric monoidal, closed symmetric monoidal, module 2000 MSC: 18D10, 16D90
Abstract
, 906
"... In this paper, we classify additive closed symmetric monoidal structures on the category of left Rmodules by using Watts ’ theorem. An additive closed symmetric monoidal structure is equivalent to an Rmodule ΛA,B equipped with two commuting right Rmodule structures represented by the symbols A an ..."
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In this paper, we classify additive closed symmetric monoidal structures on the category of left Rmodules by using Watts ’ theorem. An additive closed symmetric monoidal structure is equivalent to an Rmodule ΛA,B equipped with two commuting right Rmodule structures represented by the symbols A and B, an Rmodule K to serve as the unit, and certain isomorphisms. We use this result to look at simple cases. We find rings R for which there are no additive closed symmetric monoidal structures on Rmodules, for which there is exactly one (up to isomorphism), for which there are exactly seven, and for which there are a proper class of isomorphism classes of such structures. We also prove some general structual results; for example, we prove that the unit K must always be a finitely generated Rmodule. Key words: symmetric monoidal, closed symmetric monoidal, module 2000 MSC: 18D10, 16D90
Universal Properties of Impure Programming Languages
"... We investigate impure, callbyvalue programming languages. Our first language only has variables and letbinding. Its equational theory is a variant of Lambek’s theory of multicategories that omits the commutativity axiom. We demonstrate that type constructions for impure languages — products, sums ..."
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We investigate impure, callbyvalue programming languages. Our first language only has variables and letbinding. Its equational theory is a variant of Lambek’s theory of multicategories that omits the commutativity axiom. We demonstrate that type constructions for impure languages — products, sums and functions — can be characterized by universal properties in the setting of ‘premulticategories’, multicategories where the commutativity law may fail. This leads us to new, universal characterizations of two earlier equational theories of impure programming languages: the premonoidal categories of Power and Robinson, and the monadbased models of Moggi. Our analysis thus puts these earlier abstract ideas on a canonical foundation, bringing them to a new, syntactic level. F.3.2 [Semantics of Pro