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On PropertyLike Structures
, 1997
"... A category may bear many monoidal structures, but (to within a unique isomorphism) only one structure of "category with finite products". To capture such distinctions, we consider on a 2category those 2monads for which algebra structure is essentially unique if it exists, giving a precis ..."
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Cited by 13 (3 self)
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A category may bear many monoidal structures, but (to within a unique isomorphism) only one structure of "category with finite products". To capture such distinctions, we consider on a 2category those 2monads for which algebra structure is essentially unique if it exists, giving a precise mathematical definition of "essentially unique" and investigating its consequences. We call such 2monads propertylike. We further consider the more restricted class of fully propertylike 2monads, consisting of those propertylike 2monads for which all 2cells between (even lax) algebra morphisms are algebra 2cells. The consideration of lax morphisms leads us to a new characterization of those monads, studied by Kock and Zoberlein, for which "structure is adjoint to unit", and which we now call laxidempotent 2monads: both these and their colaxidempotent duals are fully propertylike. We end by showing that (at least for finitary 2monads) the classes of propertylikes, fully propertylike...
Categorical Term Rewriting: Monads and Modularity
 University of Edinburgh
, 1998
"... Term rewriting systems are widely used throughout computer science as they provide an abstract model of computation while retaining a comparatively simple syntax and semantics. In order to reason within large term rewriting systems, structuring operations are used to build large term rewriting syste ..."
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Cited by 11 (6 self)
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Term rewriting systems are widely used throughout computer science as they provide an abstract model of computation while retaining a comparatively simple syntax and semantics. In order to reason within large term rewriting systems, structuring operations are used to build large term rewriting systems from smaller ones. Of particular interest is whether key properties are modular, thatis,ifthe components of a structured term rewriting system satisfy a property, then does the term rewriting system as a whole? A body of literature addresses this problem, but most of the results and proofs depend on strong syntactic conditions and do not easily generalize. Although many specific modularity results are known, a coherent framework which explains the underlying principles behind these results is lacking. This thesis posits that part of the problem is the usual, concrete and syntaxoriented semantics of term rewriting systems, and that a semantics is needed which on the one hand elides unnecessary syntactic details but on the other hand still possesses enough expressive power to model the key concepts arising from
COMPACTLY ACCESSIBLE CATEGORIES AND QUANTUM KEY DISTRIBUTION
"... Abstract. Compact categories have lately seen renewed interest via applications to quantum physics. Being essentially finitedimensional, they cannot accomodate (co)limitbased constructions. For example, they cannot capture protocols such as quantum key distribution, that rely on the law of large n ..."
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Cited by 2 (2 self)
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Abstract. Compact categories have lately seen renewed interest via applications to quantum physics. Being essentially finitedimensional, they cannot accomodate (co)limitbased constructions. For example, they cannot capture protocols such as quantum key distribution, that rely on the law of large numbers. To overcome this limitation, we introduce the notion of a compactly accessible category, relying on the extra structure of a factorisation system. This notion allows for infinite dimension while retaining key properties of compact categories: the main technical result is that the choiceofduals functor on the compact
ON PROPERTYLIKE STRUCTURES G. M. KELLY
"... ABSTRACT. A category may bear many monoidal structures, but (to within a unique isomorphism) only one structure of “category with finite products”. To capture such distinctions, we consider on a 2category those 2monads for which algebra structure is essentially unique if it exists, giving a precis ..."
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ABSTRACT. A category may bear many monoidal structures, but (to within a unique isomorphism) only one structure of “category with finite products”. To capture such distinctions, we consider on a 2category those 2monads for which algebra structure is essentially unique if it exists, giving a precise mathematical definition of “essentially unique ” and investigating its consequences. We call such 2monads propertylike. We further consider the more restricted class of fully propertylike 2monads, consisting of those propertylike 2monads for which all 2cells between (even lax) algebra morphisms are algebra 2cells. The consideration of lax morphisms leads us to a new characterization of those monads, studied by Kock and Zöberlein, for which “structure is adjoint to unit”, and which we now call laxidempotent 2monads: both these and their colaxidempotent duals are fully propertylike. We end by showing that (at least for finitary 2monads) the classes of propertylikes, fully propertylikes, and laxidempotents are each coreflective among all 2monads. 1.
THE HEART OF A COMBINATORIAL MODEL CATEGORY
"... Abstract. We show that every small model category that satisfies certain size conditions can be completed to yield a combinatorial model category, and conversely, every combinatorial model category arises in this way. We will also see that these constructions preserve right properness and compatibi ..."
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Abstract. We show that every small model category that satisfies certain size conditions can be completed to yield a combinatorial model category, and conversely, every combinatorial model category arises in this way. We will also see that these constructions preserve right properness and compatibility with simplicial enrichment. Along the way, we establish some technical results on the index of accessibility of various constructions on accessible categories, which may be of independent interest.