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24
Categories of Containers
 In Proceedings of Foundations of Software Science and Computation Structures
, 2003
"... Abstract. We introduce the notion of containers as a mathematical formalisation of the idea that many important datatypes consist of templates where data is stored. We show that containers have good closure properties under a variety of constructions including the formation of initial algebras and f ..."
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Cited by 40 (7 self)
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Abstract. We introduce the notion of containers as a mathematical formalisation of the idea that many important datatypes consist of templates where data is stored. We show that containers have good closure properties under a variety of constructions including the formation of initial algebras and final coalgebras. We also show that containers include strictly positive types and shapely types but that there are containers which do not correspond to either of these. Further, we derive a representation result classifying the nature of polymorphic functions between containers. We finish this paper with an application to the theory of shapely types and refer to a forthcoming paper which applies this theory to differentiable types. 1
Operads In HigherDimensional Category Theory
, 2004
"... The purpose of this paper is to set up a theory of generalized operads and multicategories and to use it as a language in which to propose a definition of weak ncategory. Included is a full explanation of why the proposed definition of ncategory is a reasonable one, and of what happens when n <= 2 ..."
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Cited by 32 (2 self)
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The purpose of this paper is to set up a theory of generalized operads and multicategories and to use it as a language in which to propose a definition of weak ncategory. Included is a full explanation of why the proposed definition of ncategory is a reasonable one, and of what happens when n <= 2. Generalized operads and multicategories play other parts in higherdimensional algebra too, some of which are outlined here: for instance, they can be used to simplify the opetopic approach to ncategories expounded by Baez, Dolan and others, and are a natural language in which to discuss enrichment of categorical structures.
Reflexive Graphs and Parametric Polymorphism
, 1993
"... this paper is to understand why that is a parametric categorical model. In [10] Ma and Reynolds propose a parametricity hypothesis for a functor between categorical models of polymorphism which essentially requires that there is an extension of (a certain form of) an identity relation functor which ..."
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Cited by 18 (0 self)
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this paper is to understand why that is a parametric categorical model. In [10] Ma and Reynolds propose a parametricity hypothesis for a functor between categorical models of polymorphism which essentially requires that there is an extension of (a certain form of) an identity relation functor which preserve the model structure. There is no mention in the paper of any case when the parametricity hypothesis is satified, nor if there is a canonical completion of a category to one which satisfies the hypothesis. We shall suggest how the construction of a PLcategory of relations on a given category presented in [10] can be viewed as a "parametric completion". We shall also follow the suggestion of Ma in [9] that subtyping is a kind of parametricity requirement and show how to fit subtyping in the same setup. The basic idea is to use reflexive graphs of categories as in [12]. We shall employ their construction to present a kind of parametric completion of a given category. We also give a different presentation of the RELconstruction in [10], and use it to discuss some examples. We show in particular that the RELconstruction acts (essentially) in the same way on a category and on its completion. Hence it follows that the identity functor on the completion satisfies the parametricity hypothesis. Discussions with Eugenio Moggi, Peter O'Hearn, Edmund Robinson, and Thomas Streicher were very useful. Paul Taylor's beutiful diagram macros were used for typesetting all the diagrams in the text. 1 Graphs of categories
The Euler Characteristic of a Category
 DOCUMENTA MATH.
, 2008
"... The Euler characteristic of a finite category is defined and shown to be compatible with Euler characteristics of other types of object, including orbifolds. A formula is proved for the cardinality of a colimit of sets, generalizing the classical inclusionexclusion formula. Both rest on a generali ..."
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Cited by 10 (3 self)
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The Euler characteristic of a finite category is defined and shown to be compatible with Euler characteristics of other types of object, including orbifolds. A formula is proved for the cardinality of a colimit of sets, generalizing the classical inclusionexclusion formula. Both rest on a generalization of Rota’s Möbius inversion from posets to categories.
General operads and multicategories
 Eprint math.CT/9810053
, 1997
"... Notions of ‘operad ’ and ‘multicategory ’ abound. This work provides a single framework in which many of these various notions can be expressed. Explicitly: given a monad ∗ on a category S, we define the term (S, ∗)multicategory, subject to certain conditions on S and ∗. Different choices ofS and ..."
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Cited by 9 (3 self)
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Notions of ‘operad ’ and ‘multicategory ’ abound. This work provides a single framework in which many of these various notions can be expressed. Explicitly: given a monad ∗ on a category S, we define the term (S, ∗)multicategory, subject to certain conditions on S and ∗. Different choices ofS and ∗ give some of the existing notions. We then describe the algebras for an (S, ∗)multicategory and, finally, present a tentative selection of further developments. Our approach makes possible concise descriptions of Baez and Dolan’s opetopes and Batanin’s operads; both of these are included.
Data Categories
 Computing: The Australasian Theory Symposium Proceedings
, 1996
"... Data categories and functors, and the strong natural transformations between them provide a universe in which to model parametric polymorphism. Data functors are distinguished by being decomposable into shape and data, i.e. they represent types that store data. Every strong transformation between tw ..."
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Cited by 7 (5 self)
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Data categories and functors, and the strong natural transformations between them provide a universe in which to model parametric polymorphism. Data functors are distinguished by being decomposable into shape and data, i.e. they represent types that store data. Every strong transformation between two such is given by a uniform algorithm, and so may represent a polymorphic term. The data functors are closed under composition, finite products and sums, exponentiation by an object, final coalgebras and initial algebras. For any two such, the collection of strong natural transformations between them is representable by an object. The covariant type system supports parametric polymorphism on data types, and can be modelled in a data category. Since the category of sets is a data category, it follows that parametric polymorphism can have a settheoretic model. Keywords data categories covariance parametric polymorphism. 1 Introduction This paper introduces data functors, the data categor...
Toposes are adhesive
 In International Conference on Graph Tranformation, icgt’06, volume 4178 of Lect. Notes Comput. Sc
, 2006
"... Abstract. Adhesive categories have recently been proposed as a categorical foundation for facets of the theory of graph transformation, and have also been used to study techniques from process algebra for reasoning about concurrency. Here we continue our study of adhesive categories by showing that ..."
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Cited by 7 (2 self)
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Abstract. Adhesive categories have recently been proposed as a categorical foundation for facets of the theory of graph transformation, and have also been used to study techniques from process algebra for reasoning about concurrency. Here we continue our study of adhesive categories by showing that toposes are adhesive. The proof relies on exploiting the relationship between adhesive categories, Brown and Janelidze’s work on generalised van Kampen theorems as well as Grothendieck’s theory of descent.
Types are weak ωgroupoids
, 2008
"... We define a notion of weak ωcategory internal to a model of MartinLöf type theory, and prove that each type bears a canonical weak ωcategory structure obtained from the tower of iterated identity types over that type. We show that the ωcategories arising in this way are in fact ωgroupoids. ..."
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Cited by 7 (0 self)
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We define a notion of weak ωcategory internal to a model of MartinLöf type theory, and prove that each type bears a canonical weak ωcategory structure obtained from the tower of iterated identity types over that type. We show that the ωcategories arising in this way are in fact ωgroupoids.
HOMOTOPY THEORY OF WELLGENERATED ALGEBRAIC TRIANGULATED CATEGORIES
, 2007
"... Abstract. For every regular cardinal α, we construct a cofibrantly generated Quillen model structure on a category whose objects are essentially DG categories which are stable under suspensions, cosuspensions, cones and αsmall sums. Using results of Porta, we show that the category of wellgenerate ..."
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Cited by 3 (0 self)
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Abstract. For every regular cardinal α, we construct a cofibrantly generated Quillen model structure on a category whose objects are essentially DG categories which are stable under suspensions, cosuspensions, cones and αsmall sums. Using results of Porta, we show that the category of wellgenerated (algebraic) triangulated categories in the sense of Neeman is naturally enhanced by our Quillen model category.