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Representable Multicategories
 Advances in Mathematics
, 2000
"... We introduce the notion of representable multicategory , which stands in the same relation to that of monoidal category as bration does to contravariant pseudofunctor (into Cat). We give an abstract reformulation of multicategories as monads in a suitable Kleisli bicategory of spans. We describe ..."
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Cited by 33 (6 self)
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We introduce the notion of representable multicategory , which stands in the same relation to that of monoidal category as bration does to contravariant pseudofunctor (into Cat). We give an abstract reformulation of multicategories as monads in a suitable Kleisli bicategory of spans. We describe representability in elementary terms via universal arrows . We also give a doctrinal characterisation of representability based on a fundamental monadic adjunction between the 2category of multicategories and that of strict monoidal categories. The first main result is the coherence theorem for representable multicategories, asserting their equivalence to strict ones, which we establish via a new technique based on the above doctrinal characterisation. The other main result is a 2equivalence between the 2category of representable multicategories and that of monoidal categories and strong monoidal functors. This correspondence extends smoothly to one between bicategories and a se...
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.
Automatic Analysis of Pointer Aliasing for Untyped Programs
 Science of Computer Programming
, 1999
"... Interpretation that discovers potential sharing relationships among the data structures created by an imperative program. The analysis is able to distinguish between elements in inductively defined structures and does not require any explicit data type declaration by the programmer. In order to ..."
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Cited by 16 (5 self)
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Interpretation that discovers potential sharing relationships among the data structures created by an imperative program. The analysis is able to distinguish between elements in inductively defined structures and does not require any explicit data type declaration by the programmer. In order to construct the abstract interpretation we introduce a new class of abstract domains: the cofibered domains.
Categorified algebra and quantum mechanics, Theory and Applications of Categories 16
, 2006
"... Abstract. Interest in combinatorial interpretations of mathematical entities stems from the convenience of the concrete models they provide. Finding a bijective proof of a seemingly obscure identity can reveal unsuspected significance to it. Finding a combinatorial model for some mathematical entity ..."
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Cited by 13 (1 self)
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Abstract. Interest in combinatorial interpretations of mathematical entities stems from the convenience of the concrete models they provide. Finding a bijective proof of a seemingly obscure identity can reveal unsuspected significance to it. Finding a combinatorial model for some mathematical entity is a particular instance of the process called “categorification”. Examples include the interpretation of as the Burnside rig of the category of finite sets with product and coproduct, and the interpretation of [x] as the category of combinatorial species. This has interesting applications to quantum mechanics, and in particular the quantum harmonic oscillator, via Joyal’s “species”, a new generalization called “stuff types”, and operators between these, which can be represented as rudimentary Feynman diagrams for the oscillator. In quantum mechanics, we want to represent states in an algebra over the complex numbers, and also want our Feynman diagrams to carry more structure than these “stuff operators ” can do, and these turn out to be closely related. We will show how to construct a combinatorial model for the quantum harmonic
Opmonoidal Monads
, 2002
"... Hopf monads are identified with monads in the 2category OpMon of monoidal categories, opmonoidal functors and transformations. Using EilenbergMoore objects, it is shown that for a Hopf monad S, the categories Alg(Coalg(S)) and Coalg(Alg(S)) are canonically isomorphic. The monadic arrows OpMon are ..."
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Cited by 6 (0 self)
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Hopf monads are identified with monads in the 2category OpMon of monoidal categories, opmonoidal functors and transformations. Using EilenbergMoore objects, it is shown that for a Hopf monad S, the categories Alg(Coalg(S)) and Coalg(Alg(S)) are canonically isomorphic. The monadic arrows OpMon are then characterized. Finally, the theory of multicategories and a generalization of structure and semantics are used to identify the categories of algebras of Hopf monads.
Symmetric monoidal categories model all connective spectra. Theory and
 Applications of Categories, 1:78118
, 1995
"... Abstract. The classical in nite loopspace machines in fact induce an equivalence of categories between a localization of the category of symmetric monoidal categories and the stable homotopy category of1connective spectra. ..."
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Cited by 5 (0 self)
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Abstract. The classical in nite loopspace machines in fact induce an equivalence of categories between a localization of the category of symmetric monoidal categories and the stable homotopy category of1connective spectra.
A 2categories companion
"... Abstract. This paper is a rather informal guide to some of the basic theory of 2categories and bicategories, including notions of limit and colimit, 2dimensional universal algebra, formal category theory, and nerves of bicategories. 1. Overview and basic examples This paper is a rather informal gu ..."
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Cited by 4 (0 self)
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Abstract. This paper is a rather informal guide to some of the basic theory of 2categories and bicategories, including notions of limit and colimit, 2dimensional universal algebra, formal category theory, and nerves of bicategories. 1. Overview and basic examples This paper is a rather informal guide to some of the basic theory of 2categories and bicategories, including notions of limit and colimit, 2dimensional universal algebra, formal category theory, and nerves of bicategories. As is the way of these things, the choice of topics is somewhat personal. No attempt is made at either rigour or completeness. Nor is it completely introductory: you will not find a definition of bicategory; but then nor will you really need one to read it. In keeping with the philosophy of category theory, the morphisms between bicategories play more of a role than the bicategories themselves. 1.1. The key players. There are bicategories, 2categories, and Catcategories. The latter two are exactly the same (except that strictly speaking a Catcategory should have small homcategories, but that need not concern us here). The first two are nominally different — the 2categories are the strict bicategories, and not every bicategory is strict — but every bicategory is biequivalent to a strict one, and biequivalence is the right general notion of equivalence for bicategories and for 2categories. Nonetheless, the theories of bicategories, 2categories, and Catcategories have rather different flavours.
Symmetric Monoidal Categories Model All Connective Spectra
 Applications of Categories, 1:78118
, 1995
"... . The classical infinite loopspace machines in fact induce an equivalence of categories between a localization of the category of symmetric monoidal categories and the stable homotopy category of1connective spectra. Introduction Since the early seventies it has been known that the classifying ..."
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. The classical infinite loopspace machines in fact induce an equivalence of categories between a localization of the category of symmetric monoidal categories and the stable homotopy category of1connective spectra. Introduction Since the early seventies it has been known that the classifying spaces of small symmetric monoidal categories are infinite loop spaces, the zeroth space in a spectrum, a sequence of spaces X i ; i 0 with given homotopy equivalences to the loops on the succeeding space X i \Gamma!\Omega X i+1 . Indeed, many of the classical examples of infinite loop spaces were found as such classifying spaces ( e.g. [Ma2], [Se]). These infinite loop spaces and spectra are of great interest to topologists. The homotopy category formed by inverting the weak equivalences of spectra is the stable homotopy category, much better behaved than but still closely related to the usual homotopy category of spaces (e.g., [Ad] III ). One has in fact classically a functor Spt from t...
Double clubs
, 2008
"... Abstract. We develop a theory of double clubs which extends Kelly’s theory of clubs to the pseudo double categories of Paré and Grandis. We then show that the club for symmetric strict monoidal categories on Cat extends to a ‘double club ’ on the pseudo ..."
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Abstract. We develop a theory of double clubs which extends Kelly’s theory of clubs to the pseudo double categories of Paré and Grandis. We then show that the club for symmetric strict monoidal categories on Cat extends to a ‘double club ’ on the pseudo
unknown title
, 705
"... Abstract. Coherence theorems for covariant structures carried by a category have traditionally relied on the underlying term rewriting system of the structure being terminating and confluent. While this holds in a variety of cases, it is not a feature that is inherent to the coherence problem itself ..."
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Abstract. Coherence theorems for covariant structures carried by a category have traditionally relied on the underlying term rewriting system of the structure being terminating and confluent. While this holds in a variety of cases, it is not a feature that is inherent to the coherence problem itself. This is demonstrated by the theory of iterated monoidal categories, which model iterated loop spaces and have a coherence theorem but fail to be confluent. We develop a framework for expressing coherence problems in terms of term rewriting systems equipped with a two dimensional congruence. Within this framework we provide general solutions to two related coherence theorems: Determining whether there is a decision procedure for the commutativity of diagrams in the resulting structure and determining sufficient conditions ensuring that “all diagrams commute”. The resulting coherence theorems rely on neither the termination nor the confluence of the underlying rewriting system. We apply the theory to iterated monoidal categories and obtain a new, conceptual proof of their coherence theorem. 1.