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19
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 51 (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 < ..."
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Cited by 38 (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.
Categorified algebra and quantum mechanics
, 2006
"... 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 par ..."
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Cited by 21 (3 self)
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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
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 18 (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.
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 9 (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.
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 8 (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.
Monoidal indeterminates and categories of possible worlds
 In Proc. of MFPS XXV
, 2009
"... Given any symmetric monoidal category C, a small symmetric monoidal category Σ and a strong monoidal functor j:Σ C, we construct C[x: jΣ], the polynomial category with a system of (freely adjoined) monoidal indeterminates x: I j(w), natural in w ∈ Σ. As a special case, we construct the free coaffin ..."
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Given any symmetric monoidal category C, a small symmetric monoidal category Σ and a strong monoidal functor j:Σ C, we construct C[x: jΣ], the polynomial category with a system of (freely adjoined) monoidal indeterminates x: I j(w), natural in w ∈ Σ. As a special case, we construct the free coaffine category (symmetric monoidal category with initial unit) on a given small symmetric monoidal category. We then exhibit all the known categories of “possible worlds ” used to treat languages that allow for dynamic creation of “new ” variables, locations, or names as instances of this construction and explicate their associated universality properties. As an application of the resulting characterisation of O(W), Oles’s category of possible worlds, we present an O(W)indexed Lawvere theory of manysorted storage, generalizing the singlesorted one introduced by J. Power, and we describe explicitly an associated
POLYNOMIALS IN CATEGORIES WITH PULLBACKS
"... Abstract. The theory developed by Gambino and Kock, of polynomials over a locally cartesian closed category E, is generalised for E just having pullbacks. The 2categorical analogue of the theory of polynomials and polynomial functors is given, and its relationship with Street’s theory of fibration ..."
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Abstract. The theory developed by Gambino and Kock, of polynomials over a locally cartesian closed category E, is generalised for E just having pullbacks. The 2categorical analogue of the theory of polynomials and polynomial functors is given, and its relationship with Street’s theory of fibrations within 2categories is explored. Johnstone’s notion of “bagdomain data ” is adapted to the present framework to make it easier to completely exhibit examples of polynomial monads. 1.