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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.
From Coherent Structures to Universal Properties
 J. Pure Appl. Algebra
, 1999
"... Given a 2category K admitting a calculus of bimodules, and a 2monad T on it compatible with such calculus, we construct a 2category L with a 2monad S on it such that: • S has the adjointpseudoalgebra property. • The 2categories of pseudoalgebras of S and T are equivalent. Thus, coh ..."
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Cited by 13 (2 self)
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Given a 2category K admitting a calculus of bimodules, and a 2monad T on it compatible with such calculus, we construct a 2category L with a 2monad S on it such that: • S has the adjointpseudoalgebra property. • The 2categories of pseudoalgebras of S and T are equivalent. Thus, coherent structures (pseudoTalgebras) are transformed into universally characterised ones (adjointpseudoSalgebras). The 2category L consists of lax algebras for the pseudomonad induced by T on the bicategory of bimodules of K. We give an intrinsic characterisation of pseudoSalgebras in terms of representability. Two major consequences of the above transformation are the classifications of lax and strong morphisms, with the attendant coherence result for pseudoalgebras. We apply the theory in the context of internal categories and examine monoidal and monoidal globular categories (including their monoid classifiers) as well as pseudofunctors into Cat.
Inferring Type Isomorphisms Generically
 Proceedings of the 7th International Conference on Mathematics of Program Construction, MPC 2004, volume 3125 of LNCS
"... Datatypes which di#er inessentially in their names and structure are said to be isomorphic; for example, a ternary product is isomorphic to a nested pair of binary products. In some canonical cases, the conversion function is uniquely determined solely by the two types involved. ..."
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Cited by 11 (7 self)
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Datatypes which di#er inessentially in their names and structure are said to be isomorphic; for example, a ternary product is isomorphic to a nested pair of binary products. In some canonical cases, the conversion function is uniquely determined solely by the two types involved.
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.
Polarized category theory, modules and game semantics
, 2004
"... Abstract. Motivated by an analysis of AbramskyJagadeesan games, the paper considers ..."
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Cited by 9 (0 self)
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Abstract. Motivated by an analysis of AbramskyJagadeesan games, the paper considers
Polycategories via pseudodistributive laws
"... In this paper, we give a novel abstract description of Szabo’s polycategories. We use the theory of double clubs – a generalisation of Kelly’s theory of clubs to ‘pseudo ’ (or ‘weak’) double categories – to construct a pseudodistributive law of the free symmetric strict monoidal category pseudocomo ..."
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Cited by 7 (1 self)
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In this paper, we give a novel abstract description of Szabo’s polycategories. We use the theory of double clubs – a generalisation of Kelly’s theory of clubs to ‘pseudo ’ (or ‘weak’) double categories – to construct a pseudodistributive law of the free symmetric strict monoidal category pseudocomonad on Mod over itself qua pseudomonad, and show that monads in the ‘twosided Kleisli bicategory’ of this pseudodistributive law are precisely symmetric polycategories. 1
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.
Paracategories I: Internal Paracategories and Saturated Partial Algebras
 Comp. Sci
, 2002
"... Based on the monoid classi er , we give an alternative axiomatization of Freyd's paracategories, which can be interpreted in any bicategory of partial maps. Assuming furthermore a freemonoid monad T in our ambient category, and coequalisers satisfying some exactness conditions, we give an abst ..."
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Cited by 5 (1 self)
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Based on the monoid classi er , we give an alternative axiomatization of Freyd's paracategories, which can be interpreted in any bicategory of partial maps. Assuming furthermore a freemonoid monad T in our ambient category, and coequalisers satisfying some exactness conditions, we give an abstract envelope construction, putting paramonoids (and paracategories) in the more general context of partial algebras . We introduce for the latter the crucial notion of saturation, which characterises those partial algebras which are isomorphic to the ones obtained from their enveloping algebras. We also set up a factorisation system for partial algebras, via inclusions and Kleene morphisms.
A language for multiplicativeadditive linear logic
 In Proc. Category Theory in Computer Science 2004, ENTCS 122 (2005
"... A term calculus for the proofs in multiplicativeadditive linear logic is introduced and motivated as a programming language for channel based concurrency. The term calculus is proved complete for a semantics in linearly distributive categories with additives. It is also shown that proof equivalence ..."
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Cited by 5 (2 self)
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A term calculus for the proofs in multiplicativeadditive linear logic is introduced and motivated as a programming language for channel based concurrency. The term calculus is proved complete for a semantics in linearly distributive categories with additives. It is also shown that proof equivalence is decidable by showing that the cut elimination rewrites supply a confluent rewriting system modulo equations. 0
Universal properties of Span
 in The Carboni Festschrift, Theory and Applications of Categories 13 (2005
"... Abstract. We give two related universal properties of the span construction. The first involves sinister morphisms out of the base category and sinister transformations. The second involves oplax morphisms out of the bicategory of spans having an extra property; we call these “jointed ” oplax morphi ..."
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Cited by 4 (2 self)
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Abstract. We give two related universal properties of the span construction. The first involves sinister morphisms out of the base category and sinister transformations. The second involves oplax morphisms out of the bicategory of spans having an extra property; we call these “jointed ” oplax morphisms.