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Higherdimensional algebra II: 2Hilbert spaces
"... A 2Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we define a 2Hilbert space to be an abelian category enriched over Hilb with a ∗structure, conjugatelinear on the homsets, satisfying 〈fg,h 〉 = 〈g,f ∗ h 〉 = 〈f,hg ∗ 〉. We also ..."
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Cited by 43 (13 self)
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A 2Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we define a 2Hilbert space to be an abelian category enriched over Hilb with a ∗structure, conjugatelinear on the homsets, satisfying 〈fg,h 〉 = 〈g,f ∗ h 〉 = 〈f,hg ∗ 〉. We also define monoidal, braided monoidal, and symmetric monoidal versions of 2Hilbert spaces, which we call 2H*algebras, braided 2H*algebras, and symmetric 2H*algebras, and we describe the relation between these and tangles in 2, 3, and 4 dimensions, respectively. We prove a generalized DoplicherRoberts theorem stating that every symmetric 2H*algebra is equivalent to the category Rep(G) of continuous unitary finitedimensional representations of some compact supergroupoid G. The equivalence is given by a categorified version of the Gelfand transform; we also construct a categorified version of the Fourier transform when G is a compact abelian group. Finally, we characterize Rep(G) by its universal properties when G is a compact classical group. For example, Rep(U(n)) is the free connected symmetric 2H*algebra on one even object of dimension n. 1
Modelling environments in callbyvalue programming languages
, 2003
"... In categorical semantics, there have traditionally been two approaches to modelling environments, one by use of finite products in cartesian closed categories, the other by use of the base categories of indexed categories with structure. Each requires modifications in order to account for environmen ..."
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Cited by 14 (4 self)
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In categorical semantics, there have traditionally been two approaches to modelling environments, one by use of finite products in cartesian closed categories, the other by use of the base categories of indexed categories with structure. Each requires modifications in order to account for environments in callbyvalue programming languages. There have been two more general definitions along both of these lines: the first generalising from cartesian to symmetric premonoidal categories, the second generalising from indexed categories with specified structure to κcategories. In this paper, we investigate environments in callbyvalue languages by analysing a finegrain variant of Moggi’s computational λcalculus, giving two equivalent sound and complete classes of models: one given by closed Freyd categories, which are based on symmetric premonoidal categories, the other given by closed κcategories.
A Classification of Accessible Categories
 Max Kelly volume, J. Pure Appl. Alg
, 1996
"... For a suitable collection D of small categories, we define the Daccessible categories, generalizing the #accessible categories of Lair, Makkai, and Pare; here the ..."
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Cited by 11 (0 self)
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For a suitable collection D of small categories, we define the Daccessible categories, generalizing the #accessible categories of Lair, Makkai, and Pare; here the
Closed Freyd and κCategories
, 1999
"... We give two classes of sound and complete models for the computational λcalculus, or ccalculus. For the first, we generalise the notion of cartesian closed category to that of closed Freydcategory. For the second, we generalise simple indexed categories. The former gives a direct semantics fo ..."
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Cited by 5 (0 self)
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We give two classes of sound and complete models for the computational λcalculus, or ccalculus. For the first, we generalise the notion of cartesian closed category to that of closed Freydcategory. For the second, we generalise simple indexed categories. The former gives a direct semantics for the computational λcalculus. The latter corresponds to an idealisation of stackbased intermediate languages used in some approaches to compiling.
Noncommutative geometry through monoidal categories I
"... Abstract. After introducing a noncommutative counterpart of commutative algebraic geometry based on monoidal categories of quasicoherent sheaves we show that various constructions in noncommutative geometry (e.g. Morita equivalences, HopfGalois extensions) can be given geometric meaning extending ..."
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Cited by 3 (0 self)
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Abstract. After introducing a noncommutative counterpart of commutative algebraic geometry based on monoidal categories of quasicoherent sheaves we show that various constructions in noncommutative geometry (e.g. Morita equivalences, HopfGalois extensions) can be given geometric meaning extending their geometric interpretations in the commutative case. On the other hand, we show that some constructions in commutative geometry (e.g. faithfully flat descent theory, principal fibrations, equivariant and infinitesimal geometry) can be interpreted as noncommutative geometric constructions applied to commutative objects. For such generalized geometry we define global invariants constructing cyclic objects from which we derive Hochschild, cyclic and periodic cyclic homology (with coefficients) in the standard way. Contents
Closed Freyd and kCategories
, 1999
"... . We give two classes of sound and complete models for the computational calculus, or ccalculus. For the first, we generalise the notion of cartesian closed category to that of closed Freydcategory. For the second, we generalise simple indexed categories. The former gives a direct semantics f ..."
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. We give two classes of sound and complete models for the computational calculus, or ccalculus. For the first, we generalise the notion of cartesian closed category to that of closed Freydcategory. For the second, we generalise simple indexed categories. The former gives a direct semantics for the computational calculus. The latter corresponds to an idealisation of stackbased intermediate languages used in some approaches to compiling. 1 Introduction The computational calculus, or c calculus, is a natural fragment of a callby value programming language such as ML. Its models were defined to be c  models, which consist of a small category C with finite products, and a strong monad T on C, such that T has Kleisli exponentials. The class of c models is sound and complete for the calculus, but it does not provide direct models in that a term of type X in context \Gamma is not modelled by an arrow in C from the semantics of \Gamma to the semantics of X , but by a deriv...
TENSOR PRODUCTS OF SUPLATTICES AND GENERALIZED SUPARROWS
"... Abstract. An alternative description of the tensor product of suplattices is given with yet another description provided for the tensor product in the special case of CCD suplattices. In the course of developing the latter, properties of suppreserving functions and the totally below relation are ..."
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Abstract. An alternative description of the tensor product of suplattices is given with yet another description provided for the tensor product in the special case of CCD suplattices. In the course of developing the latter, properties of suppreserving functions and the totally below relation are generalized to notnecessarilycomplete ordered sets. 1.
NONCOMMUTATIVE GEOMETRY THROUGH MONOIDAL CATEGORIES I
, 2006
"... Abstract. After introducing a noncommutative counterpart of commutative algebraic geometry based on monoidal categories of quasicoherent sheaves we show that various constructions in noncommutative geometry (e.g. Morita equivalences, HopfGalois extensions) can be given geometric meaning extending ..."
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Abstract. After introducing a noncommutative counterpart of commutative algebraic geometry based on monoidal categories of quasicoherent sheaves we show that various constructions in noncommutative geometry (e.g. Morita equivalences, HopfGalois extensions) can be given geometric meaning extending their geometric interpretations in the commutative case. On the other hand, we show that some constructions in commutative geometry (e.g. faithfully flat descent theory, principal fibrations, equivariant and infinitesimal geometry) can be interpreted as noncommutative geometric constructions applied to commutative objects. In all these considerations we lay stress on the role of the monoidal structure, and the difference between this approach and the approach using (in general nonmonoidal) abelian categories as models for categories of quasicoherent sheaves on noncommutative schemes. Contents
NONCOMMUTATIVE GEOMETRY THROUGH MONOIDAL CATEGORIES I
, 2007
"... Abstract. After introducing a noncommutative counterpart of commutative algebraic geometry based on monoidal categories of quasicoherent sheaves we show that various constructions in noncommutative geometry (e.g. Morita equivalences, HopfGalois extensions) can be given geometric meaning extending ..."
Abstract
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Abstract. After introducing a noncommutative counterpart of commutative algebraic geometry based on monoidal categories of quasicoherent sheaves we show that various constructions in noncommutative geometry (e.g. Morita equivalences, HopfGalois extensions) can be given geometric meaning extending their geometric interpretations in the commutative case. On the other hand, we show that some constructions in commutative geometry (e.g. faithfully flat descent theory, principal fibrations, equivariant and infinitesimal geometry) can be interpreted as noncommutative geometric constructions applied to commutative objects. In all these considerations we lay stress on the role of the monoidal structure, and the difference between this approach and the approach using (in general nonmonoidal) abelian categories as models for categories of quasicoherent sheaves on noncommutative schemes. Contents
NONCOMMUTATIVE GEOMETRY THROUGH MONOIDAL CATEGORIES
, 2007
"... Abstract. After introducing a noncommutative counterpart of commutative algebraic geometry based on monoidal categories of quasicoherent sheaves we show that various constructions in noncommutative geometry (e.g. Morita equivalences, HopfGalois extensions) can be given geometric meaning extending ..."
Abstract
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Abstract. After introducing a noncommutative counterpart of commutative algebraic geometry based on monoidal categories of quasicoherent sheaves we show that various constructions in noncommutative geometry (e.g. Morita equivalences, HopfGalois extensions) can be given geometric meaning extending their geometric interpretations in the commutative case. On the other hand, we show that some constructions in commutative geometry (e.g. faithfully flat descent theory, principal fibrations, equivariant and infinitesimal geometry) can be interpreted as noncommutative geometric constructions applied to commutative objects. For such generalized geometry we define global invariants constructing cyclic objects from which we derive Hochschild, cyclic and periodic cyclic homology (with coefficients) in the standard way. Contents