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29
Hypercoherences: A Strongly Stable Model of Linear Logic
 Mathematical Structures in Computer Science
, 1993
"... We present a model of classical linear logic based on the notion of strong stability that was introduced in [BE], a work about sequentiality written jointly with Antonio Bucciarelli. ..."
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Cited by 59 (8 self)
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We present a model of classical linear logic based on the notion of strong stability that was introduced in [BE], a work about sequentiality written jointly with Antonio Bucciarelli.
Genealogy of Nonperturbative QuantumInvariants of 3Manifolds: The Surgical Family. qalg/9601021
"... Abstract: We study the relations between the invariants τRT, τHKR, and τL of ReshetikhinTuraev, HenningsKauffmanRadford, and Lyubashenko, respectively. In particular, we discuss explicitly how τL specializes to τRT for semisimple categories and to τHKR for Tannakian categories. We give arguments ..."
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Cited by 17 (5 self)
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Abstract: We study the relations between the invariants τRT, τHKR, and τL of ReshetikhinTuraev, HenningsKauffmanRadford, and Lyubashenko, respectively. In particular, we discuss explicitly how τL specializes to τRT for semisimple categories and to τHKR for Tannakian categories. We give arguments for that τL is the most general invariant that stems from an extended TQFT. We introduce a canonical, central element, Q, for a quasitriangular Hopf algebra, A, that allows us to apply the Hennings algorithm directly, in order to compute τRT, which is originally obtained from the semisimple tracesubquotient of A − mod. Moreover, we generalize Hennings ’ rules to the context of cobordisms, in order to obtain a TQFT for connected surfaces compatible with τHKR. As an application we show that, for lens spaces and A = Uq(sl2), the ratio of τHKR and τRT is the order of the first homology group. In the course of this paper we also outline the topology and the algebra that enter invariance proofs, which contain no reference to 2handle slides, but to other moves that are local. Finally, we give a list of open questions regarding cellular invariants, as defined by TuraevViro, Kuperberg, and others, their relations among
Relational Properties of Recursively Defined Domains
 In 8th Annual Symposium on Logic in Computer Science
, 1993
"... This paper describes a mixed induction/coinduction property of relations on recursively defined domains. We work within a general framework for relations on domains and for actions of type constructors on relations introduced by O'Hearn and Tennent [20], and draw upon Freyd's analysis [7] of recurs ..."
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Cited by 15 (2 self)
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This paper describes a mixed induction/coinduction property of relations on recursively defined domains. We work within a general framework for relations on domains and for actions of type constructors on relations introduced by O'Hearn and Tennent [20], and draw upon Freyd's analysis [7] of recursive types in terms of a simultaneous initiality/finality property. The utility of the mixed induction/coinduction property is demonstrated by deriving a number of families of proof principles from it. One instance of the relational framework yields a family of induction principles for admissible subsets of general recursively defined domains which extends the principle of structural induction for inductively defined sets. Another instance of the framework yields the coinduction principle studied by the author in [22], by which equalities between elements of recursively defined domains may be proved via `bisimulations'. 1 Introduction A characteristic feature of higherorder functional lan...
Operational Resolutions And State Transitions In A Categorical Setting
 Found. Phys. Letters
, 1998
"... this paper consists of lifting the  categorically  equivalent descriptions of physical systems by a (i) `state space' or a (ii) `property lattice'  see [14,20,25,26]  to an asymmetrical  i.e., not anymore isomorphic  duality on the level of: (i) ..."
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Cited by 11 (8 self)
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this paper consists of lifting the  categorically  equivalent descriptions of physical systems by a (i) `state space' or a (ii) `property lattice'  see [14,20,25,26]  to an asymmetrical  i.e., not anymore isomorphic  duality on the level of: (i)
Modules over operator algebras, and the maximal C ∗ dilation
, 1999
"... Abstract. We continue our study of the general theory of possibly nonselfadjoint algebras of operators on a Hilbert space, and modules over such algebras, developing a little more technology to connect ‘nonselfadjoint operator algebra ’ with the C ∗ −algebraic framework. More particularly, we make u ..."
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Cited by 8 (6 self)
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Abstract. We continue our study of the general theory of possibly nonselfadjoint algebras of operators on a Hilbert space, and modules over such algebras, developing a little more technology to connect ‘nonselfadjoint operator algebra ’ with the C ∗ −algebraic framework. More particularly, we make use of the universal, or maximal, C ∗ −algebra generated by an operator algebra, and C ∗ −dilations. This technology is quite general, however it was developed to solve some problems arising in the theory of Morita equivalence of operator algebras, and as a result most of the applications given here (and in a companion paper) are to that subject. Other applications given here are to extension problems for module maps, and characterizations of C ∗ −algebras. * Supported by a grant from the NSF. The contents of this paper were announced at the January 1999 meeting of the American Mathematical Socety. 1 2 DAVID P. BLECHER 1. Introduction Modules
A Unified Scheme for Generalized Sectors based on Selection Criteria. I. Thermal situations, unbroken symmetries and criteria as classifying categorical adjunctions
"... Continuing the analysis in a unified scheme for treating generalized superselection sectors based upon the notion of selection criteria for states of relevance in quantum physics, we extend the DoplicherRoberts superselection theory for recovering the field algebra and the gauge group (of the first ..."
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Cited by 8 (6 self)
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Continuing the analysis in a unified scheme for treating generalized superselection sectors based upon the notion of selection criteria for states of relevance in quantum physics, we extend the DoplicherRoberts superselection theory for recovering the field algebra and the gauge group (of the first kind) from the data of group invariant observables to the situations with spontaneous symmetry breakdown: in use of the machinery proposed, the basic structural features of the theory with spontaneously broken symmetry, are clarified in a clearcut way, such as the degenerate vacua parametrized by the variable belonging to the relevant homogeneous space, the Goldstone modes and condensates. 1
A Morita theorem for algebras of operators on Hilbert Space
"... Abstract. We show that two operator algebras are strongly Morita equivalent (in the sense of Blecher, Muhly and Paulsen) if and only if their categories of operator modules are equivalent via completely contractive functors. Moreover, any such functor is completely isometrically isomorphic to the Ha ..."
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Cited by 7 (2 self)
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Abstract. We show that two operator algebras are strongly Morita equivalent (in the sense of Blecher, Muhly and Paulsen) if and only if their categories of operator modules are equivalent via completely contractive functors. Moreover, any such functor is completely isometrically isomorphic to the Haagerup tensor product ( = interior tensor product) with a strong Morita equivalence bimodule.
The Fundamental Groupoid as a Topological Groupoid
 Proc. Edinburgh Math. Soc
, 1975
"... Let X be a topological space. Then we may define the fundamental groupoid nX and also the quotient groupoid (nX)/N for N any wide, totally disconnected, normal subgroupoid N of nX (1). The purpose of this note is to show that if X is locally pathconnected and semilocally 1connected, then the topo ..."
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Cited by 7 (4 self)
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Let X be a topological space. Then we may define the fundamental groupoid nX and also the quotient groupoid (nX)/N for N any wide, totally disconnected, normal subgroupoid N of nX (1). The purpose of this note is to show that if X is locally pathconnected and semilocally 1connected, then the topology of X
A cartesian closed category of approximable concept structures
 Proceedings of the International Conference On Conceptual Structures
, 2004
"... Abstract. Infinite contexts and their corresponding lattices are of theoretical and practical interest since they may offer connections with and insights from other mathematical structures which are normally not restricted to the finite cases. In this paper we establish a systematic connection betwe ..."
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Cited by 6 (4 self)
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Abstract. Infinite contexts and their corresponding lattices are of theoretical and practical interest since they may offer connections with and insights from other mathematical structures which are normally not restricted to the finite cases. In this paper we establish a systematic connection between formal concept analysis and domain theory as a categorical equivalence, enriching the link between the two areas as outlined in [25]. Building on a new notion of approximable concept introduced by Zhang and Shen [26], this paper provides an appropriate notion of morphisms on formal contexts and shows that the resulting category is equivalent to (a) the category of complete algebraic lattices and Scott continuous functions, and (b) a category of information systems and approximable mappings. Since the latter categories are cartesian closed, we obtain a cartesian closed category of formal contexts that respects both the context structures as well as the intrinsic notion of approximable concepts at the same time. 1
THE MCCORD MODEL FOR THE TENSOR PRODUCT OF A SPACE AND A COMMUTATIVE RING SPECTRUM
, 2002
"... We begin this paper by noting that, in a 1969 paper in the Transactions, M.C.McCord introduced a construction that can be interpreted as a model for the categorical tensor product of a based space and a topological abelian group. This can be adapted to Segal’s very special Γ–spaces indeed this is ..."
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Cited by 6 (3 self)
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We begin this paper by noting that, in a 1969 paper in the Transactions, M.C.McCord introduced a construction that can be interpreted as a model for the categorical tensor product of a based space and a topological abelian group. This can be adapted to Segal’s very special Γ–spaces indeed this is roughly what Segal did and then to a more modern situation: K ⊗ R where K is a based space and R is a unital, augmented, commutative, associative S–algebra. The model comes with an easytodescribe filtration. If one lets K = S n, and then stabilize with respect to n, one gets a filtered model for the Topological André–Quillen Homology of R. When R = Ω ∞ Σ ∞ X, one arrives at a filtered model for the connective cover of a spectrum X, constructed from its 0 th space. Another example comes by letting K be a finite complex, and R the S–dual of a finite complex Z. Dualizing again, one arrives at G. Arone’s model for the Goodwillie tower of the functor sending Z to Σ ∞ MapT (K, Z). Applying cohomology with field coefficients, one gets various spectral sequences for deloopings with known E1–terms. A few nontrivial examples are given. In an appendix, we describe the construction for unital, commutative, associative S–algebras not necessarily augmented.