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Quantum logic in dagger kernel categories
 Order
"... This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest, such as categories of relations, partial inject ..."
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This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest, such as categories of relations, partial injections, Hilbert spaces (also modulo phase), and Boolean algebras, and (2) have interesting categorical/logical/ordertheoretic properties, in terms of kernel fibrations, such as existence of pullbacks, factorisation, orthomodularity, atomicity and completeness. For instance, the Sasaki hook and andthen connectives are obtained, as adjoints, via the existentialpullback adjunction between fibres. 1
THE KAROUBI ENVELOPE AND LEE’S DEGENERATION OF KHOVANOV HOMOLOGY
"... Abstract. We give a simple proof of Lee’s result from [5], that the dimension of the Lee variant of the Khovanov homology of an ccomponent link is 2 c, regardless of the number of crossings. Our method of proof is entirely local and hence we can state a Leetype theorem for tangles as well as for k ..."
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Cited by 8 (1 self)
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Abstract. We give a simple proof of Lee’s result from [5], that the dimension of the Lee variant of the Khovanov homology of an ccomponent link is 2 c, regardless of the number of crossings. Our method of proof is entirely local and hence we can state a Leetype theorem for tangles as well as for knots and links. Our main tool is the “Karoubi envelope of the cobordism category”, a certain enlargement of the cobordism category which is mild enough so that no information is lost yet
Galois coverings, Morita equivalence and smash extensions of categories over a field
, 2005
"... We consider categories over a field k in order to prove that smash extensions and Galois coverings with respect to a finite group coincide up to Morita equivalence of kcategories. For this purpose we describe processes providing Morita equivalences called contraction and expansion. We prove that co ..."
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Cited by 5 (3 self)
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We consider categories over a field k in order to prove that smash extensions and Galois coverings with respect to a finite group coincide up to Morita equivalence of kcategories. For this purpose we describe processes providing Morita equivalences called contraction and expansion. We prove that composition of these processes provides any Morita equivalence, a result which is related with the karoubianisation (or idempotent completion) and additivisation of a kcategory.
Information Categories
 Applied Categorical Structures
, 1991
"... \Information systems" have been introduced by Dana Scott as a convenient means of presenting a certain class of domains of computation, usually known as Scott domains. Essentially the same idea has been developed, if less systematically, by various authors in connection with other classes of dom ..."
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Cited by 3 (3 self)
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\Information systems" have been introduced by Dana Scott as a convenient means of presenting a certain class of domains of computation, usually known as Scott domains. Essentially the same idea has been developed, if less systematically, by various authors in connection with other classes of domains. In previous work, the present authors introduced the notion of an Icategory as an abstraction and enhancement of this idea, with emphasis on the solution of domain equations of the form D = F (D), with F a functor. An important feature of the work is that we are not conned to domains of computation as usually understood; other classes of spaces, more familiar to mathematicians in general, become also accessible. Here we present the idea in terms of what we call information categories, which are concrete Icategories in which the objects are structured sets of \tokens" and morphisms are relations between tokens. This is more in the spirit of information system work, and...
Extended TQFT’s and Quantum Gravity
, 2007
"... Abstract. This paper gives a definition of an extended topological quantum field theory (TQFT) as a weak 2functor Z: nCob2→2Vect, by analogy with the description of a TQFT as a functor Z: nCob→Vect. We also show how to obtain such a theory from any finite group G. This theory is related to a topolo ..."
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Abstract. This paper gives a definition of an extended topological quantum field theory (TQFT) as a weak 2functor Z: nCob2→2Vect, by analogy with the description of a TQFT as a functor Z: nCob→Vect. We also show how to obtain such a theory from any finite group G. This theory is related to a topological gauge theory, the DijkgraafWitten model. To give this definition rigorously, we first define a bicategory of cobordisms between cobordisms. We also give some explicit description of a highercategorical version of Vect, denoted 2Vect, a bicategory of 2vector spaces. Along the way, we prove several results showing how to construct 2vector spaces of Vectvalued presheaves on certain kinds of groupoids. In particular, we use the case when these are groupoids whose objects are connections, and whose morphisms are gauge transformations, on the manifolds on which the extended TQFT is to be defined. On cobordisms between these manifolds, we show how a construction of “pullback and pushforward ” of presheaves gives both the morphisms and 2morphisms in 2Vect for the extended TQFT, and that these
WHAT DOES IT TAKE TO PROVE FERMAT’S LAST THEOREM? GROTHENDIECK AND THE LOGIC OF NUMBER THEORY
, 2009
"... Abstract. This paper explores the set theoretic assumptions used in the current published proof of Fermat’s Last Theorem, how these assumptions figure in the methods Wiles uses, and the currently known prospects for a proof using weaker assumptions. Does the proof of Fermat’s Last Theorem (FLT) go b ..."
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Abstract. This paper explores the set theoretic assumptions used in the current published proof of Fermat’s Last Theorem, how these assumptions figure in the methods Wiles uses, and the currently known prospects for a proof using weaker assumptions. Does the proof of Fermat’s Last Theorem (FLT) go beyond Zermelo Fraenkel set theory (ZFC)? Or does it merely use Peano Arithmetic (PA) or some weaker fragment of that? The answers depend on what is meant by “proof ” and “use, ” and are not entirely known. This paper surveys the current state of these questions and briefly sketches the methods of cohomological number theory used in the existing proof. The existing proof of FLT is Wiles [1995] plus improvements that do not yet change its character. Far from selfcontained it has vast prerequisites merely introduced in the 500 pages of [Cornell et al., 1997]. We will say that the assumptions explicitly used in proofs that Wiles cites as steps in his own are “used in fact in the published proof. ” It is currently unknown what assumptions are “used in principle ” in the sense of being prooftheoretically indispensable to FLT. Certainly much less than ZFC is used in principle, probably nothing beyond PA, and perhaps much less than that. The oddly contentious issue is universes, often called Grothendieck universes. 1 On ZFC foundations a universe is an uncountable transitive set U such that 〈U, ∈ 〉 satisfies the ZFC axioms in the nicest way: it contains the powerset of each of its elements, and for any function from an element of U to U the range is also an element of U. This is much stronger than merely saying 〈U, ∈ 〉 satisfies the ZFC axioms. We do not merely say the powerset axiom “every set has a powerset ” is true with all quantifiers relativized to U. Rather, we require “for every set x ∈ U, the powerset of x is also in U ”
A Theory of Adjoint Functors —with some Thoughts about their Philosophical Significance
, 2005
"... The question “What is category theory ” is approached by focusing on universal mapping properties and adjoint functors. Category theory organizes mathematics using morphisms that transmit structure and determination. Structures of mathematical interest are usually characterized by some universal map ..."
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The question “What is category theory ” is approached by focusing on universal mapping properties and adjoint functors. Category theory organizes mathematics using morphisms that transmit structure and determination. Structures of mathematical interest are usually characterized by some universal mapping property so the general thesis is that category theory is about determination through universals. In recent decades, the notion of adjoint functors has moved to centerstage as category theory’s primary tool to characterize what is important and universal in mathematics. Hence our focus here is to present a theory of adjoint functors, a theory which shows that all adjunctions arise from the birepresentations of “chimeras ” or “heteromorphisms ” between the objects of different categories. Since representations provide universal mapping properties, this theory places adjoints within the framework of determination through universals. The conclusion considers some unreasonably effective analogies between these mathematical
Data Structures in Natural Computing: Databases as Weak or Strong Anticipatory Systems
"... Abstract. Information systems anticipate the real world. Classical databases store, organise and search collections of data of that real world but only as weak anticipatory information systems. This is because of the reductionism and normalisation needed to map the structuralism of natural data on t ..."
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Abstract. Information systems anticipate the real world. Classical databases store, organise and search collections of data of that real world but only as weak anticipatory information systems. This is because of the reductionism and normalisation needed to map the structuralism of natural data on to idealised machines with von Neumann architectures consisting of fixed instructions. Category theory developed as a formalism to explore the theoretical concept of naturality shows that methods like sketches arising from graph theory as only nonnatural models of naturality cannot capture realworld structures for strong anticipatory information systems. Databases need a schema of the natural world. Natural computing databases need the schema itself to be also natural. Natural computing methods including neural computers, evolutionary automata, molecular and nanocomputing and quantum computation have the potential to be strong. At present they are mainly at the stage of weak anticipatory systems.
Contents
"... Abstract. We discuss the representation theory of Hf, which is a deformation of the symplectic oscillator algebra sp(2n) ⋉ hn, where hn is the ((2n + 1)dimensional) Heisenberg algebra. We first look at a more general algebra with a triangular decomposition. Assuming the PBW theorem, and ..."
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Abstract. We discuss the representation theory of Hf, which is a deformation of the symplectic oscillator algebra sp(2n) ⋉ hn, where hn is the ((2n + 1)dimensional) Heisenberg algebra. We first look at a more general algebra with a triangular decomposition. Assuming the PBW theorem, and
Geometric Theory of Machine Awareness for Legal Information Retrieval and Reasoning
, 1995
"... This report 1 considers that links in hypertext are representable as links in thought by covariant arrows between categories. Taken in dynamic context, the rightexactness of the Heyting implication A ) B corresponds to inference and the next document in a nonlinear trail through hypermedia. Aw ..."
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This report 1 considers that links in hypertext are representable as links in thought by covariant arrows between categories. Taken in dynamic context, the rightexactness of the Heyting implication A ) B corresponds to inference and the next document in a nonlinear trail through hypermedia. Awareness is provided by the dual contravariant arrows with the important special case of the intensionextension relationship. The corresponding leftexactness is the closure limit that invokes consciousness. About the author Michael Heather is senior lecturer in law where he has been responsible for computers and law since 1979. Nick Rossiter is lecturer in the Department of Computing Science with particular interests in databases and systems analysis. Suggested Keywords hypertext, legal reasoning, information retrieval, category theory, adjoints, Heyting algebra. 1 The work on geometric logic and law in this report was presented at the 17th IVR World Congress June 16th21st 1995 Chall...