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Categorical Logic of Names and Abstraction in Action Calculi
, 1993
"... ion elimination Definition 3.1. A monoidal category where every object has a commutative comonoid structure is said to be semicartesian. An action category is a K\Omega category with a distinguished admissible commutative comonoid structure on every object. A semicartesian category is cartesi ..."
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Cited by 27 (14 self)
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ion elimination Definition 3.1. A monoidal category where every object has a commutative comonoid structure is said to be semicartesian. An action category is a K\Omega category with a distinguished admissible commutative comonoid structure on every object. A semicartesian category is cartesian if and only if each object carries a unique comonoid structure, and such structures form two natural families, \Delta and !. The naturality means that all morphisms of the category must be comonoid homomorphisms. In action categories, the property of semicartesianness is fixed as structure: on each object, a particular comonoid structure is chosen. This choice may be constrained by some given graphic operations, with respect to which the structures must be admissible. The proof of proposition 2.6 shows that such structures determine the abstraction operators, and are determined by them. This is the essence of the equivalence of action categories and action calculi. As the embodiment of 2...
Geometry of abstraction in quantum computation
"... Quantum algorithms are sequences of abstract operations, performed on nonexistent computers. They are in obvious need of categorical semantics. We present some steps in this direction, following earlier contributions of Abramsky, Coecke and Selinger. In particular, we analyze function abstraction i ..."
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Quantum algorithms are sequences of abstract operations, performed on nonexistent computers. They are in obvious need of categorical semantics. We present some steps in this direction, following earlier contributions of Abramsky, Coecke and Selinger. In particular, we analyze function abstraction in quantum computation, which turns out to characterize its classical interfaces. Some quantum algorithms provide feasible solutions of important hard problems, such as factoring and discrete log (which are the building blocks of modern cryptography). It is of a great practical interest to precisely characterize the computational resources needed to execute such quantum algorithms. There are many ideas how to build a quantum computer. Can we prove some necessary conditions? Categorical semantics help with such questions. We show how to implement an important family of quantum algorithms using just abelian groups and relations.
Laminations, or How to Build a QuantumLogieValued Model of Set Theory
, 1979
"... An explicit construction of the coli mit of a filtered diagram in the category of topoi and logical morphisms is given and then used to construct a family of topoi with a fixed Boolean algebra of truth values but with varying amounts of cocompleteness. This same construction, when applied to the dia ..."
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An explicit construction of the coli mit of a filtered diagram in the category of topoi and logical morphisms is given and then used to construct a family of topoi with a fixed Boolean algebra of truth values but with varying amounts of cocompleteness. This same construction, when applied to the diagram of complete Boolean algebras in a quantum logic Q gives a partial topos, a noncategory which is a close to being a model of set theory with algebra of truth values Q as a noncategory can be. This investigation concerns the construction of topoi or topos like noncategories which have specified algebras for their propositional logics. It is well known that one can construct topoi with given complete Heyting algebras as propositional logics by taking sheaves for the can onical topology, but the construction for noncomplete Boolean algebras appears in the literature only in the exercises of Johnstone's book (4) p. 331. That construction is a filtered colimit in the category of topoi. The existenc e of such colimits is asserted by Freyd (2) to follow from the essentially algebraic nature of the theory of topoi. No explicit construction exists in the literature. Freyd gives several examples of non Grothendieck topoi constructed using colimits; for his purposes other descriptions give the