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A Unified SheafTheoretic Account Of NonLocality and Contextuality
, 2011
"... A number of landmark results in the foundations of quantum mechanics show that quantum systems exhibit behaviour that defies explanation in classical terms, and that cannot be accounted for in such terms even by postulating “hidden variables” as additional unobserved factors. Much has been written o ..."
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A number of landmark results in the foundations of quantum mechanics show that quantum systems exhibit behaviour that defies explanation in classical terms, and that cannot be accounted for in such terms even by postulating “hidden variables” as additional unobserved factors. Much has been written on these matters, but there is surprisingly little unanimity even on basic definitions or the interrelationships among the various concepts and results. We use the mathematical language of sheaves and monads to give a very general and mathematically robust description of the behaviour of systems in which one or more measurements can be selected, and one or more outcomes observed. We say that an empirical model is extendable if it can be extended consistently to all sets of measurements, regardless of compatibility. A hiddenvariable model is factorizable if, for each value of the hidden variable, it factors as a product of distributions on the basic measurements. We prove that an empirical model is extendable if and only if there is a factorizable hiddenvariable model which realizes it. From this we are able to prove generalized versions of wellknown NoGo theorems. At the conceptual level, our equivalence result says that the existence of incompatible measurements is the essential ingredient in nonlocal and contextual behavior in quantum mechanics.
On Fixpoint Objects and Gluing Constructions
, 1997
"... This article 1 has two parts: In the first part, we present some general results about fixpoint objects. The minimal categorical structure required to model soundly the equational type theory which combines higher order recursion and computation types (introduced by [4]) is shown to be precisely a ..."
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This article 1 has two parts: In the first part, we present some general results about fixpoint objects. The minimal categorical structure required to model soundly the equational type theory which combines higher order recursion and computation types (introduced by [4]) is shown to be precisely a letcategory possessing a fixpoint object. Functional completeness for such categories is developed. We also prove that categories with fixpoint operators do not necessarily have a fixpoint object. In the second part, we extend Freyd's gluing construction for cartesian closed categories to cartesian closed letcategories, and observe that this extension does not obviously apply to categories possessing fixpoint objects. We solve this problem by giving a new gluing construction for a limited class of categories with fixpoint objects; this is the main result of the paper. We use this categorytheoretic construction to prove a typetheoretic conservative extension result. A version of this pap...
The Convex Powerdomain in a Category of Posets Realized By Cpos
 In Proc. Category Theory and Computer Science
, 1995
"... . We construct a powerdomain in a category whose objects are posets of data equipped with a cpo of "intensional" representations of the data, and whose morphisms are those monotonic functions between posets that are "realized" by continuous functions between the associated cpos. ..."
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. We construct a powerdomain in a category whose objects are posets of data equipped with a cpo of "intensional" representations of the data, and whose morphisms are those monotonic functions between posets that are "realized" by continuous functions between the associated cpos. The category of cpos is contained as a full subcategory that is preserved by lifting, sums, products and function spaces. The construction of the powerdomain uses a cpo of binary trees, these being intensional representations of nondeterministic computation. The powerdomain is characterized as the free semilattice in the category. In contrast to the other type constructors, the powerdomain does not preserve the subcategory of cpos. Indeed we show that the powerdomain has interesting computational properties that differ from those of the usual convex powerdomain on cpos. We end by considering the solution of recursive domain equations. The surprise here is that the limitcolimit coincidence fails. Nevertheless, ...
TENSORS, MONADS AND ACTIONS Dedicated to the memory of Pawel Waszkiewicz
"... We exhibit sufficient conditions for a monoidal monad T on a monoidal ..."
Universal Properties of Impure Programming Languages
"... We investigate impure, callbyvalue programming languages. Our first language only has variables and letbinding. Its equational theory is a variant of Lambek’s theory of multicategories that omits the commutativity axiom. We demonstrate that type constructions for impure languages — products, sums ..."
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We investigate impure, callbyvalue programming languages. Our first language only has variables and letbinding. Its equational theory is a variant of Lambek’s theory of multicategories that omits the commutativity axiom. We demonstrate that type constructions for impure languages — products, sums and functions — can be characterized by universal properties in the setting of ‘premulticategories’, multicategories where the commutativity law may fail. This leads us to new, universal characterizations of two earlier equational theories of impure programming languages: the premonoidal categories of Power and Robinson, and the monadbased models of Moggi. Our analysis thus puts these earlier abstract ideas on a canonical foundation, bringing them to a new, syntactic level. F.3.2 [Semantics of Pro
CMCS 2010 Categorifying Computations into Components via Arrows as Profunctors
"... The notion of arrow by Hughes is an axiomatization of the algebraic structure possessed by structured computations in general. We claim that an arrow also serves as a basic component calculus for composing statebased systems as components—in fact, it is a categorified version of arrow that does so. ..."
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The notion of arrow by Hughes is an axiomatization of the algebraic structure possessed by structured computations in general. We claim that an arrow also serves as a basic component calculus for composing statebased systems as components—in fact, it is a categorified version of arrow that does so. In this paper, following the second author’s previous work with Heunen, Jacobs and Sokolova, we prove that a certain coalgebraic modeling of components—which generalizes Barbosa’s—indeed carries such arrow structure. Our coalgebraic modeling of components is parametrized by an arrow A that specifies computational structure exhibited by components; it turns out that it is this arrow structure of A that is lifted and realizes the (categorified) arrow structure on components. The lifting is described using the first author’s recent characterization of an arrow as an internal strong monad in Prof, the bicategory of small categories and profunctors.
Vol. XXIII. 19.72 113 Strong Functors and Monoidal Monads By
"... In [4] we proved that a commutative monad on a symmetric monoidal closed category carries the structure of a symmetric monoidal monad ([4], Theorem 3.2). We here prove the converse, so that, taken together, we have: there is a 11 correspondence between commutative monads and symmetric monoidal mona ..."
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In [4] we proved that a commutative monad on a symmetric monoidal closed category carries the structure of a symmetric monoidal monad ([4], Theorem 3.2). We here prove the converse, so that, taken together, we have: there is a 11 correspondence between commutative monads and symmetric monoidal monads (Theorem 2.3 below). The main computational work needed consists in constructing an equivalence between possible strengths 8tA,B: A c ~ B+ A T ~ B T on a functor, and possible "tensorial stren~hs " on T t"X,B: X ( ~ BT> (X ( ~ B) T; T is assumed to be a functor between categories tensored over a monoidal closed category 3~'. The equivalence is stated in Theorem 1.3. (There is a similar theorem for the notion of eotensorial strength Ax,B: (Xt ~ B) T+ Xr B T, which we do not include in this note.) As an application of the theory here, we construct strength on certain functors related to the power set monad. If ~r is a 3~category, we use t ~ to denote the homfunctor ~r x ~r as well as to denote the homfunctor of 3r ~ itself. 1. Making a functor strong. Let ~r and ~ be categories tensored over the symmetric monoidal closed ~r [3]. Let T: ~0> ~0 be a functor between the underlying categories. To a family of maps (1.1) 8tA,A,: Ac~A'> A Tc~A ' T we associate a family of maps (1.2) t"X,A: X (D A T> ( X @ A) T by commutativity of (1.3) ua
Commutative monads, distributions, and differential categories
"... Abstract We describe the relationship between the theory of commutative monads on a cartesian closed category, and distribution theory (in the sense of Schwartz) inside this category. We also indicate how differential categories grow out of suitable commutative monads. The only data assumed for the ..."
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Abstract We describe the relationship between the theory of commutative monads on a cartesian closed category, and distribution theory (in the sense of Schwartz) inside this category. We also indicate how differential categories grow out of suitable commutative monads. The only data assumed for the theory presented is: a strong monad on a cartesian
Theory and Applications of Categories, Vol. 5, No. 10, pp. 251–265. ASPECTS OF FRACTIONAL EXPONENT FUNCTORS
"... ABSTRACT. We prove that certain categories arising from atoms in a Grothendieck topos are themselves Grothendieck toposes. We also investigate enrichments of these categories over the base topos; there are in fact often two distinct enrichments. 1. ..."
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ABSTRACT. We prove that certain categories arising from atoms in a Grothendieck topos are themselves Grothendieck toposes. We also investigate enrichments of these categories over the base topos; there are in fact often two distinct enrichments. 1.