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Free Products of Higher Operad Algebras
, 909
"... Abstract. One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product of 2categories. In this paper we continue the developments of [3] and [2] by understanding the natural generalisations of Gray’s little brother, ..."
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Abstract. One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product of 2categories. In this paper we continue the developments of [3] and [2] by understanding the natural generalisations of Gray’s little brother, the funny tensor product of categories. In fact we exhibit for any higher categorical structure definable by an noperad in the sense of Batanin [1], an analogous tensor product which forms a symmetric monoidal closed structure
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.
COMMUTATIVE MONADS AS A THEORY OF DISTRIBUTIONS
"... Abstract. It is shown how the theory of commutative monads provides an axiomatic framework for several aspects of distribution theory in a broad sense, including probability distributions, physical extensive quantities, and Schwartz distributions of compact support. Among the particular aspects cons ..."
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Abstract. It is shown how the theory of commutative monads provides an axiomatic framework for several aspects of distribution theory in a broad sense, including probability distributions, physical extensive quantities, and Schwartz distributions of compact support. Among the particular aspects considered here are the notions of convolution, density, expectation, and conditional probability.
Coalgebraic Walks, in Quantum and Turing Computation
, 2010
"... Abstract. The paper investigates nondeterministic, probabilistic and quantum walks, from the perspective of coalgebras and monads. Nondeterministic and probabilistic walks are coalgebras of a monad (powerset and distribution), in an obvious manner. It is shown that also quantum walks are coalgebras ..."
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Abstract. The paper investigates nondeterministic, probabilistic and quantum walks, from the perspective of coalgebras and monads. Nondeterministic and probabilistic walks are coalgebras of a monad (powerset and distribution), in an obvious manner. It is shown that also quantum walks are coalgebras of a new monad, involving additional control structure. This new monad is also used to describe Turing machines coalgebraically, namely as controlled ‘walks ’ on a tape. 1
Probabilities, Distribution Monads, and Convex Categories
"... Probabilities are understood abstractly as forming a monoid in the category of effect algebras. They can be added, via a partial operation, and multiplied. This generalises key properties of the unit interval [0, 1]. Such effect monoids can be used to define a probability distribution monad, again g ..."
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Probabilities are understood abstractly as forming a monoid in the category of effect algebras. They can be added, via a partial operation, and multiplied. This generalises key properties of the unit interval [0, 1]. Such effect monoids can be used to define a probability distribution monad, again generalising the situation for [0, 1]probabilities. It will be shown that there are translations backandforth, in the form of an adjunction, between effect monoids and “convex ” monads. This convexity property is formalised, both for monads and for categories. In the end this leads to “triangles of adjunctions ” (in the style of Coumans and Jacobs) relating all the three relevant structures: probabilities, monads, and categories. 1
TENSORS, MONADS AND ACTIONS Dedicated to the memory of Pawel Waszkiewicz
"... We exhibit sufficient conditions for a monoidal monad T on a monoidal ..."
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
Kleene Algebra with Products and Iteration Theories
"... We develop a typed equational system that subsumes both iteration theories and typed Kleene algebra in a common framework. Our approach is based on cartesian categories endowed with commutative strong monads to handle nondeterminism. ..."
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We develop a typed equational system that subsumes both iteration theories and typed Kleene algebra in a common framework. Our approach is based on cartesian categories endowed with commutative strong monads to handle nondeterminism.