Girard Couples of Quantales

Egger, J. M.; Kruml, David

doi:10.1007/s10485-008-9138-3

Cited by 4 publications

(8 citation statements)

References 14 publications

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“…The reason is that the category

C^{*} Alg

C^{*}

‐algebras is not (or at least not known to be) a category of *‐algebras in a suitable involutive closed symmetric monoidal category. There are proposals in the literature to replace

C^{*} Alg

by the category of *‐algebras in the involutive closed symmetric monoidal category of operator spaces in order to obtain a categorical approach to the theory of operator algebras. To the best of our knowledge, such an approach has not been applied to AQFT yet.…”

Section: Aqft From An Algebraic Perspectivementioning

confidence: 99%

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Higher Structures in Algebraic Quantum Field Theory

Benini

Schenkel

2019

Fortschritte der Physik

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“…The reason is that the category

C^{*} Alg

C^{*}

‐algebras is not (or at least not known to be) a category of *‐algebras in a suitable involutive closed symmetric monoidal category. There are proposals in the literature to replace

C^{*} Alg

Section: Aqft From An Algebraic Perspectivementioning

confidence: 99%

“…The reason is that the category C * Alg of C * -algebras is not (or at least not known to be) a category of * -algebras in a suitable involutive closed symmetric monoidal category. There are proposals in the literature [26,27] to replace…”

mentioning

confidence: 99%

Higher Structures in Algebraic Quantum Field Theory

Benini

Schenkel

2019

Fortschritte der Physik

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“…Since every sup-lattice is a unital 2-module, we get a triad (S op , 2, S). Quantale Q(S) of all suplattice endomorphisms [11] of S and quantale C(S) = S ⊗ S op are clearly solutions of the triad [4].…”

Section: Definitionmentioning

confidence: 99%

“…The solution Q 0 is realized by tensor product R ⊗ T L while Q 1 is a generalization of a quantale of endomorphisms. The results are based on a special case studied in [4] and further communication with J. Egger and the idea of P. Resende [12] who constructed Q 1 for the case of Galois connections. The quantales Q 0 , Q 1 reflects two aspects of the quantization of topology -it is a non-commutative intersection represented by multiplication on Q 0 , and transitivity of states represented by actions of Q 1 on L and R.…”

Section: Introductionmentioning

confidence: 99%

Quantum triads: an algebraic approach

Kruml

2008

Preprint

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“…It has been shown in [6] that S 0 and S 1 form a so-called couple of quantales (see also [4]) and that S is a solution iff S 0 → S 1 factorizes through S.…”

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confidence: 99%

Generalizing the concept of quantum triads

Šlesinger

EPiC Series in Computing

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The concept of quantum triad has been introduced by D. Kruml [6], where for a given pair of quantale 1 modules L, R over a common quantale Q, endowed with a bimorphism (a 'bilinear map') to Q, a construction equipping L and R with additional module structure and another bimorphism, both compatible with the existing bimorphism and action of the quantale, was presented. As the original concept was only defined in a specific setting of categories of quantale modules, we extend it to a more universal one, which can be applied to other common algebraic structures.In what follows, we assume that all the categories are concrete (via the forgetful functor | − | into Set). Let V = (V, ⊗ V , I V ) be a closed symmetric monoidal category and C be a subcategory of V, enriched in V. Suppose M is a monoid in V (viewed as a V-category with a single object M ). We call an objectDefinition. Let V, C be as above, L, R be a left and a right module over a V-monoid T . Further, let τ be a C-bimorphism from L × R to T . Then the tuple (L, R, τ, T ) is called a triad.If there exists a monoid S in V together with a V-bimorphism σ : R × L → S which makes L a T, S-bimodule and R a S, T -bimodule, and is compatible with τ (this means, for instance 'RLR': for any l ∈ L and r, r ∈ R, σ(r, l) · r = r · τ (l, r ), and a few similar conditions), we call (S, σ) a solution of the triad.Existence of solutions together with additional properties can be proved when certain assumptions on the category C are satisfied:Proposition. 1. Let C have tensor products over T , i.e., it has coequalizers of morphisms R ⊗ T ⊗ L ⇒ R ⊗ L obtained from T acting on R and L, respectively. Then the universal property of the tensor product provides a solution (S 0 , σ 0 ) given by R ⊗ T L and σ 0 : (r, l) → r ⊗ T l, which is initial -for any solution (S, σ) there is a morphism s 0 : S 0 → S such that σ = s 0 • σ 0 . In this case, the solution belongs to C. Multiplication in S 0 and action of S 0 on R are as follows:for any l ∈ L, r ∈ R} with σ 1 : (l, r) → ((−l)r, l(r−)) is another solution. It is terminal, since any monoid acting on L (R) and satisfying the compatibility conditions can be represented in S 1 . Multiplication and action of S 1 are following:1 A quantale is a complete join-semilattice equipped with associative binary multiplication distributing over arbitrary joins, or, a semigroup in the category of complete join-semilattices. For more information on quantales and their modules, see e. g. [7,8,9].

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Girard Couples of Quantales

Cited by 4 publications

References 14 publications

Higher Structures in Algebraic Quantum Field Theory

Higher Structures in Algebraic Quantum Field Theory

Quantum triads: an algebraic approach

Generalizing the concept of quantum triads

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