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Elements of enriched and quantum category theory
... This is a rather philosophical consideration which corresponds to many proper mathematical approaches or methods for establishing "quantum category theory" (e.g.Rennela, Staton 2018;Moskaliuk, Moskaliuk 2013;Chikhladze 2011;Davis 2006; Holdsworth 1977, etc.). ...
Lewis Carroll, both logician and writer, suggested a logical paradox containing furthermore two connotations (connotations or metaphors are inherent in literature rather than in mathematics or logics). The paradox itself refers to implication demonstrating that an intermediate implication can be always inserted in an implication therefore postponing its ultimate conclusion for the next step and those insertions can be iteratively and indefinitely added ad lib, as if ad infinitum. Both connotations clear up links due to the shared formal structure with other well-known mathematical observations: (1) the paradox of Achilles and the Turtle; (2) the transitivity of the relation of equality. Analogically to (1), one can juxtapose the paradox of the Liar (for Lewis Carroll’s paradox) and that of the arrow (for “Achilles and the Turtle”), i.e. a logical paradox, on the one hand, and an aporia of motion, on the other hand, suggesting a shared formal structure
... This is a rather philosophical consideration which corresponds to many proper mathematical approaches or methods for establishing "quantum category theory" (e.g.Rennela, Staton 2018;Moskaliuk, Moskaliuk 2013;Chikhladze 2011;Davis 2006; Holdsworth 1977, etc.). ...
Lewis Carroll, both logician and writer, suggested a logical paradox containing furthermore two connotations (connotations or metaphors are inherent in literature rather than in mathematics or logics). The paradox itself refers to implication demonstrating that an intermediate implication can be always inserted in an implication therefore postponing its ultimate conclusion for the next step and those insertions can be iteratively and indefinitely added ad lib, as if ad infinitum. Both connotations clear up links due to the shared formal structure with other well-known mathematical observations: (1) the paradox of Achilles and the Turtle; (2) the transitivity of the relation of equality. Analogically to (1), one can juxtapose the paradox of the Liar (for Lewis Carroll’s paradox) and that of the arrow (for “Achilles and the Turtle”), i.e. a logical paradox, on the one hand, and an aporia of motion, on the other hand, suggesting a shared formal structure of both, which can be called “ontological”, on which basis “motion” studied by physics and “conclusion” studied by logic can be unified being able to bridge logic and physics philosophically in a Hegelian manner: even more, the bridge can be continued to mathematics in virtue of (2), which forces the equality (for its property of transitivity) of any two quantities to be postponed analogically ad lib and ad infinitum. The paper shows that Hilbert arithmetic underlies naturally Lewis Carroll’s paradox admitting at least three interpretations linked to each other by it: mathematical, physical and logical. Thus, it can be considered as both generalization and solution of his paradox therefore naturally unifying the completeness of quantum mechanics (i.e. the absence of hidden variables) and eventual completeness of mathematics as the same and isomorphic to the completeness of propositional logic in relation to set theory as a first-order logic (in the sense of Gödel (1930)’s completeness theorems).
Alain Bruguieres, in his talk [1], announced his work [2] with Alexis
Virelizier and the second author which dealt with lifting closed structure on a
monoidal category to the category of Eilenberg-Moore algebras for an opmonoidal
monad. Our purpose here is to generalize that work to the context internal to
an autonomous monoidal bicategory. The result then applies to quantum
categories and bialgebroids.
A weak bialgebra is known to be a special case of a bialgebroid. In this paper we study the relationship of this fact with the Tannaka theory of bialgebroids as developed in [4]. We obtain a Tannaka representation theorem with respect to a separable Frobenius fiber functor. Comment: 7 pages
There are various generalizations of bialgebras to their ''many object'' versions, such as quantum categories, bialgebroids and weak bialgebras. These can also be thought of as quantum analogues of small categories. In this paper we study modules over these structures, which are quantum analogues of profunctors (also called distributors) between small categories. Comment: 23 pages
Quantum categories were introduced in [5] as generalizations of both bi(co)algebroids and small categories. We clarify details of that work. In particular, we show explicitly how the monadic definition of a quantum category unpacks to a set of axioms close to the definitions of a bialgebroid in the Hopf algebraic literature. We introduce notions of functor and natural transformation for quantum categories and consider various constructions on quantum structures. © Dimitri Chikhladze, 2011. Permission to copy for private use granted.
We generalize Barr's embedding theorem for regular categories to the context of enriched categories. Comment: 11 pages