The categorical theory of relations and quantizations

• Main Authors: Jakobsen, Per K, Lychagin, Valentin
• Format: Journal Article
• Language: English
• Published: 29-10-2001
• Subjects: Mathematics - Category Theory; Mathematics - Quantum Algebra
• DOI: 10.48550/arxiv.math/0110311

In this paper we develope a categorical theory of relations and use this
formulation to define the notion of quantization for relations. Categories of
relations are defined in the context of symmetric monoidal categories. They are
shown to be symmetric monoidal categories in their own right and are found to
be isomorphic to certain categories of $A-A$ bicomodules. Properties of
relations are defined in terms of the symmetric monoidal structure. Equivalence
relations are shown to be commutative monoids in the category of relations.
Quantization in our view is a property of functors between monoidal categories.
This notion of quantization induce a deformation of all algebraic structures in
the category, in particular the ones defining properties of relations like
transitivity and symmetry.