Monoidal Quantaloids
Gejza Jenča, Bert Lindenhovius
TL;DR
This work develops the theory of symmetric monoidal quantaloids, with dagger compact quantaloids (notably qRel and V-Rel) serving as noncommutative analogues of Rel. It introduces Matr(Q) as the universal biproduct completion, enabling internalization of classical structures such as maps, preorders, and power objects within a dagger framework, and shows how orthomodularity arises from dagger kernels. The paper then builds internal maps and internal preorders, analyzes embeddings of sets, and establishes existence/reconstruction results for power objects and internal homs, linking these notions to a quantum analogue of power allegories/topoi. The resulting framework provides a robust categorical setting for quantizing mathematical structures and for exploring quantum analogues of classical order-theoretic concepts, with qRel and qSet as central, guiding examples and potential applications to quantum topology and logic.
Abstract
We investigate how to add a symmetric monoidal structure to quantaloids in a compatible way. In particular, dagger compact quantaloids turn out to have properties that are similar to the category Rel of sets and binary relations. Examples of such quantaloids are the category qRel of quantum sets and binary relations, and the category V-Rel of sets and binary relations with values in a commutative quantale V. For both examples, the process of internalization structures is of interest. Discrete quantization, a process of generalization of mathematical structures to the noncommutative setting can be regarded as the process of internalizing these structures in qRel, whereas fuzzification, the process of introducing degrees of truth or membership to concepts that are traditionally considered either true or false, can be regarded as the process of internalizing structures in V-Rel. Hence, we investigate how to internalize power sets and preordered structures in dagger compact quantaloids.