generalise research on regular categories to Typeside

Even Patterson in his 2017 paper "Knowledge Representation in Bicategories of Relations" makes the case that RDF and OWL can be be seen to emerge out functorial semantics mapping a small bicategory of relations (the Schema) onto Rel, and taking the Grothendieck Construction of that functor. See also Regular and relational categories: Revisiting ‘Cartesian bicategories I’.

One interest of Functorial Semantics is that it allows one to work on mappings between schemas. (How much of this still works in regular categories is not yet clear to me). But one problem is that such a mapping could easily transform numbers and literals into other isomorphic sets of objects. Spivak et al have found a number of answers to this.

• The 2012 paper Functorial Data Migration ends with a section on "Data types and filtering" that expalins the problem.
• Improving on Database Queries and Constraints via Lifting Problems the paper Algebraic Data Integration adds Algebraic types to solve this problem.

So the question would be: how to apply typeside to regular categories?

This could be useful to the RDF community as the evolution of the specs seem to have created some complications in this area. See Antoine Zimmermann response on the semantic web mailing list to a thread on the topic of regular logic.