The way to run conjunctive SQL queries forward and backward is described in this paper, https://www.cambridge.org/core/journals/journal-of-functiona... , (also available on the arxiv), where they are referred to as query 'evaluation' and 'co-evaluation', respectively. We never would have been able to discover co-evaluation if not for category theory! The previous link includes this paper, and many others.
Bi-directional data exchange has many uses. For example, given a set of conjunctive queries Q, because coeval_Q is left adjoint to eval_Q, the composition coeval_Q o eval_Q forms a monad, whose unit can be used to quantify the extent to which the original query Q is "information preserving" on a particular source (so query/data quality). As another example, we use the technique to load data into OWL ontologies from SQL sources, by specifying an OWL to SQL projection query (tends to be easy) and then running it in reverse (tends to be hard). But no doubt more applications await!
Can you point to some examples of owl/sql transforms being flipped? I have trouble believing that an invertible transformation is hard (presumably each step is invertible, right), and certainly "never would have been able to discover" seems inconceivable to me.
Looking at the paper it is very dense and abstract, also 50 pages long.
Edit: on reflection I am doing a bit of sealioning which was not my intention but it does look that way. I'll try to read your paper but if you assure me cat theory really allowed you to do those things you claim, I'll accept you at your word.
You might try pages 8-16 of this presentation: https://www.categoricaldata.net/cql/lambdaconf.pdf . The examples are relational to relational and simplistic but they do illustrate running the same transformation both forward and backward, as well as show the "unit" of such a "monad". We implemented everything in public software, so hopefully the software is even better than my word! As for loading SQL to RDF specifically, I'd be happy to share that technique, but it isn't public yet- please ping me at ryan@conexus.com.
Could this possibly be explained to the average programmer, who doesn't have the foggiest notion what conjunctive queries, coevaluation, or monads are?
