Yes, for example the dual of join (called 'sigma'), which only exists for databases in categorical normal form. There are many academic papers and an open-source project ('categorical query language') here: http://www.categoricaldata.net.
Sounds interesting! I had a look, but couldn't find out what sigma does. This is not quite the same level as, say, the PostgreSQL documentation:
http://www.categoricaldata.net/help/instancesigma.html
Is there somewhere i can go for a worked example of using sigma?
Sure, here is an example that uses sigma to compute an equivalence relation:
So is this application productive? It's nice to be able to describe something that already exists in terms of category theory, and even nicer if those descriptions are concise and satisfying. But does that description enable anything new? Did we get any new migration operations, normal forms, etc?