We normally construct theories by presuming underlying sets, such as sets of space time points, or sets of vectors in a Hilbert space. I think you can show that leads to weaknesses of realism (see Hardy's Paradox, or Bell's theorem, or Kochen Spekker).
It would be nice to take operationalism as a starting point, in an abstract sense, and then use our operational findings in a lab as a basis to construct theories. A theory generally boils down to some mathematical structure or category such as Hilbert spaces or manifolds.
Recently, Abramsky has been using Chu spaces as a place to talk about operationalism, but I see operationalism as more palatable when we eschew spaces for just the structured categories and the diagrams we find therein. Much like Abramsky's idea, would it be possible to take a categorical viewpoint to operationalism, where we define an arbitrary category as defining our interface to our system under test (he uses Chu spaces)? Then we perform transformations in the laboratory which we consider as more abstract diagrams rather than morphisms who's relevance is based on the structure type we assign to the lab setup. In Abramsky's case, he would say that a laboratory setup and a result set is a Chu space, but we are more interested in the diagram types we generate during our experiments. We then use those diagram types as a foundation to start to build a theory of the "underlying system". Note the vast difference between this approach and the normal realist approach where you say "Okay, here's my set of states and here's my set of observations."
Can anyone agree or disagree that this approach is possible and will also likely eliminate problems of local realism?