Categorical Universal Algebra, Fibrations, Bidirectional Transformations, and Categorical Universal Algebra again

Abstract

Categorical universal algebra, Grothendieck fibrations and double categories were all developed in a remarkably fertile short period in the early 1960s.  This talk will review them and show how they are unexpectedly intertwined in recent work on bidirectional transformations, a mathematical concept designed to model and support interoperating systems. 

Categorical universal algebra turns out to provide a new and unexpectedly simple model for the specification and  analysis of information systems (think databases or business processes).  In the context of that model, the study of fibrations provides a rigorous solution to the long-standing view update problem, and that solution supports the development of interoperating systems.  Meanwhile, other workers have attempted to mathematically analyse interoperating systems and they developed the bidirectional transformations framework and supporting technologies.  In both my category theoretic work developing interoperating systems, and in the bidirectional transformation work, solutions weaker than fibrations, and apparently ad hoc, have sometimes seemed to be needed.  We'll discuss this, and the mathematical solution offered by another category theorist, Zinovy Diskin, called delta lenses.  Finally, to complete the cycle we'll see how despite my initial scepticism, work by my student Bryce Clarke resolved the elegant mathematical nature of delta lenses and raised new questions in double category theory.

The talk includes joint work with Kit Dampney, and with Robert Rosebrugh and independent work done by Bryce Clarke.