In categorical proof theory, propositions and proofs are presented as objects and arrows in a category. It offers an answer to the notorious question of the equality of proofs. Diagram chasing comes about in algebra of proofs. The corresponding categorical structure is regular fibration. The examples include stable factorisations, sites, triposes.
Regular logic is exactly what is needed to talk about maps, as total and single-valued relations.
This is a report on a mathematician's effort to understand some concurrency theory. The starting point is a logical interpretation of Nielsen and Winskel's account of the basic models of concurrency. Upon the obtained logical structures, we build a calculus of relations which yields, when cut down by bisimulations, Abramsky's interaction category of synchronous processes. It seems that all interaction categories arise in this way. The obtained presentation uncovers some of their logical contents and perhaps sheds some more light on the original idea of processes as relations extended in time.