Structured Co-spans: An Algebra of Interaction Protocols

We extend the theory of (co-)spans as a means of providing an algebraic approach to complex interactions as they arise in software-intensive systems. In order to make interconnections independent of the nature of components involved, interaction protocols are formalised not in terms of morphisms (i.e. part-of relationships) but a generalised notion of (co-)span in which the arms are structured morphisms – the head (the glue of the protocol) and the hands (the interfaces of the protocol) belong to different categories, the category of glues being coordinated over that of the interfaces. The proposed generalisation sheds some additional light into adjunctions in bicategories, namely on the factorisation of left adjoint 2-sided enrichments.