Given a category C with finite limits, it is well-known that one may form a category with morphisms spans in C, and composition given by pullback. Furthermore, when an epi-mono factorisation system is available, one may form a category with morphisms jointly-monic spans in C. In the category Set this construction gives relations between sets. Dually, one may talk of categories of cospans and corelations.
Given a cospan or corelation X→N←Y, we may equip the so-called apex N of the cospan with additional structure, such as a finite graph with vertices N, or a real valued function on N. We call this additional structure a decoration, and the cospan a decorated cospan. We shall discuss the conditions under which this construction gives a category, and how to construct functors between such categories.
As an example, we shall show that a class of electrical circuit diagrams can be considered as morphisms in a decorated cospan category, and that the semantics of these circuit diagrams may be view as a functor from this category to the category of linear relations.
This is an expanded version of my Octoberfest 2015 talk. Joint work with John Baez (UC Riverside).