### Cospan Span (Graph)

We (Nicoletta and I) have been lecturing at the University of Milan at Bicocca this fortnight on our work on an algebra of automata for application to concurrent, distributed, hierarchical systems partly described in this post.

One thought occurred to me before the lecture today which I do not usually emphasize in lectures to computer scientists: the algebra we describe is really standard categorical algebra invented for completely different reasons.

For example the (bi)category of spans (and hence also cospans) was invented in 1967 (if I remember correctly) by Jean Benabou. Followers of the n-category cafe will see its use in other fields. The structure we use on Span(Graph) was introduced by Carboni and me in studying relations but its roots goes back to Frobenius (google Frobenius algebras).

The structure which includes both spans and cospans seems new though quite in line with other developments. The computer science interpretation is due to us and collaborators.

I mention all this because similar models introduced by Computer Scientists, such as Asynchronous automata, automata of Arnold and Nivat, Team automata etc seem to me to be ad hoc constructions with little connection with other parts of science.

Of course using standard mathematics might be a way of forcing mathematics on an application. We don't believe that is so in this case.

One thought occurred to me before the lecture today which I do not usually emphasize in lectures to computer scientists: the algebra we describe is really standard categorical algebra invented for completely different reasons.

For example the (bi)category of spans (and hence also cospans) was invented in 1967 (if I remember correctly) by Jean Benabou. Followers of the n-category cafe will see its use in other fields. The structure we use on Span(Graph) was introduced by Carboni and me in studying relations but its roots goes back to Frobenius (google Frobenius algebras).

The structure which includes both spans and cospans seems new though quite in line with other developments. The computer science interpretation is due to us and collaborators.

I mention all this because similar models introduced by Computer Scientists, such as Asynchronous automata, automata of Arnold and Nivat, Team automata etc seem to me to be ad hoc constructions with little connection with other parts of science.

Of course using standard mathematics might be a way of forcing mathematics on an application. We don't believe that is so in this case.

Labels: category theory, computing

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