Dear Steve and all, I heard Philip Higgins talk to the BMC at Liverpool in 1960 or so. This work was published as Algebras with a scheme of operators}.Math. Nach. {27} (1963) 115--132. and it is about algebras whose operations are partially defined. We discussed something like this, in his home, in 1967, after a seminar I gave on van Kampen's theorem with a set of base points, discussing possible cubically defined compositions, but joint work did not get going till 1974. See my "philosophy" paper "Modelling and computing homotopy types: I" for the 2018 Brouwer honor volume of Indigationes. My interest in this started in 1965 partly on reading a few pages of Ehresmann's "Cat\'egories et Structure" and saw that it seems double groupoids could possibly be " more nonabelian" than groups. What then could be a homotopy double groupoid? Nine years later ...! On submitting a joint paper, I was told that "It had been seen by two international authorities and editors of a journal should not embarrass one another. " I replied "Tell me what is wrong" and after a 3rd referee it finally appeared much cut. This use for local-to-global questions I now give the tag "Algebraic inverses to subdivision" and it is done cubically since I can't do it simplicially (although a polyhedral groupoid version was David Jones 1984 thesis "Poly T-complexes" (available from Rosprawy Math 266). Hope that helps. Ronnie groupoids.org/uk ------ Original Message ------ From: "Steve Vickers" <s.j.vickers@cs.bham.ac.uk> To: "John Baez" <baez@math.ucr.edu>; "categories" <categories@mta.ca> Sent: Tuesday, 20 Aug, 2019 At 09:55 Subject: categories: Re: only_marketing_? Dear John, Those are rather pertinent examples, as the dagger closed and hypergraph categories show up a weakness in my question. I asked about seeking objects, morphisms, identities and associative composition, which seems very natural because it's the basic definition of category. Everything has a domain and a codomain, an input and an output, and composition is malformed unless it's domain with codomain. This leads many of our category theoretic intuitions to be based on thinking of objects and morphisms as being, at some level of abstraction, like sets and functions. Once you have set up the structure of what is input and what is output, it takes some effort to forget it. Dagger closed and the associated string diagrams provide a mechanism for doing that. A good example is Rel. A morphism from X1 x ... x Xm to Y1 x ... x Yn is just a subset of X1 x ... x Xm x Y1 x ... x Yn, in the light of which it is perhaps perverse to impose domain and codomain structure - unless, perhaps you want to carry on to say which relations are functional. As you propose, this certainly looks like a good way to analyse networks, and open systems where there is an interface between internal structure and external behaviour, an interface along which we must compose components. I've heard Jamie Vicary and others use the word "compositionality" as something not quite the same as category theory. Is this what they mean, letting go of the strict domain-codomain discipline? All the best, Steve. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]