Re: only_marketing_?

[Steve Vickers <s.j.vickers@cs.bham.ac.uk>]

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.


> On 17 Aug 2019, at 04:44, John Baez <baez@math.ucr.edu> wrote:
> 
> Hi -
> 
> Steve wrote:
> 
>> So, to return to John Baez's interview, how might we look for category
> theory
>> helping to understand the world's problems? We must first look for
> objects and
>> morphisms,with identities and associative composition, so what are the
> real-world
>> prototypes of what we are trying to do there? What is the first step
> beyond the
>> vague aspirations?
> 
> The interviewer didn't give me a chance to say much. Personally I've been
> trying to understand the various kind of "networks" that come up in
> electrical engineering:
> 
> https://arxiv.org/abs/1504.05625
> 
> control theory:
> 
> https://arxiv.org/abs/1405.6881
> 
> chemistry:
> 
> https://arxiv.org/abs/1704.02051
> 
> and the study of Markov processes:
> 
> https://arxiv.org/abs/1508.06448
> 
> Researchers in these and many other subjects use diagrams to describe
> the networks they're working with.   These diagrams are morphisms in
> various symmetric monoidal categories.   So there are already plenty of
> symmetric monoidal categories being put to work in applied math.
> 
> But which ones, exactly?  That's what my papers are about.  These
> categories
> turn out to be beautiful and not always familiar; trying to understand them
> is
> making my students and me come up with new ideas.   So, right now, I'd say
> researchers in these subjects have more to teach category theorists than
> vice
> versa.
> 
> Best,
> jb
> 


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