Re: only_marketing_?

[Bob Coecke <bob.coecke@cs.ox.ac.uk>]

Hi Steve, John and everyone, 

From the perspective of quantum theory there of course is still a clear input-output thing going on, as these morphisms are subject to the causality of the world in which we live, captured by the symmetric monoidal structure that caries that causal structure.  Then the key thing is not to be Cartesian i.e. the whole is not the sum of the parts.   Parts may even be pure noise while the whole is a sharp state.  In that context, compact closure is a beautifiul thing given that, just like being Cartesian, it is more a property than it is a structure, due to being `maximally wholistic’.

Compactness seems to appear everywhere when you look around in the real world, for example in language which is also compact closed if one adopts Lambek’s pregroups for the grammar, and then attaching meaning to things gives one a thick (non-symmetric) compact closed category.  Here, inputs and outputs truly become meaningless it seems.  

—

As a pure category theory problem, there is still something interesting to be done to better characterise compactness as a property.  We made some (admittedly very modest) baby steps here:

https://arxiv.org/abs/1803.00708 <https://arxiv.org/abs/1803.00708>
https://arxiv.org/abs/1805.12088 <https://arxiv.org/abs/1805.12088>

I believe that compact closure deserves the same attention as the Cartesian structures has gotten, if not more.  The reason to say more is that this case is further away from how me (are made to) think, so having some maths in place is particularly useful.

—

On a related note, in 2001 I was about to leave Montreal for Oxford, but then 9/11 happened, and the planes didn’t go for a while.  A number of other people turned out to be in Montreal too those days, including Bill Lawvere, and there were some talks at UDM (can’t remember if this was because of being stuck, or planned).  Bill Lawvere explained then very nicely the `real world motivation’ for many of the decissions he made for developing certain parts of Category theory, mostly drawn from how to organise society .  That was an eye-opener.  

One typically thinks of computer science as the main domain to drive parts of the development of category theory from an applied perspective, but some of the fathers of the field clearly also had other real world motivations beyond pure mathematics, which makes perfect sense.  Category theory deserves to be about the real world, and hence play an important role beyond pure maths, just like many other developments in math like geometry and analysis.  That’s why some of us started with the dedicated ACT conference + school series.

The thing to do now is not to just get category theory about the real world, but also into the real world, and that’s not going to happen by just writing papers.  The investment hunger seems to be there now, so let’s do it!   Personally, I am going to give up part of my academic position to do just that.

Cheers, Bob.

> On 20 Aug 2019, at 09:55, Steve Vickers <s.j.vickers@cs.bham.ac.uk> wrote:
> 
> 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.

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