Re: only_marketing_?

[Scott Morrison <scott@tqft.net>]

(On the subject of "letting go of the strict domain-codomain discipline"...)

In my work with Kevin Walker (e.g. https://arxiv.org/abs/1009.5025,
particularly the disklike n-categories of \S 6) we talk about
n-categories for which notions of input and output are solely "in the
eye of the beholder", and argue that this is a convenient and natural
language for the role of categories in TFT.

As an application of this, in a very recent paper with Kevin and Paul
Wedrich, https://arxiv.org/abs/1907.12194, we give what is arguably
the "first interesting example" of a 4-category, built out of Khovanov
homology. Formalising this gadget as a disklike 4-category, we can say
enough to produce invariants of oriented 4-manifolds. Attempting
instead to formalise this gadget as a "conventional" "domain-codomain"
4-category, we were much less satisfied --- we can check the axioms
for a braided monoidal 2-category, but after that it's not
particularly clear which duality properties one would need to check
(in Lurie's language, perhaps this is working out what an SO(4)-fixed
point structure actually is?) in order to continue on to building
4-manifold invariants.

best regards,
Scott

On Wed, Aug 21, 2019 at 3:56 AM 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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