Re: only_marketing_?

[Alexander Kurz <axhkrz@gmail.com>]

> 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?

Wrt real-world prototypes of morphism, John Baez in his recent SYCO 4 talk [1] suggested that the idea of a morphism as a component of a network (and composition as "wiring up'' components) could have applications in many areas outside of computer science, including engineering, biology and ecology.

This is still vague, so I am not pretending to answer Steve's question, but I think it points in a direction that is promising.

All the best,

Alexander

[1] http://events.cs.bham.ac.uk/syco/4/slides/Baez.pdf

Btw, slides of (almost) all talks of SYCO 4 are linked from the schedule at 

http://events.cs.bham.ac.uk/syco/4/

> 
> Steve Vickers
> 
>> On 12 Aug 2019, at 01:03, Vaughan Pratt <pratt@cs.stanford.edu> wrote:
>> 
>> Formulating the *future *with category theory?  My understanding is that
>> the *present *is already formulated less with set theory than with category
>> theory, at least its morphisms under composition, and has been for a long
>> time.
>> 
>> In 1969 Jack Schwartz introduced the programming language SETL based on set
>> theory.  However programmers found it much easier to write programs as
>> functions composed from simpler functions than to implement them with sets
>> based on membership, and SETL never caught on.
>> 
>> There have also been sporadic attempts to introduce set theory into K-12
>> mathematics, under such rubrics as "New maths", starting with the
>> operations of union and intersection, but these have not caught on either.
>> 
>> Category theory is an abstract formulation of functions taking composition
>> as the primitive operation.  Functions are in wide use in both mathematics
>> and software.  Whenever a function calls another function from within it,
>> that's composition.
>> 
>> In recent years I've been promoting a viewpoint of algebra that emphasizes
>> the associativity of composition as the root of not just algebra but
>> bialgebra in the sense of typed Chu spaces.  The defining properties of
>> both homomorphisms and Chu transforms follow from associativity.  My most
>> recent talk on this was at FMCS in June, the "ten" slides are here
>> <http://boole.stanford.edu/pub/fmcs.pdf>.  (They're less cryptic when the
>> speaker is there to explain them, especially with an audience of only one
>> or two and with no time pressure.)
>> 
>> One might call this "category light" by virtue of working in the *class*
>> CAT, i.e. just categories, no functors etc.  By Yoneda (unintended pun
>> there, it's actually "biYoneda")  the functors are there, they're just
>> "under the hood".  Just as you only need to know what sort of engine is in
>> the car you're driving if you have to service it, you only need to know
>> about functors, natural transformations, etc. when you become a "category
>> mechanic" so to speak.
>> 
>> Sets are automatic because they arise wherever morphisms gather together in
>> those spaces we call homsets.
>> 
>> Vaughan



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