* Re: only_marketing_?
2019-08-08 7:01 ` only_marketing_? Patrik Eklund
@ 2019-08-12 0:03 ` Vaughan Pratt
2019-08-16 10:40 ` only_marketing_? Steve Vickers
2019-08-12 8:32 ` only_marketing_? John Baez
2019-08-20 23:32 ` only_marketing_? Bob Coecke
2 siblings, 1 reply; 15+ messages in thread
From: Vaughan Pratt @ 2019-08-12 0:03 UTC (permalink / raw)
To: categories
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
On Sun, Aug 11, 2019 at 7:20 AM Patrik Eklund <peklund@cs.umu.se> wrote:
> It's an article that names the company Risk Group LLC (with John Baez as
> co-founder).
>
> The website of Risk Group has 'warfare' as the first menu item in the
> menu row. Under other menus there is AI and such related things that
> industry more in general is interested in. There are relations to health
> care solutions etc, so the ambition is broad.
>
> "Is it pure marketing?" I don't know about 'pure', but it looks like
> marketing, at least partly, and why not? I for one wish Risk Group as a
> company all the best. If category theory becomes more in focus in
> applications serving the public and private sectors in general, also as
> promoted by Risk Group LLC, this is fine.
>
> Possible relations to military application I wouldn't personally
> support, but that's my personal choice. I'm a European, and work e.g.
> with health care aspects as related also to the European Commission.
>
> All the best,
>
> Patrik
>
> www.glioc.com
>
>
>
> On 2019-08-06 17:37, Eduardo J. Dubuc wrote:
>> Hi, I will like to hear opinions of members of this list about the
>> following link:
>>
>>
> https://www.forbes.com/sites/cognitiveworld/2019/07/29/the-future-will-be-formulated-using-category-theory/
>>
>> Is it pure marketing ?
>>
>> Is it serious ?
>>
>> The points in this post can attire more funds to category theory
>> research in general ?
>>
>> The point in this post will absorb for fake category theory research
>> the
>> existing funds in detriment to serious category theory research ?
>>
>> etc etc
>>
>> eduardo dubuc
>>
>>
>
>
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
>
--000000000000658884058fe044a6
Content-Type: text/html; charset="UTF-8"
Content-Transfer-Encoding: quoted-printable
<div dir=3D"ltr"><div>Formulating the <i>future </i>with category theory?=
=C2=A0 My understanding is that the <i>present </i>is already formulated le=
ss with set theory than with category theory, at least its morphisms under =
composition, and has been for a long time.<br></div><div><br></div><div>In =
1969 Jack Schwartz introduced the programming language SETL based on set th=
eory.=C2=A0 However programmers found it much easier to write programs as f=
unctions composed from simpler functions than to implement them with sets b=
ased on membership, and SETL never caught on.</div><div><br></div><div>Ther=
e have also been sporadic attempts to introduce set theory into K-12 mathem=
atics, under such rubrics as "New maths", starting with the opera=
tions of union and intersection, but these have not caught on either.<br></=
div><div><br></div><div>Category theory is an abstract formulation of funct=
ions taking composition as the primitive operation.=C2=A0 Functions are in =
wide use in both mathematics and software.=C2=A0 Whenever a function calls =
another function from within it, that's composition.<br></div><div><br>=
</div><div>In recent years I've been promoting a viewpoint of algebra t=
hat emphasizes the associativity of composition as the root of not just alg=
ebra but bialgebra in the sense of typed Chu spaces.=C2=A0 The defining pro=
perties of both homomorphisms and Chu transforms follow from associativity.=
=C2=A0 My most recent talk on this was at FMCS in June, <a href=3D"http://b=
oole.stanford.edu/pub/fmcs.pdf">the "ten" slides are here</a>.=C2=
=A0 (They're less cryptic when the speaker is there to explain them, es=
pecially with an audience of only one or two and with no time pressure.)<br=
></div><div><br></div><div> One might call this "category light" =
by virtue of working in the <i>class</i> CAT, i.e. just categories, no func=
tors etc.=C2=A0 By Yoneda (unintended pun there, it's actually "bi=
Yoneda")=C2=A0 the functors are there, they're just "under th=
e hood".=C2=A0 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 kno=
w about functors, natural transformations, etc. when you become a "cat=
egory mechanic" so to speak.</div><div><br></div><div>Sets are automat=
ic because they arise wherever morphisms gather together in those spaces we=
call homsets.</div><div><br></div><div>Vaughan<br></div></div><br><div cla=
ss=3D"gmail_quote"><div dir=3D"ltr" class=3D"gmail_attr">On Sun, Aug 11, 20=
19 at 7:20 AM Patrik Eklund <<a href=3D"mailto:peklund@cs.umu.se">peklun=
d@cs.umu.se</a>> wrote:<br></div><blockquote class=3D"gmail_quote" style=
=3D"margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding=
-left:1ex">It's an article that names the company Risk Group LLC (with =
John Baez as<br>
co-founder).<br>
<br>
The website of Risk Group has 'warfare' as the first menu item in t=
he<br>
menu row. Under other menus there is AI and such related things that<br>
industry more in general is interested in. There are relations to health<br=
>
care solutions etc, so the ambition is broad.<br>
<br>
"Is it pure marketing?" I don't know about 'pure', bu=
t it looks like<br>
marketing, at least partly, and why not? I for one wish Risk Group as a<br>
company all the best. If category theory becomes more in focus in<br>
applications serving the public and private sectors in general, also as<br>
promoted by Risk Group LLC, this is fine.<br>
<br>
Possible relations to military application I wouldn't personally<br>
support, but that's my personal choice. I'm a European, and work e.=
g.<br>
with health care aspects as related also to the European Commission.<br>
<br>
All the best,<br>
<br>
Patrik<br>
<br>
<a href=3D"http://www.glioc.com" rel=3D"noreferrer" target=3D"_blank">www.g=
lioc.com</a><br>
<br>
<br>
<br>
On 2019-08-06 17:37, Eduardo J. Dubuc wrote:<br>
> Hi, I will like to hear opinions of members of this list about the<br>
> following link:<br>
><br>
> <a href=3D"https://www.forbes.com/sites/cognitiveworld/2019/07/29/the-=
future-will-be-formulated-using-category-theory/" rel=3D"noreferrer" target=
=3D"_blank">https://www.forbes.com/sites/cognitiveworld/2019/07/29/the-futu=
re-will-be-formulated-using-category-theory/</a><br>
><br>
> Is it pure marketing ?<br>
><br>
> Is it serious ?<br>
><br>
> The points in this post can attire more funds to category theory<br>
> research in general ?<br>
><br>
> The point in this post will absorb for fake category theory research<b=
r>
> the<br>
> existing funds in detriment to serious category theory research ?<br>
><br>
> etc etc<br>
><br>
> eduardo dubuc<br>
><br>
><br>
<br>
<br>
[For admin and other information see: <a href=3D"http://www.mta.ca/~cat-dis=
t/" rel=3D"noreferrer" target=3D"_blank">http://www.mta.ca/~cat-dist/</a> ]=
<br>
</blockquote></div>
--000000000000658884058fe044a6--
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
^ permalink raw reply [flat|nested] 15+ messages in thread
* Re: only_marketing_?
2019-08-12 0:03 ` only_marketing_? Vaughan Pratt
@ 2019-08-16 10:40 ` Steve Vickers
2019-08-16 16:30 ` only_marketing_? Alexander Kurz
2019-08-17 3:44 ` only_marketing_? John Baez
0 siblings, 2 replies; 15+ messages in thread
From: Steve Vickers @ 2019-08-16 10:40 UTC (permalink / raw)
To: categories; +Cc: Vaughan Pratt
Vaughan has hit the nail on the head as far as computer science goes. The object-morphism analysis is such a good match for types and functions in programming languages, especially functional programming, that there is now a lot of applied category theory going on in computer science departments. The universal characterizations of category theory are a perfectly natural way to think of type constructors.
What I missed in John's interview was any sense of a similar match between categorical structure and structure in the world's many problems.
In physics there are certainly some applications with a similar direct correspondence.
The Abramsky-Coecke group at Oxford, clearly influenced by the computer science, took categorical ideas of quantum computation which then spilled over into more general foundational ideas.
But we can see another principle in play that's more profound that simply saying "types are objects, functions are morphisms".
Category theory abstracts away from the internal structure of objects. This is in stark contrast with set-theoretical foundations, which tend to say "everything is a set, so just say which set you are using". Category theory instead stresses how objects interact with each other, using universal characterizations. Computer science takes this as a practical necessity. We don't want to know the "implementation dependent" internal structure of a type or function. That's both impractical and useless, as well as institutionalizing bugs so they become difficult to correct. What we need is the "Application Programmer Interface" (API) of how we can use them. As long as the API is stable, we don't mind if the implementation is changed to fix bugs or make it more efficient. The idea of API matches universal properties in category theory.
For the Oxford group, the programmers' "types are objects, functions are morphisms" mutates into the questions like "How do we really use finite dimensional Hilbert spaces?", with the answer expressed categorically.
This engineering approach applies to mathematics too. Grothendieck's discovery of toposes came out of a categorical analysis of sheaves. Instead of asking what a sheaf is, you ask how they interact with each other, how we use them. In particular, how do we get sheaf cohomology and other topological invariants out of them? Then it doesn't matter what sheaves are, as long as the category of them has the right properties.
I think some of this is part of - to take as an example someone I know - Chris Isham's interest in categories. The scandal of physics is the incompatibility of general relativity and quantum physics, but in both cases it is hard to modify the mathematical structures used without breaking them. It therefore becomes important to ask how we actually use the structures, what is their API, and the hope is that category theory can help.
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?
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
>
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
^ permalink raw reply [flat|nested] 15+ messages in thread
* Re: only_marketing_?
2019-08-16 10:40 ` only_marketing_? Steve Vickers
@ 2019-08-16 16:30 ` Alexander Kurz
2019-08-17 3:44 ` only_marketing_? John Baez
1 sibling, 0 replies; 15+ messages in thread
From: Alexander Kurz @ 2019-08-16 16:30 UTC (permalink / raw)
To: Steve Vickers; +Cc: categories
> 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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
^ permalink raw reply [flat|nested] 15+ messages in thread
* Re: only_marketing_?
2019-08-16 10:40 ` only_marketing_? Steve Vickers
2019-08-16 16:30 ` only_marketing_? Alexander Kurz
@ 2019-08-17 3:44 ` John Baez
2019-08-20 8:55 ` only_marketing_? Steve Vickers
1 sibling, 1 reply; 15+ messages in thread
From: John Baez @ 2019-08-17 3:44 UTC (permalink / raw)
To: categories
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
^ permalink raw reply [flat|nested] 15+ messages in thread
* Re: only_marketing_?
2019-08-17 3:44 ` only_marketing_? John Baez
@ 2019-08-20 8:55 ` Steve Vickers
2019-08-21 3:30 ` only_marketing_? Scott Morrison
0 siblings, 1 reply; 15+ messages in thread
From: Steve Vickers @ 2019-08-20 8:55 UTC (permalink / raw)
To: John Baez, categories
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
>
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
^ permalink raw reply [flat|nested] 15+ messages in thread
* Re: only_marketing_?
2019-08-20 8:55 ` only_marketing_? Steve Vickers
@ 2019-08-21 3:30 ` Scott Morrison
0 siblings, 0 replies; 15+ messages in thread
From: Scott Morrison @ 2019-08-21 3:30 UTC (permalink / raw)
To: Steve Vickers; +Cc: John Baez, categories
(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.
>
>
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
^ permalink raw reply [flat|nested] 15+ messages in thread
* Re: only_marketing_?
2019-08-08 7:01 ` only_marketing_? Patrik Eklund
2019-08-12 0:03 ` only_marketing_? Vaughan Pratt
@ 2019-08-12 8:32 ` John Baez
2019-08-20 23:32 ` only_marketing_? Bob Coecke
2 siblings, 0 replies; 15+ messages in thread
From: John Baez @ 2019-08-12 8:32 UTC (permalink / raw)
Cc: Categories list
Hi -
On Sun, Aug 11, 2019 at 10:20 PM Patrik Eklund <peklund@cs.umu.se> wrote:
> It's an article that names the company Risk Group LLC (with John Baez as
> co-founder).
>
I am not a co-founder of this company. The article does not say that I
am. I have no involvement with this company other than that its founder,
Jayshree Pandya, interviewed me.
Best,
John Baez
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
^ permalink raw reply [flat|nested] 15+ messages in thread
* Re: only_marketing_?
2019-08-08 7:01 ` only_marketing_? Patrik Eklund
2019-08-12 0:03 ` only_marketing_? Vaughan Pratt
2019-08-12 8:32 ` only_marketing_? John Baez
@ 2019-08-20 23:32 ` Bob Coecke
2 siblings, 0 replies; 15+ messages in thread
From: Bob Coecke @ 2019-08-20 23:32 UTC (permalink / raw)
To: Steve Vickers, categories, John Baez
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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