John Baez wrote, "some companies are starting to hire people in applied category theory". Actually, IBM hired Joe Goguen in 1971: http://cseweb.ucsd.edu/~goguen/pps/beatcs-adj.ps And engineers at Galois Inc (founded 1999) and Kestrel Technologies (founded 2000) have been using category theory in industry for quite some time. I'm sure there are many other examples. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Indeed there are many. I tried out a posting on real world applications of CT some year ago. I did receive answers, but only a few, and they were stylishly scattered, so I never summarized that situation. It could be done again in some form, I wold like to believe. Kind of an inventory, or similar. In the end, the CT community being able to point at some success stories in these respect, would be useful in many ways and for many, I can imagine. And they will inspire to come up with more. --- Microsoft could be mentioned. They shouted 'Bayesian' around 1995-96 (was it?) in Los Angeles Times, pointing finger towards Denmark. Later on, many of us know how monads have been manipulated into functional, F#, LINQ, etc by Microsoft Research at Cambridge. LINQ is favoured by many DBMS programmers using the Visual Studio environment, even if probably only a few are really well versed in how CT and monads support it. --- My own take on the practicality of CT is having a category as kind of canvas for information representation. Expressions and the term functor is then a good example. Being over Set, expressions are just traditional, but being over something else, expressions become annotated with bits and pieces taken from that underlying category. We have been trying out applications using classification and terminology in health care. Expressions using such terminologies, do they come with hidden structures not easily to be identified and recovered if going over Set only? That Microsoft/monad thing is just over Set, as are basically all programming models. --- Best, Patrik On 2019-08-13 22:53, Vaughan Pratt wrote:

"... and Kestrel Technologies(founded 2000) have been using category theory in industry for quite some time"

That would be Kestrel Institute in Palo Alto, founded by Cordell Green in 1981, who've been getting help with category theory from the likes of Dusko Pavlovic, Samson Abramsky, etc. There's a Kestrel Technologies LTD in the UK but they're a family business machining parts for prototypes and low volume production in the automotive and aerospace industries.

Vaughan

On Tue, Aug 13, 2019 at 11:38 AM David Espinosa <david@davidespinosa.net> wrote:

...

John Baez wrote, "some companies are starting to hire people in applied category theory".

Actually, IBM hired Joe Goguen in 1971: http://cseweb.ucsd.edu/~goguen/pps/beatcs-adj.ps

And engineers at Galois Inc (founded 1999) and Kestrel Technologies (founded 2000) have been using category theory in industry for quite some time.

I'm sure there are many other examples.

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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-fo...

...

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/ ]

</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=

--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= 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/ ]

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/ ]

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/ ]

(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/ ]