I would add something between 2 and 3 about Triples (allright, monads) and Equational theories. Here is an example of the sort of thing we are up against. A colleague called me this morning because a student had taken a set of notes (in French) on his course and was interested in publishing it. My colleague had an objection because in describing conformal isomorphism from the complex plane (or maybe sphere) to itself, the student had used the word "towards" (vers) instead of "on". His objection was that a conformal isomorphism was something between two spaces, not from one to the other. My answer was a specific such map was a map from one to the other. His reply essentially was, "Oh, it's category theory language. Well, I won't allow any of that in MY notes. No analyst would use that language." Michael On Mon, 21 Dec 2009, Joyal, André wrote:

In my message to John Baez, I wrote:

...

I can distinguish approximatly 6 major currents:

...

1) Algebraic topology and homological algebra 2) Abelian categories 3) Algebraic Geometry and topos theory 4) Logic and elementary topos theory 5) Category theory and computer science 6) Higher categories with homotopy theory

The list is too restrictive. I would like to expand it further:

1) Algebraic topology and homological algebra 2) Abelian categories 3) Algebraic geometry and topos theory 4) General cartesian algebra 5) Categorical logic 6) Homotopical algebra 7) Elementary topos theory and set theory 8) Monoidal categories and enriched category theory 9) General tensor algebra and coalgebra 10) Category theory and computer science 11) Quantum field theory 12) Higher categories and homotopy theory

Algebraic theories and limit sketches are included in (4). Multicategories, operads are included in (9).

I have included Quillen homotopical algebra in (6).

Best, André

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On Mon, 21 Dec 2009 07:20:29 PM EST, Michael Barr <barr@math.mcgill.ca> reminisced:

... His reply essentially was, "Oh, it's category theory language. Well, I won't allow any of that in MY notes. No analyst would use that language."

Amusing tale, reminding me of Sammy's response when I approached him, oh so many decades ago, about supervising my proposed thesis on functorial measure theory: "Measure theory? That's analysis, isn't it? Go ask an analyst." Cheers, and Seasons' Greetings, -- FEJ Linton [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Dear Michael, 2009/12/21 Michael Barr <barr@math.mcgill.ca>

... His reply essentially was, "Oh, it's category theory language. Well, I won't allow any of that in MY notes. No analyst would use that language."

There's an easy reply to people infected with such silliness -- ask them if Terry Tao is an analyst, to which they'd probably reply "of course", and then tell them go to Tao's blog http://terrytao.wordpress.com/ and do a search for "category" (the search bar on Terry's page is on the left just below "recent comments"). The results will speak for themselves. Mark Weber [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Responding to my tale of Sammy's having waved me away with his "Measure theory? That's analysis, isn't it? Go ask an analyst." Michael Barr quite correctly remembered, in a private email, "But you did ask an analyst, as I recall. Lorch was your advisor." True. But only after Sammy's "rejection," on the grounds given, and Dick Kadison's subsequent rebuff after I approached him: "Measure Theory? Integration? That's Functional Analysis. I do Operator Theory. Go find a functional analyst." Edgar Raymond Lorch was rather more ... umm ... open-minded :-) . Though at one point, after I handed him the nth installment of my draft thesis to peruse, he did echo the words of Jesus, on the cross, when he was given a cup of vinegar to quench his thirst: "Oh, Lord, must I drink of this?" Cheers, -- FEJ Linton [For admin and other information see: http://www.mta.ca/~cat-dist/ ] Status: RO

Michael Barr wrote:

I would add something between 2 and 3 about Triples (allright, monads) and Equational theories.

I agree. Let me expand the list further: 1) Algebraic topology and homological algebra 2) Abelian categories 3) Adjoint functors and monads 4) Algebraic geometry and topos theory 5) General universal algebra 6) Limit sketches and locally presentable categories 7) Categorical logic 8) Categorical model theory 9) Homotopical algebra 10) Elementary toposes theory and set theory 11) Monoidal categories and enriched category theory 12) General tensor algebras and coalgebras 13) Category theory and computer science 14) Quantum field theory 15) Higher categories and homotopy theory Best, André -------- Message d'origine-------- De: Michael Barr [mailto:barr@math.mcgill.ca] Date: lun. 21/12/2009 14:20 À: Joyal, André Cc: categories@mta.ca Objet : Re: categories: additions I would add something between 2 and 3 about Triples (allright, monads) and Equational theories. Here is an example of the sort of thing we are up against. A colleague called me this morning because a student had taken a set of notes (in French) on his course and was interested in publishing it. My colleague had an objection because in describing conformal isomorphism from the complex plane (or maybe sphere) to itself, the student had used the word "towards" (vers) instead of "on". His objection was that a conformal isomorphism was something between two spaces, not from one to the other. My answer was a specific such map was a map from one to the other. His reply essentially was, "Oh, it's category theory language. Well, I won't allow any of that in MY notes. No analyst would use that language." Michael [For admin and other information see: http://www.mta.ca/~cat-dist/ ] Status: RO

bob coecke proposed to add quantum computing to andre joyal's list of important directions of categorical research, but andre rejected it. i cannot overstate my respect for andre's work and judgement. but this left me pondering. like andre, i must confess that i am quite ignorant about quantum computing. (unlike andre, i am also ignorant about many other categorical topics on his list.) but we probably all know the following. most results in quantum computing are theorems about hilbert spaces. quantum computing is a *tensor calculus*. but it is a tensor calculus of a special kind: it attempts to describe a wildly unintuitive world. even the greatest contributors, like von neumann and feynman, deplored the gap between the quantum world, imposed on us in the lab, and the intuitions imposed on us in everyday life. now category theory often helps where the common intuitions fail. many of its applications demonstrate this. so quantum computation might be an opportunity for an effective application of *geometry of tensor calculus*. is it really wise to reject an attempt to develop this, as objectionable as it might be in any details? physicists like string diagrams, category theorists like string diagrams. most communities would actively reach out... is it just my impression, or are category theorists a little more sceptical about the value of applications than most mathematical communities? they seem to seek a recognition that categories are useful across mathematics, but then hesitate to recognize the depth and value of the applications in the other areas. --- can it be that we suffer from a superiority complex of some sort? the questions raised in the *well kept secret* thread were: 1) why are the achievements of category theory not recognized publicly? 2) what have we done to deserve the opprobium? 3) how can we convince the sceptics? please allow me to add one more: 4) how can the achievements of category theory be used to expand its future developments and applications, and not to constrain them? with best wishes, -- dusko [For admin and other information see: http://www.mta.ca/~cat-dist/ ] Status: RO

In defense of Andre's list, the explanation he gave for his original list was that subjects in category theory become hot from time to time in response to factors such as new developments outside category theory. The list was supposed to be a list of categorical subjects, not a list of the respective developments that have inspired their use and advancement. The current use of category theory in quantum foundations is clearly an interesting development, and has inspired new work in category theory. But I would still be comfortable, for the time being, in classifying this new work as falling within the existing subjects of "monoidal categories" and "category theory and computer science" on Andre's growing list. Recently also "topos theory" due to the work of Andreas Doering, Klaas Landsman, and others on topos models for basic physics. As for the completeness result that Thorsten mentioned, the reference is: M. Hasegawa, M. Hofmann and G. Plotkin, "Finite dimensional vector spaces are complete for traced symmetric monoidal categories". In Pillars of Computer Science: Essays Dedicated to Boris (Boaz) Trakhtenbrot on the Occasion of His 85th Birthday, Springer LNCS 4800, pages 367-385, February 2008. The result is that an equation holds in all traced symmetric monoidal categories if and only if it holds in finite dimensional vector spaces. An immediate corollary is that the analogous result holds for compact closed categories. A simplified proof, and extension to dagger compact closed categories (w.r.t. finite dimensional Hilbert spaces), can be found here: P. Selinger, "Finite dimensional Hilbert spaces are complete for dagger compact closed categories". Extended abstract, to appear in Proceedings of the 5th International Workshop on Quantum Physics and Logic (QPL 2008), Reykjavik, 2010. Merry Christmas to all, -- Peter

Dear Mike, Of course, in functional programming the applications of categories go far beyond lambda terms. (1) At a fairly elementary level, the treatment of list types in functional programming languages gives a good lead in to universal properties (e.g. list type = free monoid or free (empty, cons)- algebra). Things you do with the universal properties are present as well known tools in functional programming with lists. The universal properties can then be used to motivate the abstract structure of categories: they describe data types by their external interfaces with the rest of the world rather than by their concrete internal structure, and the morphisms play the role of saying what the external interactions are. Expositionally, for Mac Lane universal properties were an important example where the working mathematician had been doing category theory all along without knowing it. (2) More advanced, Haskell has made important use of monads as a programming technique for bringing side-effects, I/O etc into functional programming in an elegant way. (The way this came about is that it has long been more or less self-evident that categories are just what you need for describing the semantics, and the categorical experience of the semanticists led to the practical application of monads.) So it could be that the best way forward is to teach them Haskell first. (I gave a short introduction to categories at Imperial, in the Compujting Department, and I exploited heavily the fact that they had all done Miranda, a predecessor of Haskell.) Regards, Steve Vickers. On 22 Dec 2009, at 00:39, Mike Stay wrote:

On Mon, Dec 21, 2009 at 12:43 AM, Joyal, André <joyal.andre@uqam.ca> wrote:

...

In my message to John Baez, I wrote:

...

I can distinguish approximatly 6 major currents:

...

5) Category theory and computer science

I'm trying to expose my fellow programmers to the joys of category theory, but none of them have a math or physics background (or even a funcitonal programming background), which is where most of my experience with CT has been.

What have been the major applications of category theory to computer science that have affected programmers? Are there new algorithms? Are there really nice ways of solving certain problems? The fact that data types with equivalence classes of lambda terms between them form a cartesian closed category doesn't seem to inspire them very much. -- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com

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

Dear Mike,

Thanks, everyone for your replies! Many of you suggested the same approach as Steve, functional programming and monads. At Google, however, we use Java, C++ and Python (collectively "JCP") for programs that run on our servers and JavaScript for programs that run in our webpages. So there's not a lot of call for learning a functional programming language either.

It might be worth noting that JavaScript is a functional language. (It has a lambda operator ("function"), closures, and can pass functions as parameters and return values.) However, because it has eager evaluation, the whole monad business does not apply, at least not in the way it applies to Haskell. In fact, JavaScript is probably the most widely used functional language on the planet. But there are two strange phenomena: - Functional-programming experts keep on overlooking JavaScript (probably because it is so ugly from a theorists point of view) - Most professional JavaScript programmers fail to see the enormous functional potential of JavaScript. It is a very strange situation: the whole world uses a functional language and almost nobody is aware of it. Anyway, even though I am very category-prone, I must admit that category theory might be a very tough sell for the JavaScript crowd :) Best, Carsten [For admin and other information see: http://www.mta.ca/~cat-dist/ ] Status: RO

Hi Steve & Mike, I completely agree with Steve, I'd like to add that instead of functional programming we could have said "mathematically structured programming". But then indeed the two terms are very closely related. Category Theory helps us to structure abstractions. In Computer Science and in other areas (e.g. Physics). Some people seem to think that abstractions don't buy your anything concrete. E.g. they don't deliver faster algorithms or new physical theories. These people often overlook that everything they do relies essentially on abstractions which have been established a while ago. Hence, while it is hard to measure the impact of abstractions exactly, IMHO it is almost impossible to underestimate their value. Cheers, Thorsten On 23 Dec 2009, at 11:19, Steve Vickers wrote:

Dear Mike,

Of course, in functional programming the applications of categories go far beyond lambda terms.

(1) At a fairly elementary level, the treatment of list types in functional programming languages gives a good lead in to universal properties (e.g. list type = free monoid or free (empty, cons)- algebra). Things you do with the universal properties are present as well known tools in functional programming with lists. The universal properties can then be used to motivate the abstract structure of categories: they describe data types by their external interfaces with the rest of the world rather than by their concrete internal structure, and the morphisms play the role of saying what the external interactions are. Expositionally, for Mac Lane universal properties were an important example where the working mathematician had been doing category theory all along without knowing it.

(2) More advanced, Haskell has made important use of monads as a programming technique for bringing side-effects, I/O etc into functional programming in an elegant way. (The way this came about is that it has long been more or less self-evident that categories are just what you need for describing the semantics, and the categorical experience of the semanticists led to the practical application of monads.)

So it could be that the best way forward is to teach them Haskell first. (I gave a short introduction to categories at Imperial, in the Compujting Department, and I exploited heavily the fact that they had all done Miranda, a predecessor of Haskell.)

Regards,

Steve Vickers.

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