September, 2017. Fred E. J. Linton. For most of the time I knew him, I kept secret what E. and J. stand for, being of the impression that almost nobody knew and it was not my place to tell. I think it is OK to tell you now. Ernest and Julius. Fred spent his whole teaching career at Wesleyan University in Middletown Connecticut. I entered the graduate program there in 1963-64, at the tender age of 20. I do not recall meeting Fred during that first year. In my second year, I developed an interest in categories from my coursework. But Fred wasn't there. He was visiting Mac Lane in Chicago. By the end of the second semester, I knew I wanted to meet him. By good fortune, we both attended the NSF summer program (six weeks long!) in homological algebra at Bowdoin College (principal lecturer Ernst Snapper of Dartmouth). We met right away, and immediately played tennis. I have two memories regarding Fred from this time. (1) Everybody ate lunch and dinner in the dining hall. At one lunch, they served chili. In order to break up the crackers into small pieces, Fred put the packet on the table and applied great force with his elbow. The second time, Alex Rosenberg held his ears. (2) As many of you have noticed, Fred often napped in between talks at a conference. Perhaps you took this as a sign of age. Not so. Fred always did this. At the Bowdoin conference 52 years ago, there was also a music camp with many prominent musicians in residence. Two violinists came over to Fred, asleep on a chair in the lounge, and played Brahm's Lullibye. It didn't wake him up. In Middletown, Fred was very active in folk dancing. His group was very professional and gave quite astounding performances at local venues. This was a major interest in his life. In 1966, we arranged with Wesleyan that I could follow Fred to Zurich in order to attempt to write a thesis. About a week before we were to leave, I asked him about the research focus of the group of category theorists visiting the ETH. He simply replied "triples" with no further explanation. I wondered what on Earth I was getting into. Fred often began a conversation with word play, even if he hadn't seen you for years. His puns drew on English, German, French and Italian. Once when the two of us were trying to negotiate downtown San Juan (at one of Jon Beck's conferences) with neither of us knowing Spanish, he asked for information in Italian; he got strange looks, but it seemed to work. I'm not sure how I settled on "A Triple Miscellany" for a thesis title, but Fred preferred several variants. His favorite was "A Missal tripleary". As recently as a few months ago in Schenectady, Fred chided me for not using ..Missal... Beginning 1969-1970, Fred and I joined a lively group of category theorists for postdocs at Dalhousie University. The one shortcoming for Fred was the lack of a good folk dancing group, so he started one for amateurs. He somehow convinced my wife and I to join. We had fun, but I never learned the Miserlou. Recently I mentioned the names of one or two from that group to Fred. He had kept up with them. I am a graduate of Los Angeles High School and so I grew up steeped in the culture of fixing cars. Fred turned to me for advice on various car problems. One time, his engine just wouldn't start and he asked for help. There was no fuel coming through to the carburetor. Now any California kid knows that either the fuel pump was shot (usually the problem) or the fuel line to the pump was clogged (unlikely). To eliminate the second case, I explained to Fred that if he removed the fuel cap and blew into the tank, I could watch the fuel line (which I had removed from the fuel pump) to see if gasoline was coming out. As it turns out, the fuel line was indeed plugged. As a result the back pressure sprayed gasoline in Fred's eye and he had to visit the emergency room. Fred was a very kind person and often would expend considerable effort to do something good for somebody else. It was his style not to end up in a situation where plans he had promised did not materialize. When he attempted to do things for me, the first I heard about it was when it happened. As an example, I had mentioned to him early on that the Wesleyan stipend might be difficult to live on in Switzerland. He said nothing. After attending the first lecture (very memorable for me --Jon Beck had defined what a triple is), and after the tea and cookies that followed, Fred told me to follow him. We were joined by a gentleman I didn't know. He and Fred spoke in German and it was more complicated than I could follow. Then we proceeded to walk for ten minutes in a basement labyrinth that equaled any in a big city hospital, eventually coming to a very dark alcove with a small "cage". The first gentleman spoke at some length with the gentleman inside the cage; this was in Swiss German which neither Fred nor I could follow. Eventually a piece of paper was produced which I was asked to sign. They then put in my hand the biggest pile of cash I have ever personally held in this life. It was enough for my wife and I to eat on for the balance of the year. (Of course, Beno Eckmann gets credit for that too). My mathematical career was jump-started by the fact that Saunders Mac Lane convened a seminar at Chicago based on my thesis, only four months after it became clear I would finish. I realize now that Fred must have played a substantial role in making this happen. Damn! I miss him. Ernie Manes [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Even though I was one of the dozen students, along with Ross Street and Brian Day, who took Max Kelly's course in category theory at the University of Sydney in 1965, unlike them I went in other directions thereafter.?? It is therefore a bit surprising that I kept bumping into Fred Linton, who turned out to have other interests that kept bringing me into contact with him over the past four decades in matters arguably unrelated to category theory: computer software, Jonsson-Tarski algebras, electrical engineering, 3D rendering of knots, etc. But it is Fred's foundational work on monads that I want to comment on here.???? At UACT, the Universal Algebra and Category Theory meeting at MSRI organized by respectively Ralph McKenzie and Saunders Mac Lane in 1992, there were back-to-back talks in a late-morning two-talk session on what I like to think of today as the foundations of equational model theory, EML.?? These were given by Walt Taylor and Fred Linton in that order. Ok, so who here noticed these two talks were both on EML??? Not me, I was a computer scientist still getting acclimated to such abstractions.?? Maybe some people, but if so the connection passed entirely without comment at the time, like ships passing in the night, and we all headed off for lunch. At lunch I sat with George McNulty, Walt's coauthor along with Ralph McKenzie of the classic UA text /Algebras, lattices, varieties/, Volume 1, 1987, the book that took two pages to explain why (for any given signature with no constants) the empty algebra was a bad idea. As a result of my much earlier work on dynamic algebras George and I went back several years and he was keen to understand what Fred had been on about in his just-ended talk.?? So with the fervor of a missionary I launched into monad theory, which I'd been teaching at Stanford for several years. No luck.?? In retrospect what I should have done instead was try to make some sort of connection between Walt's and Fred's two back-to-back talks on EML. In my mind, whether fairly or unfairly, what distinguished Fred from his fellow category theorists at UACT was that he was the natural CT representative of EML. There is one other anecdote about UACT, nothing to do with Fred, that I have always loved.?? In the course of MSRI director Bill Thurston's opening remarks, he said words to the effect that the notion of the opposite of a category made him nauseous.?? This was the only meeting I have ever attended where fully half the attendees drew in enough breath to drop the air pressure by an audible amount. ??Vaughan Pratt [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
There is one other anecdote about UACT, nothing to do with Fred, that I have always loved. In the course of MSRI director Bill Thurston's opening remarks, he said words to the effect that the notion of the opposite of a category made him nauseous. This was the only meeting I have ever attended where fully half the attendees drew in enough breath to drop the air pressure by an audible amount.
I’ll confess that the idea of an opposite category appearing as the codomain of a functor also makes me somewhat nauseated (the domain of course is no problem). But this said, in the interest of full disclosure, I should admit that in a joint paper with Cheng and Gurski someone — Eugenia, I believe? — convinced us that the easiest way to think of a functor C x D —> E admitting right adjoints in both variables is as a functor C x D —> (E^op)^op because in this way (writing E’ for E^op) the other two adjoints also have the form D x E’ —> C^op and E’ x C —> D^op. Such two-variable adjunctions form the vertical binary morphisms in a “cyclic double multi category” of multivariable adjunctions and parametrized mates: https://arxiv.org/abs/1208.4520 Regards, Emily — Assistant Professor, Dept. of Mathematics Johns Hopkins University www.math.jhu.edu/~eriehl [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
1) Two days ago by chance I come across an article of Bill Thurston: https://arxiv.org/pdf/math/9404236.pdf and seeing his name mentioned in this thread it occurs to me that everybody in this list should read it. In my opinion it is an extraordinary document about mathematics, mathematical activity and mathematicians. 2) Respect to to subject of this thread, the formal opposite of a category, denoted "op", is simply a notation very useful to work with functors which are contravariant in some variables, either with the "op" in the domain or the codomain of the functor arrow. Notations are important, and the "op" notation is essential in the language of categories and functors. 3) Finally, concerning Fred Linton, his death sadness me, he did important work in the early days of category theory, but more important, he was one of us, it was always a pleasure to encounter him, an he was a good guy. all the best e.d. On 07/09/17 14:03, Emily Riehl wrote:
There is one other anecdote about UACT, nothing to do with Fred, that I have always loved. In the course of MSRI director Bill Thurston's opening remarks, he said words to the effect that the notion of the opposite of a category made him nauseous. This was the only meeting I have ever attended where fully half the attendees drew in enough breath to drop the air pressure by an audible amount.
I?ll confess that the idea of an opposite category appearing as the codomain of a functor also makes me somewhat nauseated (the domain of course is no problem).
But this said, in the interest of full disclosure, I should admit that in a joint paper with Cheng and Gurski someone ? Eugenia, I believe? ? convinced us that the easiest way to think of a functor
C x D ?> E
admitting right adjoints in both variables is as a functor
C x D ?> (E^op)^op
because in this way (writing E? for E^op) the other two adjoints also have the form
D x E? ?> C^op
and
E? x C ?> D^op.
Such two-variable adjunctions form the vertical binary morphisms in a ?cyclic double multi category? of multivariable adjunctions and parametrized mates:
https://arxiv.org/abs/1208.4520
Regards, Emily
? Assistant Professor, Dept. of Mathematics Johns Hopkins University www.math.jhu.edu/~eriehl
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Eduardo, Thank you for recalling this remarkable article by Thurston. It contains profound observations on the role of *communities* in the creation of mathematics. Mathematical research is about developing *human understanding* of mathematics. Thurston does not mention category theory. I remember trying to learn algebraic topology by reading the "Foundations of Algebraic Topology" by Eilenberg and Steenrod. It is a great book, but not the right place to learn the subject. I also tried to learn algebraic geometry by reading the "Elements de Geometrie Algebrique" by Grothendieck and Dieudonné. I never became an algebraic-geometer. It is very difficult to learn anything without direct access to the people who knows. Best, André ________________________________________ From: Eduardo J. Dubuc [edubuc@dm.uba.ar] Sent: Friday, September 08, 2017 12:03 PM To: Emily Riehl; categories@mta.ca Subject: categories: "op"_Fred_and_Thurston 1) Two days ago by chance I come across an article of Bill Thurston: https://arxiv.org/pdf/math/9404236.pdf and seeing his name mentioned in this thread it occurs to me that everybody in this list should read it. In my opinion it is an extraordinary document about mathematics, mathematical activity and mathematicians. 2) Respect to to subject of this thread, the formal opposite of a category, denoted "op", is simply a notation very useful to work with functors which are contravariant in some variables, either with the "op" in the domain or the codomain of the functor arrow. Notations are important, and the "op" notation is essential in the language of categories and functors. 3) Finally, concerning Fred Linton, his death sadness me, he did important work in the early days of category theory, but more important, he was one of us, it was always a pleasure to encounter him, an he was a good guy. all the best e.d. On 07/09/17 14:03, Emily Riehl wrote:
There is one other anecdote about UACT, nothing to do with Fred, that I have always loved. In the course of MSRI director Bill Thurston's opening remarks, he said words to the effect that the notion of the opposite of a category made him nauseous. This was the only meeting I have ever attended where fully half the attendees drew in enough breath to drop the air pressure by an audible amount.
I?ll confess that the idea of an opposite category appearing as the codomain of a functor also makes me somewhat nauseated (the domain of course is no problem).
But this said, in the interest of full disclosure, I should admit that in a joint paper with Cheng and Gurski someone ? Eugenia, I believe? ? convinced us that the easiest way to think of a functor
C x D ?> E
admitting right adjoints in both variables is as a functor
C x D ?> (E^op)^op
because in this way (writing E? for E^op) the other two adjoints also have the form
D x E? ?> C^op
and
E? x C ?> D^op.
Such two-variable adjunctions form the vertical binary morphisms in a ?cyclic double multi category? of multivariable adjunctions and parametrized mates:
https://arxiv.org/abs/1208.4520
Regards, Emily
? Assistant Professor, Dept. of Mathematics Johns Hopkins University www.math.jhu.edu/~eriehl
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Category Theorists, In view of the discussion of the Thurston article, you may be interested in a much lower key article about mathematics and its methodology, addressed also to beginners, and to educators. I have met one professor of maths who claimed "mathematics has no need of methodology", which is odd since most human activities seem to benefit from such a discussion. which can be at various levels depending on the audience. http://www.groupoids.org.uk/methmat.html I met Fred a few times at some category theory meetings, and always found him easy to get on with and with a firm idea on what the subject was about! And a nice sense of humour! Best wishes Ronnie ----Original message----
From : joyal.andre@uqam.ca Date : 09/09/2017 - 05:33 (GMTDT) To : edubuc@dm.uba.ar, eriehl@math.jhu.edu, categories@mta.ca Subject : categories: RE: "op"_Fred_and_Thurston
Dear Eduardo, Thank you for recalling this remarkable article by Thurston. It contains profound observations on the role of *communities* in the creation of mathematics. Mathematical research is about developing *human understanding* of mathematics. Thurston does not mention category theory. I remember trying to learn algebraic topology by reading the "Foundations of Algebraic Topology" by Eilenberg and Steenrod. It is a great book, but not the right place to learn the subject. I also tried to learn algebraic geometry by reading the "Elements de Geometrie Algebrique" by Grothendieck and Dieudonné. I never became an algebraic-geometer. It is very difficult to learn anything without direct access to the people who knows. Best, André ________________________________________ From: Eduardo J. Dubuc [edubuc@dm.uba.ar] Sent: Friday, September 08, 2017 12:03 PM To: Emily Riehl; categories@mta.ca Subject: categories: "op"_Fred_and_Thurston 1) Two days ago by chance I come across an article of Bill Thurston: https://arxiv.org/pdf/math/9404236.pdf and seeing his name mentioned in this thread it occurs to me that everybody in this list should read it. In my opinion it is an extraordinary document about mathematics, mathematical activity and mathematicians. 2) Respect to to subject of this thread, the formal opposite of a category, denoted "op", is simply a notation very useful to work with functors which are contravariant in some variables, either with the "op" in the domain or the codomain of the functor arrow. Notations are important, and the "op" notation is essential in the language of categories and functors. 3) Finally, concerning Fred Linton, his death sadness me, he did important work in the early days of category theory, but more important, he was one of us, it was always a pleasure to encounter him, an he was a good guy. all the best e.d. On 07/09/17 14:03, Emily Riehl wrote:
There is one other anecdote about UACT, nothing to do with Fred, that I have always loved. In the course of MSRI director Bill Thurston's opening remarks, he said words to the effect that the notion of the opposite of a category made him nauseous. This was the only meeting I have ever attended where fully half the attendees drew in enough breath to drop the air pressure by an audible amount.
I?ll confess that the idea of an opposite category appearing as the codomain of a functor also makes me somewhat nauseated (the domain of course is no problem).
But this said, in the interest of full disclosure, I should admit that in a joint paper with Cheng and Gurski someone ? Eugenia, I believe? ? convinced us that the easiest way to think of a functor
C x D ?> E
admitting right adjoints in both variables is as a functor
C x D ?> (E^op)^op
because in this way (writing E? for E^op) the other two adjoints also have the form
D x E? ?> C^op
and
E? x C ?> D^op.
Such two-variable adjunctions form the vertical binary morphisms in a ?cyclic double multi category? of multivariable adjunctions and parametrized mates:
https://arxiv.org/abs/1208.4520
Regards, Emily
? Assistant Professor, Dept. of Mathematics Johns Hopkins University www.math.jhu.edu/~eriehl
[For admin and other information see: http://www.mta.ca/~cat-dist/ ] [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Categorists - Vaughan wrote:
There is one other anecdote about UACT, nothing to do with Fred, that I have always loved. In the course of MSRI director Bill Thurston' opening remarks, he said words to the effect that the notion of the opposite of a category made him nauseous. This was the only meeting I have ever attended where fully half the attendees drew in enough breath to drop the air pressure by an audible amount.
Since "nauseous" means "causing nausea", perhaps Thurston's remark had just sickened the audience. Emily wrote:
I’ll confess that the idea of an opposite category appearing as the codomain of a functor also makes me somewhat nauseated (the domain of course is no problem).
Now here is someone well-attuned to these subtleties of English! I've always been delighted by opposite categories. Sometimes I think we live in one. For example: if you flip forward in a book you eventually reach the back, but if you go back far enough you reach the foreword... and in the ancient past everything was younger. I always tell my students that since category theory reduces all of mathematics to the study of arrows, and the only mistake you can make with an arrow is to get confused about which way it's pointing, they should expect to spend many hours confused about exactly this. Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear John, and category theorists, The fact that every category has an opposite introduces a symmetry in mathematics that would not be there otherwise. The category of sets is not self dual, but a disjoint union of sets is a coproduct, dual to a product. Thurston does not show esteem for logic. Most mathematicians are taking logic for granted; they just use it as a part of their natural language. It is obvious that human understanding depends on the the laws of thought, on logic. In a sense, category theory is a branch of mathematical logic, since it greatly improves mathematical thinking in general. A category theorist might say (not too loudly) that mathematical logic is a branch of category theory. Best, andré ________________________________________ From: John Baez [baez@math.ucr.edu] Sent: Friday, September 08, 2017 9:15 PM To: categories Subject: categories: Re: Fred Dear Categorists - Vaughan wrote:
There is one other anecdote about UACT, nothing to do with Fred, that I have always loved. In the course of MSRI director Bill Thurston' opening remarks, he said words to the effect that the notion of the opposite of a category made him nauseous. This was the only meeting I have ever attended where fully half the attendees drew in enough breath to drop the air pressure by an audible amount.
Since "nauseous" means "causing nausea", perhaps Thurston's remark had just sickened the audience. Emily wrote:
I’ll confess that the idea of an opposite category appearing as the codomain of a functor also makes me somewhat nauseated (the domain of course is no problem).
Now here is someone well-attuned to these subtleties of English! I've always been delighted by opposite categories. Sometimes I think we live in one. For example: if you flip forward in a book you eventually reach the back, but if you go back far enough you reach the foreword... and in the ancient past everything was younger. I always tell my students that since category theory reduces all of mathematics to the study of arrows, and the only mistake you can make with an arrow is to get confused about which way it's pointing, they should expect to spend many hours confused about exactly this. Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Andre, Your argument applies equally well beyond mathematics, to other sciences/practices wherever categorical structure is natural. I met Fred at my first CT, in 99, and immediately he made one feel welcome. Best wishes, Bob.
On 11 Sep 2017, at 17:19, Joyal, André <joyal.andre@uqam.ca> wrote:
Dear John, and category theorists,
The fact that every category has an opposite introduces a symmetry in mathematics that would not be there otherwise. The category of sets is not self dual, but a disjoint union of sets is a coproduct, dual to a product.
Thurston does not show esteem for logic. Most mathematicians are taking logic for granted; they just use it as a part of their natural language. It is obvious that human understanding depends on the the laws of thought, on logic. In a sense, category theory is a branch of mathematical logic, since it greatly improves mathematical thinking in general. A category theorist might say (not too loudly) that mathematical logic is a branch of category theory.
Best, andré
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On 11/09/17 13:19, Joyal, Andr? wrote:
Dear John, and category theorists,
The fact that every category has an opposite introduces a symmetry in mathematics that would not be there otherwise.
The category of sets is not self dual, but a disjoint union of sets is a coproduct, dual to a product.
Thurston does not show esteem for logic. Most mathematicians are taking logic for granted; they just use it as a part of their natural language. It is obvious that human understanding depends on the the laws of thought, on logic. In a sense, category theory is a branch of mathematical logic, since it greatly improves mathematical thinking in general. A category theorist might say (not too loudly) that mathematical logic is a branch of category theory.
Best, andr?
The opposite category (*) may look a senseless obscurity and make some people nauseous, but it seems to me it made an important contribution to the understanding of mathematics. It took a long time to form part of mathematical thinking (and still is). For example, Bourbaki treatment of limits (of sets say) define and develops basic properties of projective limits, including the universal property. Later does the same for inductive limits, and includes a proof of the dual statements !!. He had to do so since it had not incorporated categories and the opposite category. He states what a universal property is, but can not state that the respective universal properties (for limits and colimits) are one the dual of the other. (*) The axioms of a category are self dual. Another examples are abelian categories, and a very subtle one, namely, Quillen's model categories. Many categories are not self dual, and this is underneath the duality between algebra and geometry. best e.d. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
I would like to add another example to Eduardo’s. In computer science both algebras and coalgebras for an endofunctor on sets are useful structures and both initial algebras and final coalgebras play an important role in the semantics of programming languages. It is now an important feature that algebras and coalgebras over set are not dual to each other. Only the invention of the dual category reveals the underlying duality. The ensuing tension between `abstract’ duality and `concrete’ non-duality is certainly one reason why the study of set-coalgebras is fascinating. For example, whereas it is well-known that the initial sequence of a finitary set-endofunctor converges in omega steps, a result by Worrell shows that the final sequence of a finitary set-endofunctor converges in omega+omega steps. Best wishes, Alexander
On 12 Sep 2017, at 17:09, Eduardo J. Dubuc <edubuc@dm.uba.ar> wrote:
On 11/09/17 13:19, Joyal, Andr? wrote:
Dear John, and category theorists,
The fact that every category has an opposite introduces a symmetry in mathematics that would not be there otherwise.
The category of sets is not self dual, but a disjoint union of sets is a coproduct, dual to a product.
Thurston does not show esteem for logic. Most mathematicians are taking logic for granted; they just use it as a part of their natural language. It is obvious that human understanding depends on the the laws of thought, on logic. In a sense, category theory is a branch of mathematical logic, since it greatly improves mathematical thinking in general. A category theorist might say (not too loudly) that mathematical logic is a branch of category theory.
Best, andr?
The opposite category (*) may look a senseless obscurity and make some people nauseous, but it seems to me it made an important contribution to the understanding of mathematics. It took a long time to form part of mathematical thinking (and still is). For example, Bourbaki treatment of limits (of sets say) define and develops basic properties of projective limits, including the universal property. Later does the same for inductive limits, and includes a proof of the dual statements !!. He had to do so since it had not incorporated categories and the opposite category. He states what a universal property is, but can not state that the respective universal properties (for limits and colimits) are one the dual of the other.
(*) The axioms of a category are self dual. Another examples are abelian categories, and a very subtle one, namely, Quillen's model categories.
Many categories are not self dual, and this is underneath the duality between algebra and geometry.
best e.d.
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Alexander's example reminded me of something I always wanted to ask somebody and never did, since it always felt too vague to me. But now I thought - just ask. In at least five very different contexts that I know, one seeks for a nice placement of some very non-self-dual category against the background of another one, "less non-self-dual". In order of my increasing ignorance, these are: Presenting spaces/locales/frames as certain (co/)monoids in the category of sup-lattices, which is as nicely self-dual as it ever gets. Extending the duality between discrete and compact abelian groups to the self-dual category of locally compact abelian groups. There are several closely related similar dualities, like e. g. the duality for (locally?) linearly compact vector spaces by, I believe, Lefschetz. In fact I think working with Banach or Hilbert spaces is largely motivated by the desire to force infinite-dimensional vector spaces to behave more like finite-dimensional ones, which form some of the nicest self-dual categories. Passing from (unstable) to stable homotopy theory is in a sense forcing some amount of self-duality. The main feature of stable categories is that they are additive (i. e. finite coproducts are isomorphic to the corresponding products) but also much more - e. g. most of homotopy cartesian or cocartesian squares in such categories turn out to be homotopy bicartesian; this in particular implies the crucial feature that the adjunction between suspension and loop space functors becomes an equivalence (in a homotopy bicartesian square like A -> 0 | | V V 0 -> B A is (stably equivalent to) the loop space of B iff B is (stably equivalent to) the suspension of A; more generally, in a similar square A -> 0 | | V V X -> B A is the fibre of X -> B iff B is the cofibre of A -> X, etc.) The context mentioned by Alexander, which triggered this post in the first place - the phenomenon called limit-colimit coincidence: it seems that imposing on some categories certain constructivity constraints coming from computer science tends to imply certain amount of self-dual features. Like, initial algebras for endofunctors become forced to become isomorphic with final coalgebras for the same endofunctors. Or, similarly, left adjoints to some functors to become isomorphic to right adjoints to the same functors. In physics, it seems that the main motivation of various quantization procedures is to achieve certain amount of self-duality. For example, evolution of a physical system becomes time-reversible. It seems like in many cases such "self-dualization" can be formulated in terms of forcing certain objects in certain monoidal categories to become invertible but I don't know enough to tell more about it. In any case I am aware of several works by category theorists which provide appropriate formalism for such and similar constructions; the most general formalism that I know is probably the Chu construction. But, if I am not overlooking something obvious, I have only seen explanations of *how* to "increase self-dual features", not *why* do these self-dualization phenomena tend to occur in so many disparate contexts. Does anybody know any underlying *reasons*? Can this phenomenon be explained by the mere fact that "linearizing" the problem makes life easier at the expense of losing certain amount of information, or there actually exist some deeply rooted principles that force self-dual behavior in certain mathematical or physical circumstances? Sorry for this very vague and long post, but I am really eager to learn about opinions of the category-theoretic community about this question that I hardly ever managed to even formulate. Mamuka On Thu, 14 Sep 2017 15:53:21 +0100, Alexander Kurz <axhkrz@gmail.com> wrote:
I would like to add another example to Eduardo???s.
In computer science both algebras and coalgebras for an endofunctor on sets are useful structures and both initial algebras and final coalgebras play an important role in the semantics of programming languages.
It is now an important feature that algebras and coalgebras over set are not dual to each other. Only the invention of the dual category reveals the underlying duality.
The ensuing tension between `abstract??? duality and `concrete??? non-duality is certainly one reason why the study of set-coalgebras is fascinating.
For example, whereas it is well-known that the initial sequence of a finitary set-endofunctor converges in omega steps, a result by Worrell shows that the final sequence of a finitary set-endofunctor converges in omega+omega steps.
Best wishes, Alexander
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Mamuka, I am also puzzled with your examples of duality. I would like to speculate on some aspects of it. The category of finite dimensional vector spaces is the archtype of a self-dual category. In a sense, vector spaces are showing up in geometry as tangent spaces of smooth manifolds. But the tangent space of a manifold at a point is the "infinitesimal" structure of that maniifold at that point. It seems that linear structures are showing up naturally as infinitesimal structures. Stable categories are showing up as infinitesimal structure of higher toposes: https://ncatlab.org/nlab/show/tangent+%28infinity%2C1%29-category Of course, linear structures may also be obtained by other means. -André ________________________________________ From: Mamuka Jibladze [jib@rmi.ge] Sent: Saturday, September 16, 2017 12:35 PM To: Alexander Kurz Cc: categories@mta.ca Subject: categories: Re: the dual category Alexander's example reminded me of something I always wanted to ask somebody and never did, since it always felt too vague to me. But now I thought - just ask. In at least five very different contexts that I know, one seeks for a nice placement of some very non-self-dual category against the background of another one, "less non-self-dual". In order of my increasing ignorance, these are: Presenting spaces/locales/frames as certain (co/)monoids in the category of sup-lattices, which is as nicely self-dual as it ever gets. Extending the duality between discrete and compact abelian groups to the self-dual category of locally compact abelian groups. There are several closely related similar dualities, like e. g. the duality for (locally?) linearly compact vector spaces by, I believe, Lefschetz. In fact I think working with Banach or Hilbert spaces is largely motivated by the desire to force infinite-dimensional vector spaces to behave more like finite-dimensional ones, which form some of the nicest self-dual categories. Passing from (unstable) to stable homotopy theory is in a sense forcing some amount of self-duality. The main feature of stable categories is that they are additive (i. e. finite coproducts are isomorphic to the corresponding products) but also much more - e. g. most of homotopy cartesian or cocartesian squares in such categories turn out to be homotopy bicartesian; this in particular implies the crucial feature that the adjunction between suspension and loop space functors becomes an equivalence (in a homotopy bicartesian square like A -> 0 | | V V 0 -> B A is (stably equivalent to) the loop space of B iff B is (stably equivalent to) the suspension of A; more generally, in a similar square A -> 0 | | V V X -> B A is the fibre of X -> B iff B is the cofibre of A -> X, etc.) The context mentioned by Alexander, which triggered this post in the first place - the phenomenon called limit-colimit coincidence: it seems that imposing on some categories certain constructivity constraints coming from computer science tends to imply certain amount of self-dual features. Like, initial algebras for endofunctors become forced to become isomorphic with final coalgebras for the same endofunctors. Or, similarly, left adjoints to some functors to become isomorphic to right adjoints to the same functors. In physics, it seems that the main motivation of various quantization procedures is to achieve certain amount of self-duality. For example, evolution of a physical system becomes time-reversible. It seems like in many cases such "self-dualization" can be formulated in terms of forcing certain objects in certain monoidal categories to become invertible but I don't know enough to tell more about it. In any case I am aware of several works by category theorists which provide appropriate formalism for such and similar constructions; the most general formalism that I know is probably the Chu construction. But, if I am not overlooking something obvious, I have only seen explanations of *how* to "increase self-dual features", not *why* do these self-dualization phenomena tend to occur in so many disparate contexts. Does anybody know any underlying *reasons*? Can this phenomenon be explained by the mere fact that "linearizing" the problem makes life easier at the expense of losing certain amount of information, or there actually exist some deeply rooted principles that force self-dual behavior in certain mathematical or physical circumstances? Sorry for this very vague and long post, but I am really eager to learn about opinions of the category-theoretic community about this question that I hardly ever managed to even formulate. Mamuka [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Emily, many thanks for your message. It push me to precise some aspects of my link with the "op" things, with some words about your joint paper https://arxiv.org/abs/1208.4520. In fact in a talk at a the Louvain-la-Neuve's meeting in 2011, Category theory, algebra and geometry, 26-27 may 2011, I spoke on "Borromean Objects and Trijunctions". I do remember well that Eugenia was listening to this talk, and so probably her attention was attracted on the notion of a trijunction (the notion you are explaining in your message). Some times later this was published in a paper "Trijunctions and Triadic Galois Connections" (Cahier Top. Géo. Diff. Cat, LIV-1 (2013), pp. 13-28) (accessible on my site : http://rene.guitart.pagesperso-orange.fr/publications.html). From the summary, we can learn why I did so : "In this paper we introduce the notion of a trijunction, which is related to a triadic Galois connection just as an adjunction is to a Galois connection. We construct the trifibered tripod associated to a trijunction, the trijunction between toposes of presheaves associated to a discrete trifibration, and the generation of any trijunction by a bi-adjoint functor. While some examples are related to triadic Galois connections, to ternary relations, others are associated to some symmetric tensors, to toposes and algebraic universes". Now it is interesting to understand how this was achieved, in two steps: 1 - Firstly I read a paper by Biedermann, on triadic Galois connections, related to ternary relations as Galois connections of Ore between ordered sets are related to binary relations. Immediately I try to extend that from order sets to categories. 2 - Fortunately in the same time I was conducted to read again carefully the famous paper by Kan on Adjoint functor. There I observed that in fact he he is mainly working with tensors and Gom, i.e. with bifunctors ; and furthermore in the Mac Lane's book, the convenient lemma for parameterized adjunctions are reproduced. Then I notice that the perfect explicit ternary symmetry in Biedermann was in fact also implicit in Kan, but that only he "missed" to put the accent on it, by introducing the opposite of the opposite (E^op)^op in your message). So I did, and then I got the application to the descriptions of trifibrations and toposes, to the analysis clearly the system of functions or operations generated by a tripod. So you can see that my motivations (to unified Kan and Biedermann at the level n = 3), in order to produce a functional analysis of tripod), seems rather different from yours (to enter in a game of general n-multiadjunctions). Hence finally my question : to analyze the system of functions or operations generated by an n-pod, and to understand there the part play by the mysterious "op". with my friendly greetings, René. Le 7 sept. 2017 à 19:03, Emily Riehl a écrit :
There is one other anecdote about UACT, nothing to do with Fred, that I have always loved. In the course of MSRI director Bill Thurston's Galois Connections opening remarks, he said words to the effect that the notion of the opposite of a category made him nauseous. This was the only meeting I have ever attended where fully half the attendees drew in enough breath to drop the air pressure by an audible amount.
I’ll confess that the idea of an opposite category appearing as the codomain of a functor also makes me somewhat nauseated (the domain of course is no problem).
But this said, in the interest of full disclosure, I should admit that in a joint paper with Cheng and Gurski someone — Eugenia, I believe? — convinced us that the easiest way to think of a functor
C x D —> E
admitting right adjoints in both variables is as a functor
C x D —> (E^op)^op
because in this way (writing E’ for E^op) the other two adjoints also have the form
D x E’ —> C^op
and
E’ x C —> D^op.
Such two-variable adjunctions form the vertical binary morphisms in a “cyclic double multi category” of multivariable adjunctions and parametrized mates:
https://arxiv.org/abs/1208.4520
Regards, Emily
— Assistant Professor, Dept. of Mathematics Johns Hopkins University www.math.jhu.edu/~eriehl
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Emily and Ren??, Under this list, and e.g. in connection with G??del's Incompleteness Theorem (which I still call the Incompleteness Paradox), I have thrown out the idea that logic may and perhaps even should be "lative", in the sense that - first we fix sorts and operators, i.e., the signature - then we build terms based on that signature, but terms cannot change anything about the underlying signature, so the door is closed, so as to say - sentences then build upon terms, and similarly cannot manipulate or change whatever is in the term set And so on, so an entailment, building e.g. upon sentences, even if being "true" cannot be seen as a sentence, i.e., we should allow ourselves to view entailments also as sentences and throw them back into the bag of sentences. This is what G??del is doing, and this is widely accepted. I don't accept, but this is now not my message here. --- Triadic relations are potentially "illative" in this respect. Port-Royal is dyadically lative, I would say, and maybe Peirce "triadically lative", but a bit les so. Both are not formal enough, but my question is how trijunctions or multivariable adjunctions think in theses respects? Best, Patrik PS "Lative logic", yet to be better defined, is an extension of Goguen's Institutions and Meseguer's Entailment Systems. http://umu.diva-portal.org/smash/get/diva2:619702/FULLTEXT01.pdf On 2017-09-27 12:10, Ren?? Guitart wrote:
Dear Emily,
many thanks for your message. It push me to precise some aspects of my link with the "op" things, with some words about your joint paper https://arxiv.org/abs/1208.4520.
In fact in a talk at a the Louvain-la-Neuve's meeting in 2011, Category theory, algebra and geometry, 26-27 may 2011, I spoke on "Borromean Objects and Trijunctions". I do remember well that Eugenia was listening to this talk, and so probably her attention was attracted on the notion of a trijunction (the notion you are explaining in your message). Some times later this was published in a paper "Trijunctions and Triadic Galois Connections" (Cahier Top. G??o. Diff. Cat, LIV-1 (2013), pp. 13-28) (accessible on my site : http://rene.guitart.pagesperso-orange.fr/publications.html). From the summary, we can learn why I did so : "In this paper we introduce the notion of a trijunction, which is related to a triadic Galois connection just as an adjunction is to a Galois connection. We construct the trifibered tripod associated to a trijunction, the trijunction between toposes of presheaves associated to a discrete trifibration, and the generation of any trijunction by a bi-adjoint functor. While some examples are related to triadic Galois connections, to ternary relations, others are associated to some symmetric tensors, to toposes and algebraic universes". Now it is interesting to understand how this was achieved, in two steps: 1 - Firstly I read a paper by Biedermann, on triadic Galois connections, related to ternary relations as Galois connections of Ore between ordered sets are related to binary relations. Immediately I try to extend that from order sets to categories. 2 - Fortunately in the same time I was conducted to read again carefully the famous paper by Kan on Adjoint functor. There I observed that in fact he he is mainly working with tensors and Gom, i.e. with bifunctors ; and furthermore in the Mac Lane's book, the convenient lemma for parameterized adjunctions are reproduced. Then I notice that the perfect explicit ternary symmetry in Biedermann was in fact also implicit in Kan, but that only he "missed" to put the accent on it, by introducing the opposite of the opposite (E^op)^op in your message). So I did, and then I got the application to the descriptions of trifibrations and toposes, to the analysis clearly the system of functions or operations generated by a tripod.
So you can see that my motivations (to unified Kan and Biedermann at the level n = 3), in order to produce a functional analysis of tripod), seems rather different from yours (to enter in a game of general n-multiadjunctions). Hence finally my question : to analyze the system of functions or operations generated by an n-pod, and to understand there the part play by the mysterious "op".
with my friendly greetings,
Ren??.
Le 7 sept. 2017 ?? 19:03, Emily Riehl a ??crit :
There is one other anecdote about UACT, nothing to do with Fred, that I have always loved. In the course of MSRI director Bill Thurston's Galois Connections opening remarks, he said words to the effect that the notion of the opposite of a category made him nauseous. This was the only meeting I have ever attended where fully half the attendees drew in enough breath to drop the air pressure by an audible amount.
I???ll confess that the idea of an opposite category appearing as the codomain of a functor also makes me somewhat nauseated (the domain of course is no problem).
But this said, in the interest of full disclosure, I should admit that in a joint paper with Cheng and Gurski someone ??? Eugenia, I believe? ??? convinced us that the easiest way to think of a functor
C x D ???> E
admitting right adjoints in both variables is as a functor
C x D ???> (E^op)^op
because in this way (writing E??? for E^op) the other two adjoints also have the form
D x E??? ???> C^op
and
E??? x C ???> D^op.
Such two-variable adjunctions form the vertical binary morphisms in a ???cyclic double multi category??? of multivariable adjunctions and parametrized mates:
https://arxiv.org/abs/1208.4520
Regards, Emily
??? Assistant Professor, Dept. of Mathematics Johns Hopkins University www.math.jhu.edu/~eriehl
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (12)
-
Alexander Kurz -
Bob Coecke -
Eduardo J. Dubuc -
Emily Riehl -
Ernest G. Manes -
John Baez -
Joyal, André -
Mamuka Jibladze -
Patrik Eklund -
René Guitart -
RONALD BROWN -
Vaughan Pratt