Fred

Fred

[Robert Pare]

Fred was a great guy and I liked him. His love of language was manifest
in his writing as well as in his oral communication. It was from his papers  that I
learned the words “behoove” and “antepenultimate”. In one paper he explained,
in a precise and understandable way, the theory of triples (aka monads), their
algebras and how they relate to adjoints, … all in a single sentence!

I first met him when I was a graduate student at McGill in 1968. He was sleeping
on the couch in the lounge with a sign on his chest saying “Wake me at 2:30”.
Someone asked, “Who’s this?”. The reply: “The colloquium speaker.” He spoke
(at 2:35) on enriched theories and algebras. One thing he said has stuck with me.
The base category was a non-symmetric monoidal category and the algebras were
contravariant functors into it (if my memory serves me correctly). He said there
may come a time when we have to consider covariant functors as contravariant
ones on the opposite category.

He was a true original and a great loss to our community.


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Re: Fred

[Ross Street]

Preparing a stir fry for last Sunday's lunch is when I received Marta's message saying we had lost Fred. I realized that our family was introduced to this wonderful style of cooking by Fred Linton in his big New Haven house. I was on sabbatical 1976-7 at Wesleyan University in Middletown (CT) to work with Fred. The year was a hard one for Fred but he was a great host. We rented a nice University-owned house in Middletown, with bushy slope one side and a disused car park the other, so our two boys could toboggan or ride their tag-sale-acquired tricycle, according to season. Our older boy started school that year. Fred had found a microscope at a tag sale which is a gift my boys often talk about. He also found books, fruits, vegies and other things he rightly thought would interest us.

One day he took me to a liquor store in New Haven where he knew they had Australian beer. They had \Huge{cans} of Fosters. The shopkeeper, not knowing  at first my nationality, told me that, in Australia, they bought these by the 6 pack and got through many of them at their barbies! Fosters was a Melbourne company, uncommon in Sydney in those days; I gave him the benefit of  the doubt.    

I may have met Fred in November 1968 at a MidWest Category Seminar in Urbana, Illinois; there were so many of the category theorists who had just been  names on papers to me before that. However, we certainly met at the Summer  of '69 Bowdoin College, Maine, gathering. Fred gave a series of lectures on enriching the theory of triples (monads), especially over bases that were  either closed or monoidal but not necessarily both. I had been thinking about the closed monoidal case over the previous year so it was great to find  a mind who also found it worth doing that even more generally.      

The MR review Fred wrote of my ``formal theory of monads'' paper was noticed (before me I think) by our Head (Fred Chong) at the time. This was responsible in large part for my application for a full year's sabbatical at Middletown being smiled upon. Yes, Fred had a way with words!

We category theorists have enjoyed Fred's company and discussions in many meetings around the world. We all will miss his valuable contributions.

Ross

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Re: opposite category

[Peter Selinger]

Robert Pare wrote:
>
> He said there may come a time when we have to consider covariant
> functors as contravariant ones on the opposite category.

This anecdote seems to have prompted a few posts about opposite
categories, but I thought the point of the original anecdote was that
Fred said that *covariant* functors should be considered as
contravariant functors on the opposite category, i.e., that he
considered contravariant functors to be the more fundamental concept.
An interesting thought, and obviously tongue-in-cheek.

-- Peter




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Re: opposite category

[Joachim Kock]

On 14/09/2017 17:58, Peter Selinger wrote:
> Robert Pare wrote:
>>
>> He said there may come a time when we have to consider covariant
>> functors as contravariant ones on the opposite category.

Indeed: Segal's (1974) Gamma-spaces are contravariant
functors on the opposite of the category of finite pointed
sets.

Joachim.


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Re: opposite category

[Vaughan Pratt]

Well, one could look to the various power set functors for guidance there.

When you refer to the contravariant power set functor, it may be
contrary but it's reliably so and people know what you mean.

But "the" covariant power set functor??? What's that??? It's like Trump:
one day he's agreeing with the Elephants, the next with the Scientific
Consensus.

If there are more than two covariant power set functors maybe we'll see
yet another side of Trump, perhaps a side from another dimension.

Give me good old reliable contravariant.???? Mind the pence and the
pounding will take care of itself.

Oh but wait, there's profunctors,?? ??: A' x B --> V.???? Which way do /they
/go? The Elephant
<https://www.amazon.com/Sketches-Elephant-Theory-Compendium-Oxford/dp/019852496X/ref=sr_1_1?ie=UTF8&qid=1505521975&sr=8-1&keywords=sketches+of+an+elephant>
(kindle edition $4.99
<https://www.amazon.com/Draw-Animals-Step-Step-Elephants-ebook/dp/B007WKEMBE/ref=sr_1_4?ie=UTF8&qid=1505522093&sr=8-4>)
says they go from A to B.?? The Consensus, being a bunch of Deniers, says
("bunch" is singular) they go from B to A.

Who to believe??? It's enough to make anyone lose their composure. (Oh
but wait, there's left Kan extensions.)

Vaughan

PS?? How many covariant power set functors according to the Elephant???
Does every element of ?? get one?

On 09/14/17 8:58 AM, Peter Selinger wrote:
> Robert Pare wrote:
>> He said there may come a time when we have to consider covariant
>> functors as contravariant ones on the opposite category.
> This anecdote seems to have prompted a few posts about opposite
> categories, but I thought the point of the original anecdote was that
> Fred said that *covariant* functors should be considered as
> contravariant functors on the opposite category, i.e., that he
> considered contravariant functors to be the more fundamental concept.
> An interesting thought, and obviously tongue-in-cheek.
>
> -- Peter
>

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Re: opposite category

[Joyal, André]

Dear Robert, Peter and all,

We often turn covariant functors into contravariant ones:

If C is a small category, then the category [C,Set]
of covariant set valued functors on C is the topos of 
presheaves on C^{op}. 

Recall that the category \Gamma introduced by Graeme Segal 
is the opposite of the category Fin_\star of finite pointed sets.

https://ncatlab.org/nlab/show/Segal%27s+category

A Gamma-space was not defined by Segal to be a covariant functor
Fin_\star  --->Space but as a contravariant functor 
  \Gamma---->Space 

https://ncatlab.org/nlab/show/Gamma-space

-André

________________________________________
From: Peter Selinger [selinger@mathstat.dal.ca]
Sent: Thursday, September 14, 2017 11:58 AM
To: Categories List
Subject: categories: Re: opposite category

Robert Pare wrote:
>
> He said there may come a time when we have to consider covariant
> functors as contravariant ones on the opposite category.

This anecdote seems to have prompted a few posts about opposite
categories, but I thought the point of the original anecdote was that
Fred said that *covariant* functors should be considered as
contravariant functors on the opposite category, i.e., that he
considered contravariant functors to be the more fundamental concept.
An interesting thought, and obviously tongue-in-cheek.

-- Peter





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Re: Fred

[Patrik Eklund]

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


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Re: Fred

[René Guitart]

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
> 
> 

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Re: Fred

[Bob Coecke]

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
> 

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Re: Fred

[Joyal, André]

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



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Re: Fred

[John Baez]

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/ ]

Re: Fred

[Emily Riehl]

> 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



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Re: Fred

[Vaughan Pratt]

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


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Fred

[Ernest G. Manes]

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


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