Re: Fred Linton

Re: Fred Linton

[George Janelidze]

Dear Marta, Dear Friends and Colleagues,

Let me join you in asking Barbara to accept our sympathy and condolences, and in saying how sad this is to all of us. Fred was such a brilliant mathematician active in the brilliant era of category theory in USA. He stopped writing papers at some point, but never stopped thinking and never stopped being so kind to everyone...

George Janelidze


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Re: Fred Linton

[Sergei Soloviev]

Dear All,

I do rememeber Fred very well, even at our first meeting at a conference in Bulgaria in 1987...
He was very nice and gentle, and brilliant mathematician of a great generation.

Sergei Soloviev



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Fred LInton

[Stephen Urban Chase]

I was very sorry to receive the news about Fred.  My deepest sympathy to Barbara and family, and to all those in the category theory community who were close to him.


Although I spent the 1966-67 academic year at ETH Zurich, and was then in frequent contact with Fred and the other categorists who were at the Forschungsinstitut that year, my clearest memory of him is still of a very pleasant and interesting conversation we had at the 1965 category theory meeting in La Jolla.  As I recall, it was mostly about 2-categories, in which I was very much interested at the time.


Steve


Stephen U. Chase

Emeritus Professor, Department of Mathematics

Malott Hall, Cornell University

Ithaca, NY 14850



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Re: Fred Linton

[Mamuka Jibladze]

Although we only met few times with intervals of several years, I
bitterly feel loss of one of my dearest friends,
and will badly miss many good things unique to him. And I believe there
are several others feeling like me - he was that kind of person.

Mamuka


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Re: Fred Linton

[Emily Riehl]

Many of you were very generous in sharing examples with me when I started writing “Category theory in context” but my very favorite, the one I tend to lead with in conversation to whet appetites, is an application of the Yoneda lemma to high-school level matrix algebra that I learned from Fred. I’ll let him tell you about it in his own words. Because his correspondence is so charming I’ve included it in full, following an excerpted version of my original email.

My sincerest condolences to those who had the opportunity to spend more time with him than I did. He will be missed.

Emily

—
Assistant Professor, Dept. of Mathematics
Johns Hopkins University
www.math.jhu.edu/~eriehl

---

From: Emily Riehl <eriehl@math.harvard.edu>
Subject: categories: a call for examples
Date: December 28, 2014 at 4:52:55 PM EST
To: categories@mta.ca
Reply-To: Emily Riehl <eriehl@math.harvard.edu>

Hi all,

I am writing in hopes that I might pick the collective brain of the categories list. This spring, I will be teaching an undergraduate-level category theory course, entitled “Category theory in context.”

It has two aims: 

(i) To provide a thorough “Cambridge-style” introduction to the basic concepts of category theory: representability, (co)limits, adjunctions, and monads.

(ii) To revisit as many topics as possible from the typical undergraduate curriculum, using category theory as a guide to deeper understanding.

…

Over the past few months I have been collecting examples that I might use in the course, with the focus on topics that are the most “sociologically important” (to quote Tom Leinster’s talk at CT2014) and also the most illustrative of the categorical concept in question. (After all, aim (i) is to help my students internalize the categorical way of thinking!)

...

I would be very grateful to hear about other favorite examples which illustrate or are clarified by the categorical way of thinking. My view of what might be accessible to undergraduates is relatively expansive, particularly in the less-obviously-categorical areas of mathematics such as analysis.

...

Best wishes to all for a happy and productive new year.

Emily Riehl

--
Benjamin Peirce & NSF Postdoctoral Fellow
Department of Mathematics, Harvard University
www.math.harvard.edu/~eriehl <http://www.math.harvard.edu/~eriehl>


From: "Fred E.J. Linton" <fejlinton@usa.net>
Subject: Re: categories: a call for examples
Date: December 29, 2014 at 12:10:05 PM EST
To: Emily Riehl <eriehl@math.harvard.edu>

Hi, Emily,

I suppose I would be remiss not to point out all the examples your 
fellow Cambridge co-citizen David Spivak offers in his recent text,
Category Theory for the Sciences (MIT Press).

And then there's the Yoneda Lemma embodied in the classical Gaussian
row reduction observation, that a given row reduction operation (on
matrices with say k rows) being a "natural" operation (in the sense 
of natural transformations) is just multiplication (on the appropriate
side) by the effect of that operation on the k-by-k identity matrix.

And dually for column-reduction operations :-) .

Cheers, -- Fred

From: Emily Riehl <eriehl@math.harvard.edu>
Subject: Re: categories: a call for examples
Date: December 29, 2014 at 4:53:03 PM EST
To: "Fred E.J. Linton" <fejlinton@usa.net>

Fred,

> And then there's the Yoneda Lemma embodied in the classical Gaussian
> row reduction observation, that a given row reduction operation (on
> matrices with say k rows) being a "natural" operation (in the sense 
> of natural transformations) is just multiplication (on the appropriate
> side) by the effect of that operation on the k-by-k identity matrix.

I love it. Thanks :)

And I’ll check out the Spivak book.

Best,
Emily

From: "Fred E.J. Linton" <fejlinton@usa.net>
Subject: Re: categories: a call for examples
Date: December 29, 2014 at 6:37:27 PM EST
To: Emily Riehl <eriehl@math.harvard.edu>

Hi, Emily,

You're welcome.

While you're there (f.d. real/complex vector spaces and linear x-formations,
vs. real/complex matrices), exploit the connection between the Lawverian
theory (objects the natural numbers k, n, m, l, etc., and morphisms n -> k
the k-by-n matrices, with usual matrix mult'n) and the category of f.d. 
vector spaces proper, with linear x-formations as morphisms. The latter
is the category of algebras over the former, but the former is a skeleton 
("every f.d.v.sp. has a basis") of the latter, as well.

The Gauss/Yoneda observation I tend to see occurring in that skeleton.

And another example that matrices illustrate: the middle-interchange law:

think A $ B as the row-lengthening procedure taking two matrices
with = number (say k) of rows (say k-by-n and k-by-m) and delivering 
the k-rowed matrix whose rows, of length n+m, all start out being the
corresponding row of A and finish by becoming that of B; and A # A'
the column-lengthening procedure taking two matrices with = number (say n)
of columns (say k-by-n and l-by-n) and yielding the obvious (k+l)-by-n one.

(Linear algebra texts introduce those ideas implicitly when they deal
with "block decompositions".) Anyway, it's clear -- for matrices A, A',
B, B' of the matching (size/shape)s, one has:

(A $ B) # (A' $ B') = (A # A') $ (B # B').

I'm sure you'll find plenty more such illustrations here. I hope your 
Harvard kids eat them up with better appetite than my Wesleyan kids did.

Cheers, — Fred

From: Emily Riehl <eriehl@math.harvard.edu>
Subject: Re: categories: a call for examples
Date: December 30, 2014 at 4:06:22 PM EST
To: "Fred E.J. Linton" <fejlinton@usa.net>

I particularly like the Vector space — Matrix equivalence of categories. It’s one of my favorite examples. 

I have no idea what to make of this:

> And another example that matrices illustrate: the middle-interchange law:
> 
> think A $ B as the row-lengthening procedure taking two matrices
> with = number (say k) of rows (say k-by-n and k-by-m) and delivering 
> the k-rowed matrix whose rows, of length n+m, all start out being the
> corresponding row of A and finish by becoming that of B; and A # A'
> the column-lengthening procedure taking two matrices with = number (say n)
> of columns (say k-by-n and l-by-n) and yielding the obvious (k+l)-by-n one.
> 
> (Linear algebra texts introduce those ideas implicitly when they deal
> with "block decompositions".) Anyway, it's clear -- for matrices A, A',
> B, B' of the matching (size/shape)s, one has:
> 
> (A $ B) # (A' $ B') = (A # A') $ (B # B’)

But I like it.

Thanks,
Emily

From: "Fred E.J. Linton" <fejlinton@usa.net>
Subject: Re: categories: a call for examples
Date: December 31, 2014 at 1:14:02 AM EST
To: Emily Riehl <eriehl@math.harvard.edu>
Hi, Emily,

> I particularly like the Vector space — Matrix equivalence of categories.
It’s one of my favorite examples. 

Yes; it's unusual to have a variety of finitary algebras equivalent 
to its Lawverian theory, as also to have a monad whose Kleisli category 
and Eilenberg-Moore category are equivalent (!) . Well, run with it, 
for a "touchdaown" :-) .

> I have no idea what to make of this:
> 
>> And another example that matrices illustrate: the middle-interchange law:

Food for thought. You know very well that, when it comes to proper,
every-where defined binary operations with unit (call them + and &, say),
as soon as (a + b) & (a' + b') = (a & a') + (b & b'), then you soon see
the units agree, and then b & a' = a' + b = a' & b, whence also + = & 
and it's commutative.

But $ and #, below, are no longer "everywhere defined" (unless you
restrict to the case n = m = k = l = 0).

Illustrating in a tiny instance that ASCII artwork can handle easily,
and letting A, B, A', and B' be

a b

c

p q
r s

and 

x
y

respectively, A $ B becomes

a b c ,

A' $ B' becomes

p q x
r s y ,

and both 

(A $ B) # (A' $ B') and (A # A') $ (B # B') become

a b c
p q x
r s y .

>> think A $ B as the row-lengthening procedure taking two matrices
>> with = number (say k) of rows (say k-by-n and k-by-m) and delivering 
>> the k-rowed matrix whose rows, of length n+m, all start out being the
>> corresponding row of A and finish by becoming that of B; and A # A'
>> the column-lengthening procedure taking two matrices with = number (say n)
>> of columns (say k-by-n and l-by-n) and yielding the obvious (k+l)-by-n one.
>> 
>> (Linear algebra texts introduce those ideas implicitly when they deal
>> with "block decompositions".) Anyway, it's clear -- for matrices A, A',
>> B, B' of the matching (size/shape)s, one has:
>> 
>> (A $ B) # (A' $ B') = (A # A') $ (B # B’)

But these also say something interesting about an interaction between 
products and coproducts of f.d. vector spaces, no? Maybe that's why ... :

> But I like it.
> 
> Thanks,
> Emily

Enjoy :-) ! Cheers, -- Fred

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RE: Fred Linton

[Duskin, John]

I too am sad. Fred was one of the first cat theorists I met at that famous meeting in California so long ago!
________________________________________

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RE: Fred Linton

[Yefim Katsov]

Dear All,


It's, indeed, a very sad news and a great loss for everyone who had a pleasure to know Fred. He was a very dear and close friend of mine and my family---so for us, this loss is even more painful!


Definitely, Fred was a very dedicated and talented mathematician and expert  in many areas of the "classical" category theory. Fred also was a very kind and sensitive person combining the best qualities of the intelligence in the classical meaning of this concept!


I'm sure many of us will miss Fred very much.


Yefim

______________________________________

Prof. Yefim Katsov
Department of Mathematics
Hanover College
Hanover, IN 47243-0890, USA
Telephones: Office (812) 866-6119;
                   Home (812) 866-4312;
                   Fax    (812) 866-7229



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RE: Fred Linton

[Joyal, André]

[Apologies if you receive this as a duplicate]

Dear All,

Very sad news. 
Fred was a gentle person with an inexhaustible curiosity
Always ready for a discussion with good spirit.
He will be remembered for the connection between 
algebraic theories and monads.
I will miss him.


André J.



________________________________________
From: Marta Bunge [martabunge@hotmail.com]
Sent: Saturday, September 02, 2017 8:56 PM
To: categories@mta.ca
Subject: categories: Fred Linton

[Note from moderator: the message below from Barbara Mikolajewska was
forwarded by Marta Bunge]

Very sad news.=C2=A0


__________
From: Fred E.J. Linton <fejlinton@usa.net>
Sent: September 2, 2017 8:32:17 PM
To: Marta Bunge
Subject: Re: To Fred and Barbara =C2=A0
Dear Marta,

I have to inform you with great sadness
that Fred died today at 5:47 a.m.
There were no cure for his illness
in the stage when it starts to manifest
its devastating symptoms.
Hospice at least gave him some
comfort and peace in this last moments.
I discovered it is really very humanitarian
institution.

Barbara



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Re: Fred Linton

[Lawvere, F.]

Dear Marta, dear friends and colleagues,


Dear Barbara,​

Fatima joins me in sending you our heartfelt condolences for your  great loss. 

Our thoughts are with you.

 

It is with deep sadness and regret that I learned about Fred’s passing.

 

I always enjoyed his uniquely lively lectures that had a musical rhythm and showed so
much enthusiasm and passion that left us inspired.

 

For many decades Fred has been my dear friend.

He was the first person whom I encountered when I arrived at Columbia University, (in
the early days of 

the 60’s). He showed me the way to Eilenberg’s office.  Later he guided me to my
introduction to several

facets of category theory. I was a novice and I very much appreciated his kindness,
his warmth, and his friendship.

 

In a Columbia class that we both attended, he pointed out the relation between
adjoint functors and free algebras.  Later he extended that relationship to algebras
with infinitary operations. He emphasized the mutual relevance of functional analysis
and category

theory; (each still has much to teach the other)  and I know that Fred’s guidance
will continue to play a role.

 

I loved the wide-ranging discussions, when we met for the last time in  Warsaw for
Eilenberg’s 100th Birthday, and further on the phone just a few months ago.

 

I will miss him very much. 

 

Bill Lawvere

 



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Fred Linton

[Marta Bunge]

[Note from moderator: the message below from Barbara Mikolajewska was 
forwarded by Marta Bunge]

Very sad news.=C2=A0


__________
From: Fred E.J. Linton <fejlinton@usa.net>
Sent: September 2, 2017 8:32:17 PM
To: Marta Bunge
Subject: Re: To Fred and Barbara =C2=A0
Dear Marta,

I have to inform you with great sadness
that Fred died today at 5:47 a.m.
There were no cure for his illness
in the stage when it starts to manifest
its devastating symptoms.
Hospice at least gave him some
comfort and peace in this last moments.
I discovered it is really very humanitarian
institution.

Barbara



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