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

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