Re: What about biproducts?

Re: What about biproducts?

[Fred E.J. Linton]

On Sat, 14 Jan 2012 09:51:24 AM EST, George Janelidze <janelg@telkomsa.net>
wrote:

> ... What about biproducts? ...
>
> ... [snip] ...
>
> (b) if a category admits infinite biproducts, then it is indiscrete (=every
> object in it is zero) ...

Let me refute that by channeling the voice of Dana May Latch, the late 
Alex Heller's student-of-yore, who stumped me once, decades ago, by asking:

: What do you call a category where products and coproducts coincide?

I confessed I had no idea, I had not even an example of that phenomenon,
and she immediately offered the example (which I hereby share with George)
of sup-complete sup-semilattices (with bottom element (of course)), and
sup (and bottom-element) -preserving maps.

If L_i are such semilattices, with bottom elements 0_i, and L is their 
product, with projections p_i: L --> L_i, the functions j_i: L_i --> L
defined by p_n(j_i) = id_L_i (n=i), p_n(j_i) = 0_n (otherwise) display
the product L as a coproduct of the L_i.

Indeed, given a family of sup-preserving maps f_i: L_i --> T to a 
sup-complete test sup-semilattice T, the solution f: L --> T to the
associated universal mapping problem j_i(f) = f_i is given simply by

f(l) = f((..., l_i, ...)) = sup_i(f_i(l_i)) (l = (..., l_i, ...) &isin; L).

In fact, one may thus even see id_L as the sup of all the compositions
j_i(p_i), much as happens (using addition) for the biproduct of modules,
only using not addition but the infinitary "N-linear" or "semi-additive" 
structure relevant to the category of sup-complete sup-semilatiices.

Even more, as Dana May knew already back whenever that was, the
examples of &aleph;-complete sup-semilattices illustrate that one can 
have categories in which products and coproducts of up to &aleph; objects
coincide, but larger ones differ -- for pretty much any &aleph; (by
"pretty much any" should I probably mean "any regular cardinal", i.e., 
any cardinal not the sum of fewer smaller cardinals? I'm not sure).

If Dana May is lurking in the background, reading these communications,
I'd sure be glad to learn more from her what finally became of the line
of thinking these considerations were part of, and what terminology she
settled on for such "infinite biproducts" and for categories having them.

Cheers, -- Fred Linton



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Re: What about biproducts?

[Vaughan Pratt]

On 1/14/2012 2:22 PM, Fred E.J. Linton wrote:
> Let me refute [that] by channeling the voice of Dana May Latch, the late
> Alex Heller's student-of-yore, who stumped me once, decades ago, by asking:
>
> : What do you call a category where products and coproducts coincide?
>
> I confessed I had no idea, I had not even an example of that phenomenon,

Fred, you *surely* meant something else by this.

> and she immediately offered the example (which I hereby share with George)
> of sup-complete sup-semilattices (with bottom element (of course)), and
> sup (and bottom-element) -preserving maps.

Dana was sufficiently taken with CSLat as to publish a short note on
some of its properties (JPAA? AU?), prompting Peter Johnstone to write a
review of her note to the effect that she should have exploited its
self-duality to make her note even shorter.  (Peter could have set a
good example by making his review a lot shorter.  They were both young
back then, with all that implies, but come to think of it so were we
all, including you, Fred.)

One might ask what is the least change to CSLat breaking this property
while retaining most of what makes it interesting.  I found one answer
to this at

http://boole.stanford.edu/pub/es.pdf

in the course of answering a different question: is there a
non-degenerate model of linear logic that models the duality of time and
information and of events and states?  My suggestion was to leave the
(non-empty) meet-join structure of CSLat and its dual unchanged while
switching top and bottom.  The result was cute and good for a few papers
but eventually I saw the light and switched to Chu spaces which answered
my original question much better, albeit without as tight a connection
to CSLat (it's just a tiny subcategory of Chu(Set,2), and anyway these
days I work in Chu(Set,4) when not hacking climate, Euclid, etc.).

Vaughan

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Re: What about biproducts?

[Vaughan Pratt]

On 1/15/2012 3:14 PM, George Janelidze wrote:
> Many thanks to Fred, and I apologize for my "blind spot" with infinite
> biproducts.

I'm afraid I must make the same apology, I completely overlooked the
finite-infinite distinction.  I withdraw my comments on that point.
CSLat was not much in the air back then and Dana and Peter seemed to be
among the few acquainted with it.

Vaughan


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Re: What about biproducts?

[Vaughan Pratt]

On 1/13/2012 11:35 PM, Robin Cockett wrote:
> First a disclaimer:  I do not claim to be a huge expert on what linear
> logicians actually mean by "linear"!!!  Indeed, the meaning of the word --
> to linear logicians -- has definitely changed and expanded over time.
> Indeed as linear logic developed the word "linear" tended to travel with
> researchers as they focused on their favourite aspects of the logic ...
> even if it was not, perhaps, the original motivation for using the word
> (e.g "linear bicategories", "linear functors" etc.).

While the positive connotation of "linear" is "very straight," the
negative connotation is "quadratics and higher not allowed."

It's hard to know on whom to blame cartesian closed categories.  Cantor
maybe?  The genius who invented x squared (several millennia earlier)?
The diagonal functor is a sine qua non here.

The central and unchanging point with "linear" is that you aren't
allowed to use the same variable twice in the one expression.  Whether
you view doing so as the moral equivalent of drawing the last shilling
out of your bank account twice, or attempting to apply the diagonal
functor when the system protests that it's undefined, it all comes down
to the same thing.

To duplicate or not to duplicate, that is the only question.

(Which may or may not subsume "To be or not to be" depending on whether
you prefer to treat zeroary duplication separately.)

Vaughan Pratt


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Re: What about biproducts?

[rlk]

George Janelidze writes:
  > Dear All,
  >
  > Concerning categories with finite products enriched in commutative monoids:
  >
  > . . . Anyway, the main reason I am writing this message is that I think I
  > have a better name, although it is not my idea. It is
  >
  > "CATEGORY WITH BIPRODUCTS".

I want to second this.  When I earlier wrote that I was using "category with
direct sums" I did not mention that the alternative I use is "category with
biproducts".  The reason I mostly use direct sum rather than biproduct is that
it is familiar to students where biproduct is not, but "category with
biproducts" does seem a better choice for articles in category theory.

I liked Bill Lawvere's suggestion of linear category but it has at least two
alternative meanings in the literature and for my students the connection with
linear logic just causes too much confusion.

-- Bob

-- 
Robert L. Knighten
RLK@knighten.org


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Re: What about biproducts?

[Michael Barr]

Actually, I like "biproducts" at least when the category has them.  Which
it does in the situation we are working in and probably not important
otherwise.  To me "linear" would suggest a monoidal closed category, maybe
*-autonomous.

Michael

-- 
Any society that would give up a little liberty to gain a little
security will deserve neither and lose both.

             Benjamin Franklin


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What about biproducts?

[George Janelidze]

Dear All,

Concerning categories with finite products enriched in commutative monoids:

In my previous message I wrote "For instance Freyd and Scedrov call it
"half-additive" in their book, and I don't know any better name". And now I
am afraid I did not say it well: I was thinking that I don't know any better
accepted name, but would it be correct to call "half-additive" an accepted
name? Anyway, the main reason I am writing this message is that I think I
have a better name, although it is not my idea. It is

"CATEGORY WITH BIPRODUCTS".

Let me explain/recall, referring to Mac Lane's "Duality for Groups":

Assuming the existence of zero (=initial+terminal) object and therefore
having zero morphisms, Mac Lane introduces (using different notation)
free-and-direct product of two objects X and Y as a diagram of the form

X <--p-- Z --q--> Y

     --i-->      <--j--

with pi = 1, qj = 1, pj = 0, qi = 0 (cf. my previous message),

in which X <--p-- Z --q--> Y is a product (="direct product") diagram,
and X --i--> Z <--j-- Y is a coproduct (="free product") diagram.

Later "free-and-direct product" was called "biproduct" (I don't remember who
did it first, maybe I never knew that...).

This definition obviously extends to arbitrary collection of objects, not
just two (although Mac Lane does not mention that). And:

(a) the empty biproduct is nothing but the zero object; that is, defining
biproducts one actually begins with the empty one;

(b) if a category admits infinite biproducts, then it is indiscrete (=every
object in it is zero); therefore saying "category with biproducts" one
should always mean "category with finite biproducts" (and "finite" reduces
to "empty" and iterated "binary" of course).

So, a "category with biproducts" should be immediately understood as a
category satisfying the conditions of the original question of Michael.

(I hope everyone will forgive me for repeating those so well known things
above).

On the other hand, we have this attractive suggestion to say "N-linear"...
As Dominique observes, he used it in one of his papers following Lawvere and
Schanuel's "Conceptual Mathematics", and I also remember that "linear" was
Bill's idea (please correct me if I am wrong), and what Ross says sounds
very convincing (and Todd adds "initial rig", which is also good to say)...

Yet another argument - what Robin says about "globally acceptable snappy
names" is very strong too. And, as Robin says, "linear" might be in
disagreement with linear logic... My knowledge of linear logic is very close
to zero, but I would expect a protest from people working in linear algebra.
Telling them that matrices occur as morphisms from coproducts to products,
and that therefore matrices compose if and only if products and coproducts
"coincide", which (when they exist) is equivalent to the existence of "good"
addition of morphisms, might convince them to learn some category theory,
but telling them that "everything linear" is "just" about commutative
monoids might have the opposite effect. I know it is funny, but they will
tell us something like: "N is not linear, it is discrete!"

One more possibility would be to use "N-linear" for "commutative monoid
enriched", but, following Robin, I think "commutative monoid enriched" is
itself not bad, and it can be shorten as "AbM-category" (since Ab-category
is often used for "enriched in abelian groups").

Trying to summarize, I think my best answer to Michael's question would be
"category with biproducts". But I am not against "N-linear" as much as I am
against "semi-additive". I think, however, "N-linear" makes sense only if
many authors will use it. "Half-additive" is more neutral: it is very
unlikely that experts of any area of mathematics will find it contradictory
to the terminology they use.

With Old Style New Year Greetings-

George



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