Re: What about biproducts?

["Fred E.J. Linton" <fejlinton@usa.net>]

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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