Re: Limits and colimits in Rel?

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

Writing V for the category of complete join-semilattices that
Peter brought into the discussion,

> However, I don't think that the self-duality is in any sense
> responsible for the lack of (co)limits in Rel. The category of
> complete join-semilattices is self-dual, and is complete and cocomplete.

it might be worth pointing out that Uwe's category Rel is a V-category
(in exactly the sense that Eilenberg, Kelly, Street, Day, and others mean)
for exactly this choice of V.

With that in mind, the parallel

in additive categories with zero object, finitary (co-)products are
biproducts
  and
in V-categories with zero object, arbitrary (co-)products are biproducts

(first noticed, in my awareness, by Dana May Latch, in the 20th century, 
for the category V of complete join-semilattices itself), is remarkable.

Anyway, the second line of that parallel works much as it does for Rel.

And the "matrix algebra" for maps to products, from coproducts, and, most
especially, from coproducts to products, works just as it does in the case
of additive categories, when it comes to these V-categories.

Cheers, -- Fred




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