What about biproducts?

["George Janelidze" <janelg@telkomsa.net>]

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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