May I try to protest against "plurality"?
My reason of suggesting "half-" and not "semi-" is "semi-abelian". I understand that "semi-" is suggested by "semigroup", but "semi-abelian" was suggested by "semidirect products". Note that "semidirect products" are defined categorically and a semi-abelian category is abelian if and only if its semidirect products coincide with direct (that is, cartesian) products.
Similarly, if a category with finite coproducts merely has semidirect
Dear all, I completely agree with George. By the way, I studied such kind of categories (among others) in: "Intrinsic centrality and associated classifying properties" J. of Algebra, 256, 2002, 126-145. I called them "linear", following Lawvere and Schanuel's "Conceptual Mathematics". Truly yours, Dominique I agree with products, then it is additive if and only if its semidirect products coincide with direct products.
Another reason against
"semi-additive = enriched in commutative monoids + has finite products"
is that we do not want to identify monoids with semigroups, do we?
And, surely, instead of saying that
"While the category of commutative monoids is a motivating example of a
semi-additive category, the category of commutative semigroups is not semi-additive"
it is much better to say that
"Semi- refers to semidirect products and not to semigroups".
I hope to get support even from those who already made the opposite
suggestion...
George
P.S. Well, I always try to respect old terminology, but sometimes (what can we do?) it is better to change it. By the way, many years ago Dmitrii
Raikov
introduced another notion of "semi-abelian". As it turned out with help of Yaroslav Kopylov, that Raikov semi-abelian means
additive + regular + coregular
It is an important notion with interesting examples, but what we call semi-abelian today seemed to be so much more suitable to call "semi-abelian"!
-------------------------------------------------- From: "Michael Barr" <barr@math.mcgill.ca> Sent: Saturday, January 07, 2012 2:38 PM To: "Categories list" <categories@mta.ca> Subject: categories: "Semi-additive" seems to be it
Thanks for all the replies, but while there was consensus, "semi-additive" got a plurality and we will go with that. Michael -- Any society that would give up a little liberty to gain a little security will deserve neither and lose both. Benjamin Franklin [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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