On Sat, 7 Jan 2012 21:48:24 +0200, George Janelidze protested against:

"semi-additive = enriched in commutative monoids + has finite products"

I can appreciate George's motivations. I voice here only my hope that for "enriched in commutative monoids" we not retool "commutative monoidal" -) . Cheers, -- Fred [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Dear all, I completely agree with Dominique and George!!!! In response to Michael's posting I mentioned privately that Robert Seely, Rick Blute, and I (and others) had, in our work on differential categories, been using commutative monoid enriched categories and, to avoid this mouthful, had just called them "additive". Michael, as I expected did not like this at all as, of course, this means to him Abelian Group enriched. He did not like my suggestion "subtactive" as a replacement for Abelian Group enriched categories either :-) Just to mix things up: as our work is closely related to linear logic the direction of choosing "linear" was not attractive to us either: "linear" in that context means something different again! The stubborn fact is that there are only so many meaningful names. When one does a piece of work one wants to give snappy names to the important concepts in the work. Trying for a globally acceptable snappy name is almost impossible ... so I, for one, am happy to fall back on local naming conventions to replace more cumbersome formal names. And am not above poaching a name if I think it actually describes the concept well in that context. So the point is I am absolutely happy with "commutative monoid enriched category" as formal nomenclature and I am happy if an author wants to do some local naming to make things more readable. Of course, some choices are better than others! I am afraid I also shudder at semi-additive: it suggests commutative semigroup enrichment to me and relegates the concept to being a secondary one ... -robin (Robin Cockett) On Mon, Jan 9, 2012 at 1:47 AM, <bourn@lmpa.univ-littoral.fr> wrote:

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

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I also find "linear" an attractive option. Just to circumvent any confusion (e.g. with linear in the sense of linear logic, or with the question that may arise: linear over what?), one could say "N-linear" where N is of course the initial rig, as alluded to by Ross. I would hope that "N-linear category" is sufficiently unambiguous to get the meaning across, and sufficiently snappy. Best regards, Todd ----- Original Message ----- From: "Ross Street" <ross.street@mq.edu.au> To: <bourn@lmpa.univ-littoral.fr> Cc: "Categories list" <categories@mta.ca> Sent: Monday, January 09, 2012 9:35 PM Subject: categories: Re: "Semi-additive" seems to be it

Dear All

The concept of category enriched in commutative monoids is a very basic structure and it is important to find a suitable name. I must say I like the term "linear" mentioned by Dominique since the term "k-linear" is commonly used for "enriched in vector spaces over k". Hence there is no conflict if we extend to the case where k is a ring or a rig. Since the natural numbers is the basic example of a rig, we can drop the k in this case.

Best wishes, Ross

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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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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, ...) ∈ 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 ℵ-complete sup-semilattices illustrate that one can have categories in which products and coproducts of up to ℵ objects coincide, but larger ones differ -- for pretty much any ℵ (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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Dear All, Many thanks to Fred, and I apologize for my "blind spot" with infinite biproducts. I don't remember where from, but I also knew e.g. this fact about complete lattices (as we know of course, every sup-complete sup-semilattice is a complete lattice, and the only reason of mentioning "sup" is that we want morphisms to preserve arbitrary sups, but not necessarily infs), and, moreover, I believe it is a well known fact. I suddenly switched from monoids to ordinary additivity in my mind, and made that wrong statement about "indiscrete"; the correct statement would be that an additive category with infinite biproducts is indiscrete. Now, many thanks to everyone who supported the "category with biproducts" proposal. Can we still say that "biproducts" means "finite biproducts"? We certainly need the empty and binary biproducts, so we probably should say "category with finite biproducts", or, maybe, "category with (finite) biproducts". Or, if we still prefer "category with biproducts", we could go back to "Duality for Groups", and assume that biproducts (=free-and-direct products) are always binary and that they are defined only when our category category has zero object. Let me also comment to

...

"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.

from Robert Knighten's message. I agree of course, but my additional reason of not saying "direct sums" would be the situation with the theory of Grothendieck categories, where traditionally (which was started by Grothendieck himself) infinite direct sums, defined as coproducts (and they are different from products), are seriously used. Best regards, George [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]