"Semi-additive" seems to be it

"Semi-additive" seems to be it

[Michael Barr]

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


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Re: "Semi-additive" seems to be it

[George Janelidze]

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
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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Re: "Semi-additive" seems to be it

[FEJ Linton]

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



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Re: "Semi-additive" seems to be it

[Todd Trimble]

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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Re: "Semi-additive" seems to be it

[Ross Street]

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

On 09/01/2012, at 7:47 PM, bourn@lmpa.univ-littoral.fr wrote:

> 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".


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Re: "Semi-additive" seems to be it

[Robin Cockett]

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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Re: "Semi-additive" seems to be it

[bourn]

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


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






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