2 Jan 2012 12:51

### Preprint

```The following preprint is available, in pdf:

M. Grandis
Coherence and distributive lattices in homological algebra
Dip. Mat. Univ. Genova, Preprint 595 (2012)

http://www.dima.unige.it/~grandis/Lat.pdf

Abstract. Complex systems in homological algebra present problems
of coherence that can be solved by proving the distributivity of the
sublattices of subobjects generated by the system. The main applications
deal with spectral sequences, but the goal of this paper is to convey
the
importance of distributive lattices (of subobjects) in homological
algebra,
to researchers outside of this field; a parallel role played by orthodox
semigroups (of endorelations) is referred to but not developed here.

(Italy),
in September 2011.)

With best wishes

Marco Grandis

Dipartimento di Matematica
Università di Genova
Via Dodecaneso, 35
16146 Genova
```

6 Jan 2012 20:04

```Has anyone settled on a term to describe categories (such as commutative
monoids) in which finite sums and products coincide but are not quite
additive?  I guess they are commutative monoid enriched.

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/ ]

```
7 Jan 2012 03:02

### Re: Name for not-quite-additive categories

```Everything I say below is known, old, and contains no results that are mine.
In fact it goes back to Mac Lane's "Duality for Groups", even though not
everything is explicit there. So, I omitted proofs - but I shall gladly
recall any of them if anyone is interested.

1. Let us call a category pointed if it is enriched in POINTED SETS. Such an
enrichment is unique if it exists - even if the category does not have
initial (=terminal) object.

2. When C is pointed, for every two objects X and Y in C, we can obviously
define the canonical morphism I : X+Y --> XxY from the coproduct X+Y to the
product XxY (assuming or not that + and x are "chosen"). When C has binary
products and binary coproducts we say that they coincide if all such
canonical morphisms are isomorphisms.

3. The following conditions on a pointed category C are equivalent:

(a) C has binary products and binary coproducts (let us assume "not
chosen"), and they coincide;

(b) C has binary products and, for every product diagram

X <--p-- Z --q--> Y,

there exist morphisms i : X --> Z and j : Y --> Z forming a coproduct
diagram and satisfying the equalities

pi = 1, qj = 1, pj = 0, qi = 0.

(c) C has binary products and admits an enrichment in COMMUTATIVE MONOIDS.
```

7 Jan 2012 09:02

### Re: Name for not-quite-additive categories

```The term semiadditive has been used for at least 45 years[1], and is still in use today[2]. It's not used all
that often, but it has a good pedigree and  is unambiguous.

Robin

1. Barry Mitchell, Theory of Categories, 1965, section 1.18

On 6 Jan 2012, at 19:04, Michael Barr <barr <at> math.mcgill.ca> wrote:

> Has anyone settled on a term to describe categories (such as commutative
> monoids) in which finite sums and products coincide but are not quite
> additive?  I guess they are commutative monoid enriched.
>
> 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/ ]

```
7 Jan 2012 09:04

### Re: Name for not-quite-additive categories

```Michael Barr writes:
> Has anyone settled on a term to describe categories (such as commutative
> monoids) in which finite sums and products coincide but are not quite
> additive?  I guess they are commutative monoid enriched.
>
> Michael

I've been writing of direct sums when sums and products coincide and then of
categories with direct sums.

-- Bob

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

```
7 Jan 2012 12:29

### Re: Name for not-quite-additive categories

```"Semi-additive" seems a good enough name to me.

Peter Johnstone

On Fri, 6 Jan 2012, Michael Barr wrote:

> Has anyone settled on a term to describe categories (such as commutative
> monoids) in which finite sums and products coincide but are not quite
> additive?  I guess they are commutative monoid enriched.
>
> 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/ ]

```
7 Jan 2012 13:38

### "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/ ]

```
7 Jan 2012 20:48

### Re: "Semi-additive" seems to be it

```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

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...
```

8 Jan 2012 22:14

### Re: "Semi-additive" seems to be it

```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/ ]

```
9 Jan 2012 09:47

### Re: "Semi-additive" seems to be it

```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.
>
```