Re: Name for not-quite-additive categories

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

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.

(d) C has binary products and admits a unique enrichment in COMMUTATIVE
MONOIDS.

(e) C admits an enrichment in COMMUTATIVE MONOIDS and, for every two objects
X and Y in C, there exist a diagram of the form

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

    --i-->      <--j--

with pi = 1, qj = 1, pj = 0, qi = 0, and ip+jq = 1.

(One can also add the conditions dual to (b)-(d), require uniqueness of the
enrichment in (e), and deduce associativity and commutativity of the
"hom-unitary-magmas" from other axioms of course).

As I understand, the question is, how to call a pointed category with finite
products satisfying the equivalent conditions (a)-(e) above. For instance
Freyd and Scedrov call it "half-additive" in their book, and I don't know
any better name.

George

--------------------------------------------------
From: "Michael Barr" <barr@math.mcgill.ca>
Sent: Friday, January 06, 2012 9:04 PM
To: "Categories list" <categories@mta.ca>
Subject: categories: Name for not-quite-additive categories

> 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


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