Re: direct sums and products from category theory

From: Mark Adams (markjadams_at_lycos.com)
Date: 09/23/04 

Date: Thu, 23 Sep 2004 17:00:07 +0000 (UTC)

OK, after referring to MacLane and several other references, I come up
with the following, keeping in mind my goal is to use categorical
products and sums to organize a presentation of algebraic combinations
as commonly used.

- In some special categories, the categorical coproduct (also called the
categorical sum) of two objects coincides with the categorical product
of two objects; in these cases, in category theory, they are both called
the direct sum (AKA the biproduct). In particular, the categories of
abelian groups and vector spaces have categorical direct sums.

- In the case an infinite number of factors, the operations are distinct
and only the coproduct is called the direct sum, defined as consisting
of elements that have only finitely many non-identity terms.

- These definitions are commonly used for the direct sum in any
category, including those for which the categorical direct sum is a
distinct construction or does not exist. In particular, the direct sum
of non-abelian groups and algebras is defined as above, even though no
categorical direct sum exists in these categories.

The tensor product appears as a coproduct for commutative rings with
unity, but as with the direct sum this definition is extended to other
categories. An additional distinction that can be made is between an
external sum or product, where the construction of a new object is from
given constituent objects, and an internal sum or product, which is
formed from a given object by recognizing constituent objects within it.
Unfortunately, it seems there is no such consistent distinction between
the designations "inner/outer" and interior/exterior" with regard to
sums and products. In general, symbols are not always used consistently
and it is important to understand exactly what construction is meant in
a given situation.

Comments / corrections / clarifications?

"Todd Trimble" <trimble1@optonline.net> wrote in message
news:cin6n4$5vn$1@dizzy.math.ohio-state.edu...
> On 19 Sep 2004 17:45:01 -0400, Arturo Magidin wrote:
> >
> I don't know about "much mistaken", but it is a mistake: the
> tensor product over k is the coproduct in the category of
> commutative k-algebras, but not in the category of k-algebras.
>

Incidentally, I believe the tensor product over k is the coproduct only
in the category of commutative k-algebras *with unity*; in the category
of commutative k-algebras it is A O+ B O+ (A Ox B).

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