A (At) 9:09 7/12/95, "categories"I ecrivait (wrote):

Date: Thu, 7 Dec 1995 12:06:31 +0100 From: BOERGER <Reinhard.Boerger@FernUni-Hagen.de>

It is well-known that a category in which finite products exist and coincide with finite coproducts (i.e. a zero object exists and the binary product functor is naturally equivalent to the binary coproduct functor) is semi- additive, i.e. enriched over the monoidal closed category of commutative monoids. Conversely, in a semi- additive category, every (existing) finite product is automatically a finite coroduct in the canonical way (an vice versa). Now my student Claus Kirschner observed that the proof of the latter fact does not use associativity and commutativity of the addition. Strange enough, together with the first fact this means: If a category with finite products has an operation called "addition" with neutral element 0 on each hom-set satisfying f0=0f=0 and the distributive laws, then the addition is automatically associative and commutative. Has anyone ever seen this before? Of course, finite products (or coproducts) are essential; otherwise one can easily construct single-object counterexamples. On the other hand, Hilbert observes something similar but different in an appendix to his book "Grundlagen der Geometrie": For unital rings, the commutativity of addition follows from the other axioms; one can even relax the exisence of the additive inverse to cancellation conditions. The argument is as follows: x+x+y+y=(1+1)x+(1+1)y=(1+1)(x+y)=1(x+y)+1(x+y)=x+y+x+y Cancelling x on the left and y on the right, we get x+y=y+x. Does anybody know more about similar implications?

Greetings Reinhard

This interesting remark illustrates a general algebraic phenomenon that occurs when dealing with "unitary" theories or sketches. The most well-known example is about internal groups in the category Gr of groups : they are automatically commutative (in the appropriate sense). Another example : internal "non necessarily associative categories" in Gr are automatically associative. On the other hand, an internal semigroup in the category Semigr of semigroups need not be commutative; this is the reason why Semigr has many multiplicative closed structures while Gr (or Mon, the category of monoids) has none. Unitary and non-unitary algebraic theories definitely have a very different behaviour; this is one of the reasons why I think that the theory of taxonomies (i.e. "categories without identities") should be taken more seriously! Pierre Ageron