The tensor product of commutative monoids

We work throughout in the category Cmd of commutative monoids. In other words, all the monoids we meet are commutative, and consequently we say 'monoid' in place of the longer 'commutative monoid'. However, occasionally we will emphasize the commu-tativity. Almost every monoid A we meet will be written multiplicatively, that is we write xy for the compound of x, y ∈ A. In places we write something like x · y if this helps to avoid confusion. The unit (neutral element) is 1 or 1 A if the subscript helps to avoid confusion. One monoid that we use is written additively. This is the set N of natural number under addition. As we will see, this plays a special role in the constructions we describe. In the usual way for monoids B, C we write [B, C] for the set of morphism from B to C. Our first job is to turn this external set into an internal object of Cmd. To distinguish between the set and the monoid produce we introduce some notation. be the function given by (g · h)b = (gb)(hb) for b ∈ B. Let 1 B⇒C : B E C be the constant function given by 1 B⇒C b = 1 C for b ∈ B. It is routine to check that the constructed function g · h is a morphism, but this does depend on the commutativity of C. Thus for b 1 , b 2 ∈ B we have (h · g)(b 1 b 2) = (h(b 1 b 2))(g(b 1 b 2)) = (hb 1)(hb 2)(gb 1)(gb 2) = (hb 1)(gb 1)(hb 2)(gb 2) =((h · g)b 1)((h · g)b 2) where the third equality depends on the commutativity. A trivially exercise shows that the construction gives an associative and commutative operation on (B ⇒ C), and the function 1 B⇒C is the unit of this operation.