nLab Rig

Idea

Rigs and rig homomorphisms form the category Rig.

Definition

We consider rigs as having an additive unit $0$, a multiplicative unit $1$ and being such that $0.x = x.0 = 0$, as discussed in the entry rig.

We recall that a rig homomorphism $f \colon R \rightarrow S$ is a function which is a monoid homomorphism for both the additive underlying monoid and the multiplicative underlying monoid.

Properties

Proposition

(initial object)
The rig of natural numbers, $\mathbb{N}$, is an initial object in Rig.

Proof

Let $R$ be any rig. First suppose that we have a homomorphism $f \colon \mathbb{N} \rightarrow R$. Then $f(n) = n.f(1) = n.1_{R}$. This shows that if such a homomorphism exists, then it is unique. To show existence, let’s verify that we indeed define a rig homomorphism by setting $f(n)=n.1_{R}$. First, we obviously have $f(1)=1_{R}$, $f(0)=0.1_{R}=0_{R}$. Moreover, by induction on $n, p \,\in\, \mathbb{N}$ we have $f(n+p)=(n+p).1_{R}=(n.1_{R})+(p.1_{R})$ and $f(n p) = n p.1_{R}=(n.1_{R})(p.1_{R})$.

Related concepts

• Ring, CRing