nLab Rig




[[!include higher algebra - contents]]



Rigs and rig homomorphisms form the category Rig.


We consider rigs as having an additive unit 00, a multiplicative unit 11 and being such that 0.x=x.0=00.x = x.0 = 0, as discussed in the entry rig.

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



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


Let RR be any rig. First suppose that we have a homomorphism f:Rf \colon \mathbb{N} \rightarrow R. Then f(n)=n.f(1)=n.1 Rf(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 Rf(n)=n.1_{R}. First, we obviously have f(1)=1 Rf(1)=1_{R}, f(0)=0.1 R=0 Rf(0)=0.1_{R}=0_{R}. Moreover, by induction on n,pn, p \,\in\, \mathbb{N} we have f(n+p)=(n+p).1 R=(n.1 R)+(p.1 R)f(n+p)=(n+p).1_{R}=(n.1_{R})+(p.1_{R}) and f(np)=np.1 R=(n.1 R)(p.1 R)f(n p) = n p.1_{R}=(n.1_{R})(p.1_{R}).

Last revised on August 19, 2022 at 07:28:04. See the history of this page for a list of all contributions to it.