# Initial and Terminal Objects in the category of rigs

I want to determine the initial and terminal objects in $\mathsf{Rig}$, the category of rigs (provided that they exist). I can't really find a source, book or web, that deals with this category at all.

As a remainder: A rig $R$ is a ring (with identity), except ($R, +$) is only required to be a commutative monoid and we have $a0 = 0a = 0$ as an additional axiom (as it does not longer follow from the other axioms). A rig morphism $\varphi : R \rightarrow S$ is defined in the obvious way: $$\varphi(a+b) = \varphi(a)+\varphi(b)$$ $$\varphi(ab) = \varphi(a)\varphi(b)$$ for all $a,b\in R$, and: $$\varphi(0) = 0$$ $$\varphi(1) = 1$$

Every ring is rig. Also $\mathbb{N}_0$, the nonnegative integers, form a rig under addition and multiplication.

I believe, that $\mathbb{N}_0$, the nonnegative integers are initial in $\mathsf{Rig}$. The argument is analogous to the argument, that shows, that $\mathbb{Z}$ is initial in $\mathsf{Ring}$:

If $\varphi : \mathbb{N}_0 \rightarrow R$ is a rig morphism, then we necessarily have:

$$\varphi(n) = \varphi(n\cdot 1) = n\cdot \varphi(1) = n\cdot 1$$ hence, such a morphism is unique. Furthermore $\varphi : \mathbb{N}_0 \rightarrow R, n\mapsto n\cdot 1$ is indeed a rig morphism, which can be easily verified. So $\mathbb{N}_0$ is initial in $\mathsf{Rig}$.

I believe, that a trivial rig $\mathbf{1}$ (the same as a trivial ring) is terminal in $\mathsf{Rig}$, since it is terminal in the category of monoids, both as "additive" and "multiplicative" morphism, so for any rig $R$ there is a unique rig morphism $\varphi : R\rightarrow \mathbf{1}$. Therefore $\mathbf{1}$ is terminal in $\mathsf{Rig}$.

Is my reasoning correct?

(And isn't this a nice axiomatic characterization of the natural numbers ;)...)

• Yes, that looks perfectly fine. Commented Feb 27, 2015 at 20:02

As noted by Najib Idrissi in his comment, your proof is OK. You will find more under the name semiring.

Here is a conceptual explanation for "everything" concerning the category $\mathsf{SemiRing}$: It is isomorphic to $\mathsf{Mon}(\mathsf{CMon},\otimes)$. Here, $(\mathsf{CMon},\otimes)$ is the monoidal category of commutative monoids equipped with the tensor product which classifies bilinear maps in the obvious sense (similar to the case of abelian groups), unit object $\mathbb{N}$, associators as expected, etc. To every monoidal category $(\mathcal{C},\otimes)$ we may associate its category of monoid objects $\mathsf{Mon}(\mathcal{C},\otimes)$. Now here is something I would like to emphasize:

Many general statements about monoids, rings, semirings, topological monoids, Banach algebras, sheaves of rings, ring spectra, ... actually derive from general statements about monoid objects in well-behaved monoidal categories.

The first observation of this kind is that the unit object $1$ of a monoidal category $(\mathcal{C},\otimes)$ always has a canonical monoid structure (the unit is $\mathrm{id}_1 : 1 \to 1$, the multiplication is $\lambda_1=\rho_1 : 1 \otimes 1 \to 1$), and that the resulting monoid is an initial object in $\mathsf{Mon}(\mathcal{C},\otimes)$.

The next observation is that the forgetful functor $\mathsf{Mon}(\mathcal{C},\otimes) \to \mathcal{C}$ creates all limits. If you already know the proof for $\mathcal{C}=\mathsf{Set}$, just copy it. In particular, if $t \in \mathcal{C}$ is a final object, then it carries a unique monoid structure (namely the unique morphisms $1 \to t$ and $t \otimes t \to t$), and this monoid is a final object in $\mathsf{Mon}(\mathcal{C},\otimes)$.

These two general facts applied to $(\mathsf{Ab},\otimes)$ resp. $(\mathsf{CMon},\otimes)$ produce initial and final objects of $\mathsf{Ring}$ and $\mathsf{SemiRing}$ at once.

• That is a very interesting observation and I definitely keep it in mind. I just takes some time, until I have enough knowledge to understand it (I guess I understand the gist of it). Commented Apr 8, 2015 at 10:22