Regarding Q2: probably there is a way to avoid going deep into coherence conditions: instead of proving by hand the equivalence between promonoidal structures on $C$ and biclosed monoidal structures on $\hat C$, one can resort to a more conceptual pov.
What happens for pro/monoidal categories is that there is a pseudomonad $S$ on $\sf Cat$ with the property that $S$ lifts to a pseudomonad $\hat S$ on $\sf Prof$ (the Kleisli bicategory of $P=\hat{(-)} = [(-)^{op},{\sf Set}]$), and pseudo-$S$-algebra structures correspond to pseudo-$\hat S$-algebra structures (this is an equivalence of categories, in the appropriate sense; see here).
I believe a similar argument holds for every (almost every?) monad $S$ equipped with a distributive law over $P$ (the presheaf construction); this does not fall short from an equivalence
$$
\{S\text{-algebra structures on } PX\} \cong \{\hat S\text{-algebra structures on } X\}
$$ where $PX$ is regarded as an object of $\sf Cat$, and $X$ as an object of ${\sf Kl}(P)$.
Regarding Q1: have you tried to find the distributive and annullator morphisms for the putative 2-rig structure on $\widehat{C}$?
I was trying to find at least one distributive morphism, and I have no idea how to reduce $F\hat{\otimes}(H\hat{\oplus} K)$ to/from $F\hat{\otimes} H \,\hat{\oplus}\, F\hat\otimes K$, if $F,H,K : \widehat{C}$. If I'm not wrong (this is very back-of-the-envelope coend calculus),
$$\begin{align*}
F\hat\otimes H &= \int^{UA}FU\times HA\times [\_, U\otimes A]\\
F\hat\otimes K &= \int^{U'B}FU'\times KB\times [\_, U'\otimes B]
\end{align*}$$
whereas
$$\begin{align*}
F \hat\otimes \,(H\hat\oplus K) &= \int^{UV} FU \times (H\hat\oplus K)V \times [\_, U\otimes V] \\
&=\int^{UVAB} FU \times HA \times KB \times [V, A\oplus B] \times [\_, U\otimes V] \\
&=\int^{UAB} FU \times HA \times KB \times [\_, U\otimes (A\oplus B)] \\
&=\int^{UAB} FU \times HA \times KB \times [\_, U\otimes A \oplus U\otimes B] \\
\end{align*}$$
...and now we're stuck, unless we have either
- R1. a compatibility between $\oplus$ and $\times$, perhaps another distributive morphism;
- R2. a siftedness condition ensuring that
$$ \int^{UA}FU\times HA\times [\_, U\otimes A] \oplus \int^{U'B}FU'\times KB\times [\_, U'\otimes B]$$
can be reduced to an integral on just $U$.
Actually, you need both in order for the computation to proceed; but the conjunction of R1 and R2 is quite strong, as you can see.
Edit: the situation with annullators (for Laplaza, morphisms ${\bf 0}\otimes X \to {\bf 0}$ and $X\otimes {\bf 0} \to \bf 0$) is even worse!
Let's open $F \hat\otimes {\bf 0}$ recalling that in this case $\bf 0$ is the representable $y{\bf 0}$ on the additive unit of $C$:
$$\begin{align*}
\int^{UV} FU \times [V,{\bf 0}] \times [\_,U\otimes V]
&=\int^U FU \times [\_, U\otimes {\bf 0}] \\
&\overset{\rho_U}\to\int^U FU \times [\_, {\bf 0}]\\
&=\varinjlim F \times [\_, {\bf 0}]
\end{align*}$$
the cartesian structure on $\sf Set$ now entails that this is $\bf 0$ if and only if either factor is empty, but I see no way in which this can be or even map into $y{\bf 0}$ again, as it should.
I offer the following summary/interpretation of Broodryk's results.
In short, a category of algebras is coextensive if and only if there is a well behaved interpolation operation in the algebraic theory.
By "interpolation operation" what I mean is an operation $t$ of arity $2 + \kappa$, together with two $\kappa$-tuples of constants, $\vec{e}$ and $\vec{e}{}'$, satisfying the following equations:
$$\begin{aligned}
t (x, x', \vec{e}) & = x \\
t (x, x', \vec{e}{}') & = x'
\end{aligned}$$
The meaning of "well behaved" is more complicated.
Let $A$ be an algebra, let $F (0)$ be the initial algebra (= free algebra with empty generating set), and let $\iota_1 : A + F (0) \times F (0)$ and $\iota_2 : F (0) \times F (0) \to A + F (0) \times F (0)$ be the coproduct insertions.
We define a map $\delta_A : A \times A \to A + F (0) \times F (0)$ as follows:
$$\delta_A (x, x') = t (\iota_1 (x), \iota_1 (x'), \iota_2 (\vec{e}, \vec{e}{}'))$$
(Here I am abusing notation.
What I mean by $\iota_2 (\vec{e}, \vec{e}{}')$ is the image of the pair $(\vec{e}, \vec{e}{}')$ under the obvious composite $F (0)^\kappa \times F (0)^\kappa \to (F (0) \times F (0))^\kappa \to (A + F (0) \times F (0))^\kappa$.)
This is not, prima facie, a homomorphism of algebras.
We can think of it as an external pairing operation on $A$.
(This will be made precise later.)
However, if the category of algebras is coextensive, it is possible to choose $t, \vec{e}, \vec{e}{}'$ so that $\delta_A$ is an homomorphism and furthermore satisfies the following equation:
$$\delta_A (x, x) = \iota_1 (x)$$
Conversely, the existence of such $t, \vec{e}, \vec{e}{}'$ guarantees that the category of algebras is coextensive.
Example.
For the theory of commutative rigs, we can take $\kappa = 2$, $t (x, x', y, y') = x y + x' y'$, $\vec{e} = (1, 0)$, and $\vec{e}{}' = (0, 1)$.
The point is that, in $A$,
$$\begin{aligned}
t (x, x', 1, 0) & = x \\
t (x, x', 0, 1) & = x'
\end{aligned}$$
and, in $A \otimes_\mathbb{N} (\mathbb{N} \times \mathbb{N})$ (the change in operator precedence by switching from $+$ to $\otimes$ here is confusing, but I digress),
$$\delta_A (x, x') = t (x \otimes 1, x' \otimes 1, 1 \otimes (1, 0), 1 \otimes (0, 1)) = x \otimes (1, 0) + x' \otimes (0, 1)$$
i.e. $\delta_A : A \times A \to A \otimes_\mathbb{N} (\mathbb{N} \times \mathbb{N})$ is the obvious natural isomorphism.
Example.
For the theory of commutative rings we can have $\kappa = 1$.
Indeed, we can take $t (x, x', y) = x y + x' (1 - y)$, $\vec{e} = 1$, $\vec{e}{}' = 0$.
Note that $t$ is literally linear interpolation!
In more detail now:
Let $\mathcal{A}$ be a category of algebras, by which I mean a category equipped with a (strictly) monadic functor $U : \mathcal{A} \to \textbf{Set}$.
Let $F : \textbf{Set} \to \mathcal{A}$ be left adjoint to $U$ and let $T = U F$.
Broodryk implicitly only considers finitary algebraic theories (i.e. $\mathcal{A}$ is locally finitely presentable and $U : \mathcal{A} \to \textbf{Set}$ preserves filtered colimits) but I think his main results also apply to infinitary algebraic theories.
First, as a warm up:
Lemma.
If a category has an initial object that is also a strict terminal object (i.e. every morphism with domain a terminal object is an isomorphism), then every object in that category is initial/terminal/zero. ◼
A trivial observation, to be sure, but turning it around tells us something about the set $T (0)$ of constants in the algebraic theory:
Proposition.
If $\mathcal{A}$ has a strict terminal object, then either:
- $T (0)$ is empty, or
- $T (0)$ has at least two elements, or
- $\mathcal{A}$ is trivial.
Proof.
$T (0)$ is the underlying set of the initial object $F (0)$ in $\mathcal{A}$.
If $T (0)$ has exactly one element, then $F (0)$ is both an initial object and a strict terminal object in $\mathcal{A}$, in which case $\mathcal{A}$ is trivial. ◼
Next:
Proposition. If $\mathcal{A}$ has codisjoint binary products then $T (0)$ is not empty.
Proof.
Let $A$ and $B$ be objects in $\mathcal{A}$.
The product $A \times B$ is codisjoint if the following is a pushout square in $\mathcal{A}$:
$$\require{AMScd}
\begin{CD}
A \times B @>{\pi_2}>> B \\
@V{\pi_1}VV @VVV \\
A @>>> 1
\end{CD}$$
Consider the case $A = B = F (0)$.
Assume $T (0)$ is empty.
Then $U (F (0) \times F (0)) \cong T (0) \times T (0)$ is also empty, hence both projections $F (0) \times F (0) \to F (0)$ are isomorphisms.
But the pushout of an isomorphism is an isomorphism, so that implies $F (0) \to 1$ is an isomorphism, which is a contradiction.
So $T (0)$ is not empty. ◼
Corollary.
If $\mathcal{A}$ is coextensive, then either:
- $T (0)$ has at least two elements, or
- $\mathcal{A}$ is trivial.
Proof.
A coextensive category has both a strict terminal object and codisjoint binary products. ◼
So far so good – after all, a rig has two distinguished elements.
But it is not obvious how to extract binary operations from the hypothesis that $\mathcal{A}$ is coextensive.
It seems the best we can do is to get a well behaved interpolation operation.
Theorem.
$\mathcal{A}$ is coextensive if and only if the following conditions hold:
There exist a regular cardinal $\kappa$ ($\le \lambda$ if $F (0) \times F (0)$ admits a generating set of $\le \lambda$ elements), an element $t \in T (2 + \kappa)$, and $\vec{e} \in T (0)^\kappa$ and $\vec{e}{}' \in T (0)^\kappa$, such that the following equations hold in every $A$ in $\mathcal{A}$,
$$\begin{aligned}
t (x, x', \vec{e}) & = x \\
t (x, x', \vec{e}{}') & = x'
\end{aligned}$$
where $x$ and $x'$ are arbitrary elements of $A$ and (by abuse of notation) we have identified $t$ with the operation $U (A)^{2 + \kappa} \to U (A)$ it corresponds to, and $\vec{e}$ and $\vec{e}{}'$ with their images in $U (A)^\kappa$ under the unique morphism ${!} : F (0) \to A$.
For every $A$ in $\mathcal{A}$, there is a morphism $\delta_A : A \times A \to A + F (0) \times F (0)$ in $\mathcal{A}$ (whose underlying map $U (\delta_A)$ is) given by
$$\delta_A (x, x') = t (\iota_1 (x), \iota_1 (x'), \iota_2 (\vec{e}, \vec{e}{}'))$$
where $t, \vec{e}, \vec{e}{}'$ are as above and I have abused notation as in the introduction.
Moreover, $\delta_A (x, x) = \iota_1 (x)$. ◻
I will sketch just the construction of $t$.
Consider the following diagram in $\mathcal{A}$:
$$\begin{CD}
F (0) @<{\pi_1}<< F (0) \times F (0) @>{\pi_2}>> F(0) \\
@V{!_A}VV @V{\iota_2}VV @VV{!_A}V \\
A @<<{[\textrm{id}_A, \pi_1]}< A + F (0) \times F (0) @>>{[\textrm{id}_A, \pi_2]}> A \\
@| @V{\eta_A}VV @| \\
A @<<{\pi_1}< A \times A @>>{\pi_2}> A
\end{CD}$$
The two squares in the top half are pushout squares (always).
The morphism $\eta_A : A + F (0) \times F (0) \to A \times A$ is defined by the universal property of $A \times A$, and it is an isomorphism if $\mathcal{A}$ is coextensive.
Consider the case where $A = F (2)$.
Let $x$ and $x'$ be the two distinguished generators of $F (2)$.
Then $(x, x')$ is an element of $U (F (2) \times F (2))$, and $U (\eta_X) : U (F (2) + F (0) \times F (0)) \to U (F (2) \times F (2))$ is a bijection, so it has a preimage $\bar{t}$.
The hypothesis on $\kappa$ ensures there is an effective epimorphism $F (2 + \kappa) \to F (2) + F (0) \times F (0)$ and we can arrange for the first two distinguished generators of $F (2 + \kappa)$ to be mapped to the image under $\iota_1 : F (2) \to F (2) + F (0) \times F (0)$ of the two distinguished generators of $F (2)$ and the remaining $\kappa$ generators to be mapped into the image of $\iota_2 : F (0) \times F (0) \to F (2) + F (0) \times F (0)$.
Let $(\vec{e}, \vec{e}{}')$ be a preimage in $T (0)^\kappa \times T (0)^\kappa$ of the images of the $\kappa$ generators, considered as an element of $U (F (2) + F (0) \times F (0))^\kappa$.
Then, $[\textrm{id}_A, \pi_1] (\bar{t}) = t (x, x', \vec{e})$ by homomorphicity, and $[\textrm{id}_A, \pi_1] (\bar{t}) = \pi_1 (\eta_X (\bar{t})) = x$ because the bottom left square commutes, so $t (x, x', \vec{e}) = x$ as required.
Similarly, $t (x, x', \vec{e}{}') = x'$.
Since we have proved the claim for the two distinguished generators of $F (2)$, the claim follows for all pairs of elements in all algebras.
The morphism $\delta_A : A \times A \to A + F (0) \times F (0)$ is simply the inverse of $\eta_A : A + F (0) \times F (0) \to A \times A$.
This is the sense in which $\delta_A : A \times A \to A + F (0) \times F (0)$ is a pairing operation.
Remark.
We have $\vec{e} = \vec{e}{}'$ if and only if $\mathcal{A}$ is trivial.
So this is another way of seeing that $T (0)$ has at least two elements if $\mathcal{A}$ is coextensive and not trivial.
It seems difficult to come up with a novel example of an algebraic theory satisfying all three of the conditions in the theorem.
The first condition – the existence of an interpolation operation – is easy enough to arrange.
(We can just freely add such $t, \vec{e}, \vec{e}{}'$ to any existing algebraic theory and get a new one!)
The hard part seems to be in the second and third conditions.
Broodryk remarks that the first condition alone is enough to guarantee many of the properties of coextensivity.
Best Answer
It should definitely be mentioned here that one often-used categorification of the integers is the sphere spectrum, $\mathbb S$, i.e. the infinite loop space $\mathbb S = \varinjlim \Omega^n S^n$. I've heard this idea attributed to Waldhausen. There's a fun exposition in TWF 102 by John Baez.
The idea is to think of $\mathbb Z$ is the (additive) group completion of the rig $\mathbb N$, and to categorify the group completion process. A natural categorification of the rig $\mathbb N$ is the 2-rig $\mathsf{FinBij}$ of finite sets and bijections, with $\times$ distributing over $\amalg$. So we group complete $\mathsf{FinBij}$ the $\amalg$ part. The thing is, rather than creating a 1-groupoid with two monoidal structures and the appropriate universal property, it's more natural to regard $\mathsf{FinBij}$ as an $\infty$-groupoid with two monoidal structures, and to construct a group completion in the world of $\infty$-groupoids. But by the homotopy hypothesis, an $\infty$-groupoid is the same thing as a space. So we end up with a space that has two monoidal structures, one distributing over the other. The "additive" monoidal structure is appropriately commutative, so this structure can be regarded as the structure of an infinite loop space, otherwise known as a connective spectrum. And the multiplicative structure makes it a "multiplicative infinite loop space", otherwise known as a ring spectrum.
Note that $\pi_0(\mathbb S) = \mathbb Z$, so this is a categorification in the most basic sense.