The n-Category Café

I haven’t really read Hasuo, Jacobs and Sokolova’s talks yet, but they seem to suggest that for any Lawvere theory $L$, and any ‘strict categorified $L$-algebra’, say $C$, we can define an ‘$L$-algebra object in $C$’. If so, this would cover your problematic example: we should be able to define a rig object in a strict rig category.

Here’s how their suggestion seems to go, according to David. $L$ is a category with finite products. A ‘strict categorified $L$-algebra’ is a product-preserving functor

$$F : L \to Cat$$

An ‘$L$-algebra object in $C$’ is a lax natural transformation

$$\phi: 1 \Rightarrow F$$

where $1$ is the terminal categorified $L$-algebra.

I don’t know what this gives in your example, but at least it should give something.

Yes, their general definition should in particular give a definition of ‘rig in a rig category’. Of course, to make a definition is trivially easy (‘a rig in a rig category is a cheese sandwich’), but to make a sensible or useful definition is another matter.

As far as I can see, the only example of their general definition that they give in their paper is the motivating one, monoids. So I’m far from persuaded that their definition is sensible.

If I get round to working out what happens in the case of rigs, I’ll let you know.

OK, here’s what I think their definition does in the case of rigs. I’ve done this only half-rigorously, so don’t trust me too much.

Let $\mathbf{C} = (\mathbf{C}, \otimes, \oplus)$ be a rig category. (I should really mention the multiplicative and additive units too.) They only deal with strict categorical structures, so, for instance, there is an actual equality $$A \otimes (B \oplus C) = (A \otimes B) \oplus (A \otimes C)$$ for objects $A, B, C$ of $\mathbf{C}$, and similarly for morphisms. (They do make a Remark, 3.3, on how to do it weakly, but I think their suggestion goes wrong in the case of commutative monoids: it gives something stricter than symmetric monoidal categories.)

According to their definition, a rig in $\mathbf{C}$ is, I think, an object $A$ of $\mathbf{C}$ together with maps $$\cdot: A \otimes A \to A,      +: A \oplus A \to A$$ (and similarly for units, which I’ll continue to ignore). These are required to satisfy axioms:

1. $\cdot$ defines a monoid structure on $A$ (with respect to the monoidal structure $\otimes$ on $\mathbf{C}$)
2. $+$ defines a commutative monoid structure on $A$ (with respect to the strictly symmetric monoidal structure $\oplus$ on $\mathbf{C}$)
3. distributivity: the composite map $$A \otimes (A \oplus A) \stackrel{id_A \otimes +}{\to} A \otimes A \stackrel{\cdot}{\to} A$$ is equal to the composite map $$(A \otimes A) \oplus (A \otimes A) \stackrel{\cdot \oplus \cdot}{\to} A \oplus A \stackrel{+}{\to} A$$ (recalling from above that the domains of these two composites are equal), and similarly for distributivity on the other side and for nullary addition.

So, is this a useful notion? I don’t know. Here’s what happens in the examples I mentioned.

If the additive structure $\oplus$ of the rig category $\mathbf{C}$ is coproduct, as it often is, then a rig in $\mathbf{C}$ is merely a monoid in $(\mathbf{C}, \otimes)$. So, for instance, a rig in the rig category $(\mathbf{Set}, \times, +)$ is not an ordinary rig, but a monoid. And as far as I can see, there’s no rig category $\mathbf{C}$ with the property that rigs in $\mathbf{C}$ are just ordinary rigs.

If $\mathbf{C}$ is the discrete rig category corresponding to a rig $R$, then there are no rigs at all in $\mathbf{C}$ unless $R$ is the one-element rig (in which case, there is just one rig in $\mathbf{C}$).

If $\mathbf{C}$ is the rig category of (finite or not) sets and bijections, with the rig structure given by cartesian product and disjoint union of sets, then there are no rigs in $\mathbf{C}$ at all.

If $\mathbf{C}$ is the poset ${}[0, \infty)$ with the reverse of the usual ordering (so that there’s a map $1 \to 0$), and with the rig structure given by $\times$ and $+$, then there’s just one rig in $\mathbf{C}$, namely $0$.

So it doesn’t seem to give anything interesting in any of the examples of rig categories that I can think of. Maybe someone else has other ideas. Or maybe someone would like to try their definition out on the theory of groups?

(Incidentally, David and John, you’ve slightly oversimplified their definition of $\mathbf{L}$-algebra in an $\mathbf{L}$-category — or ‘$\mathbf{L}$-object in an $\mathbf{L}$-category’, as they call it.

First recall that if $\mathbf{L}$ is a Lawvere theory and $\mathbf{F}$ denotes a skeleton of the category of finite sets then there is a canonical functor $\mathbf{F}^{op} \to \mathbf{L}$, which you can think of as the trivial theory sitting inside $\mathbf{L}$. It deals with all the variable relabelling.

Now, if $\mathbf{C}$ is an $\mathbf{L}$-category, i.e. a finite-product-preserving functor $\mathbf{L} \to \mathbf{CAT}$, then an $\mathbf{L}$-algebra in $\mathbf{C}$ is not just any lax transformation $1 \to \mathbf{C}$: it’s one whose restriction to $\mathbf{F}^op$ is a strict transformation.)

Tom, I don’t see how to define a rig object in a rig-category either.

But Laplaza also considered the situation where there is only a “lax” distributivity of $\otimes$ over $\oplus$, so that one has for example maps $\delta$ $$A\otimes(B\oplus C)\to(A\otimes B)\oplus(A\oplus C)$$ In this context there does seem to be a reasonable definition of rig object: you ask that A have a monoid structure $+$ with respect to $\oplus$ and a monoid structure $\times$ with respect to $\otimes$, and then the distributivity of $\times$ over $+$ can easily be expressed using the maps $\delta$ (easily expressed, but not so easy, at least for me, to type the commutative diagram here).

The advantage of allowing the $\delta$s not to be invertible is that now we include the finite products example. If \textbf{C} is a category with finite products, then it has one of these “lax rig-category” structures with $\otimes$ and $\oplus$ both just the product, and with $\delta$ sending $(a,b,c)\in A\times (B\times C)$ to $(a,b,a,c)\in(A\times B)\times(A\times C)$. And now you get the evident notion of rig object in a category with finite products to which you alluded (as well as all the less interesting examples which you enumerated).

But, it’s only good if it actually gives something useful — not a cheese sandwich.

You remind us:

So, if F:C→D is our lax monoidal functor, we have morphisms

ϕ_a,b:F(a)⊗F(b)→F(a⊗b)

for every pair of objects a,b∈C.

so why not remind us of the coherence condition? rather than send us to wiki?