Let $A$ be a commutative rig, i.e. a commutative monoid equipped with a unital associative commutative bilinear multiplication and let $L$ be a distributive lattice. For the purposes of this question, say an oplax morphism $\lambda : A \to L$ is a map with the following properties:
- $\lambda (0)$ is the bottom element of $L$ and $\lambda (a + b) \le \lambda (a) \vee \lambda (b)$.
- $\lambda (1)$ is the top element of $L$ and $\lambda (a b) = \lambda (a) \wedge \lambda (b)$.
The dual concept was defined by Joyal in [1975, Les théorèmes de Chevalley-Tarski et remarques sur l'algèbre constructive] under the name notion of zeros. Wraith [1979, Generic Galois theory of local rings] calls an oplax morphisms (as above) a support notion.
Notice that if $A$ itself is a distributive lattice considered as a rig (i.e. $\vee$ as addition and $\wedge$ as multiplication), then oplax morphisms $A \to L$ are the same as lattice homomorphisms $A \to L$. This is essentially because multiplicative maps of lattices are automatically monotone, and monotone maps of join semilattices are automatically lax additive.
Let $A_D$ be the initial distributive lattice equipped with an oplax morphism $D : A \to A_D$, and say a finite set $\{ b_1, \ldots, b_n \} \subseteq A$ covers an element $a \in A$ if $D (a) \le D (b_1) \vee \cdots \vee D (b_n)$.
Proposition. $\{ b_1, \ldots, b_n \}$ covers $a$ if and only if there exist $c_1, \ldots, c_m$ in $A$ and a positive integer $m$ such that $b_1 c_1 + \cdots + b_n c_n = a^m$.
(The "if" direction is an easy manipulation of the relations defining $A_D$, but the "only if" direction seems to require an explicit construction of $A_D$.)
Once we have the above proposition, it is straightforward to check that $A_D$ is isomorphic to the lattice of quasicompact open subsets of the usual prime spectrum of $A$ when $A$ is a ring, with $D (a)$ identified with the standard open subset $\{ \mathfrak{p} \in \operatorname{Spec} A : a \notin \mathfrak{p} \}$. On the other hand, if $A$ is a distributive lattice, then $D : A \to A_D$ is an isomorphism. This suggests:
Question 1. Is there some reasonable (say, locally finitely presentable) concrete category $\mathcal{C}$ with the following properties?
- The category of commutative rigs is a (non-full) subcategory of $\mathcal{C}$.
- The morphisms in $\mathcal{C}$ from a commutative rig to a distributive lattice are the oplax morphisms.
- The morphisms in $\mathcal{C}$ between distributive lattices are lattice homomorphisms.
- The (full!) subcategory of distributive lattices is a reflective subcategory in $\mathcal{C}$.
Put another way, I am wondering what an oplax morphism between commutative rigs should be. Secondly:
Question 2. Is there some principled, a priori reason why we should be considering oplax morphisms in the first place?
A post hoc reason for considering oplax morphisms instead of rig homomorphisms is simply that rig homomorphisms give the "wrong" answer for rings. Indeed, the category of commutative rigs and rig homomorphisms already has the category of distributive lattices as a reflective full subcategory, but the reflector sends every ring of positive characteristic to the trivial lattice. It is also unclear to me why we "only" need to relax additivity – why not also relax multiplicativity to, say, $\lambda (a) \wedge \lambda (b) \le \lambda (a b)$? It would be nice if this turns out to be the decategorification of some reasonable notion of functor between rig categories.
