The Double Lives of the Category of Quantales

It is well known that if $M$ is a module over a commutative ring $R$, then the endofunctor $-{\otimes }_{R}M$ has a left adjoint if and only if $M$ is a finitely generated and projective. Following their ring/quantale analogy, Joyal and Tierney [1] considered monoids in the symmetric monoidal category of complete lattices and sup-preserving maps, later called quantales, and proved the above projectivity characterization for modules over a commutative quantale, without the finiteness condition. In a paper with Wood [2], we proved a general theorem characterizing the existence of a left adjoint to the functor $-{\otimes }_{R}M$ on the category of modules over a commutative monoid $R$ in a suitable symmetric monoidal categories $𝒱$, and obtained corollaries for rings and quantales.

In [4], Paré defined adjoints and Cauchy completeness for double categories and, considered the double category $\mathrm{ℝ}\mathrm{𝗂𝗇𝗀}$ of commutative rings, homomorphisms, and bimodules. There, an $(R,Q)$-bimodule $M$ has a right adjoint if and only if it is finitely generated and projective as an $R$-module, and so there are no non-trivial Cauchy complete rings. To overcome this deficiency, he replaced the homomorphisms in $\mathrm{ℝ}\mathrm{𝗂𝗇𝗀}$ by maps he called amplimorphisms, and obtained a double category $\mathrm{𝔸}\mathrm{𝗆𝗉𝗅𝗂}$ of rings in which every object is Cauchy complete. Subsequently [3], we incorporated Paré’s adjoint bimodule result into a version of the 2017 theorem with Wood, which we then applied to rings and quantales. However, the proofs of the latter were separate due to the finiteness condition for rings but not quantales, and we did not construct an $\mathrm{𝔸}\mathrm{𝗆𝗉𝗅𝗂}$-like double category for quantales.

After recalling the necessary background, we present three double categories whose objects are quantales. The first is strict, the second is pseudo, and the third is a double bicategory, in the sense of Verity [5]. The strict double category is Cauchy, i.e., every object is Cauchy complete. The pseudo one is not, but this is corrected using a Kleisi-like construction. To do so, we assume additional conditions on $𝒱$ and introduce a general notion of projective module over a monoid in $𝒱$ which, when added to the 2025 theorem, gives a single theorem which applies simultaneously to rings and quantales. Finally, we define morphisms of quantales like Paré’s amplimorphisms of rings, and construct a Cauchy double bicategory of quantales.