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I find the concept-formation for 2-rings in
particularly clear-sighted. Among other things it improves on the rationale for considering associative algebras as 2-modules/2-vector spaces and sesquialgebras as 2-rings/3-modules/3-vector spaces.
Where Baez-Dolan defined a “2-rig” to be a compatibly monoidal cocomplete category, theses authors observe that one should require a bit more and define a 2-ring to be a compatibly monoidal presentable category. (This follows Jacob Lurie’s discussion, some of which is alluded to at Pr(infinity,1)Cat).
I have now written out some of the basic definitions and statements at 2-ring in a new subsection Compatibly monoidal presentable categories. I also re-organized the full Definition section a bit, adding a lead-in discussion.
The paper says it goes to define a notion which generalizes simultaneously Grothendieck topoi and the abelian categories of quasicoherent sheaves. This is achieved similarly in a bit earlier work of Durov on vectoids (by the concept of a symmetric monoidal vectoid) see the reference there, unfortunately not cited in above work. It has a bit similar conditions to presentability or to completeness/totality, though not quite the same. The 2-category of vectoids has simpler properties than the 2-category of topoi, has interesting classifying objects (classifying vectoids) and it also explains an origin of some operad-like notions.
True, the notion of monoidal vectoid refines the Baez/Dolan notion of 2-rig and is pretty close to the Chirvasitu/Johnson-Freyd notion of 2-ring. I am addding a pointer to this now to 2-ring.
I added the reference to the earlier and quite different notion of categorical ring in a work of Jibladze and Pirashvili.
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