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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeFeb 4th 2013
   • (edited Feb 4th 2013)
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   Author: Urs
   Format: MarkdownItexI find the concept-formation for _2-rings_ in * Alexandru Chirvasitu, Theo Johnson-Freyd, _The fundamental pro-groupoid of an affine 2-scheme_ ([arXiv:1105.3104](http://arxiv.org/abs/1105.3104)) 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](http://ncatlab.org/nlab/show/2-rig#CompatiblyMonoidalPresentableCategories)_. I also re-organized the full [Definition section](http://ncatlab.org/nlab/show/2-rig#Definitions) a bit, adding a lead-in discussion.

   I find the concept-formation for 2-rings in

   • Alexandru Chirvasitu, Theo Johnson-Freyd, The fundamental pro-groupoid of an affine 2-scheme (arXiv:1105.3104)

   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.

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeFeb 4th 2013
   • (edited Feb 4th 2013)
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   Author: zskoda
   Format: MarkdownItexThe 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 [[vectoid]]s (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.

   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.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeFeb 4th 2013
   • (edited Feb 4th 2013)
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   Author: Urs
   Format: MarkdownItexTrue, 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]].

   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.

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeFeb 4th 2013
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   Author: zskoda
   Format: MarkdownItexI added the reference to the earlier and quite different notion of categorical ring in a work of Jibladze and Pirashvili.

   I added the reference to the earlier and quite different notion of categorical ring in a work of Jibladze and Pirashvili.