I am a newcomer to non-commutative geometry and really enjoying the literature. I have read several of the classic papers, assisted by the excellent lecture notes *Homological Methods in Non-commutative Geometry* by Kaledin. I have also read the papers by Reyes, Heunen, and van den Berg on the obstructions to extending the Zariski site to $\mathsf{Ring}$ in various ways. Now I see on the nLab page for noncommutative algebraic geometry there is the added statement:

This is the reason why non-commutative algebraic geometry is phrased in other terms, mostly in terms of monoidal categories “of (quasicoherent) abelian sheaves” (“2-rings”).

Are there a few names people can drop for me, or even specific papers, which are most significant for following up on this hint? Before coming across this page I worked out some things for sheaf theory on bicategories (e.g. of noncommutative algebras), but feel I must be on well-trodden ground.

]]>[Reasons for starting a new thread:

(0) This topic seems fundamental and complex enough to merit a thread of its own.

(1) This topic seems be likely to be of lasting interest to others in the nLab.

(2) The relevant threads that exist tend to be *LatestChanges* threads and so far, no change was meant on account of t*his* topic.

]

Briefly: is planarity *only-sufficient* for a rigorous formalization of pasting schemes in 2-, 3- and 4-categories, or is there something more essential that I am missing, causing mathematicians to use plane graphs when doing so?

In more detail: my understanding is that A. J. Power in “A 2-Categorical Pasting Theorem Journal of Algebra 129, 439-445 (1990), henceforth JAlg129, gave the first rigorous proof that any order in which one tries to evaluate a given finite acyclic plane pasting diagram evaluates to the same 2-cell.

It indeed seems to be the case that (telling from what I studied of work of N. Gurski and others) for 2- and 3-categories, and even (telling from what I studied of work of T. Trimble and A. E. Hoffnung, and from in particular Trimble’s diagrams hosted by J. Baez) for 4-categories, all axioms necessary to construct these structures *can* be expressed by “schemes” whose underlying graphs *happen* to be planar.

But is there a precise sense because of which one can discount the possibility that one

- might need/want pasting-scheme-equation-expressed-axioms whose underlying graphs are
*non*planar?

It seems to depend on the answer to this question whether one considers the formal definitions of “pasting diagram” and “pasting scheme”, which are plane graphs with some additional structure added, as fundamental or merely manageable expedients sufficient to rigorously formalize *those*pasting-diagram-challenges that had been thrown down so far, so to speak.

Another aspect is that some graph-theorists might disagree that Power’s proof makes “heavy use of the techniques of Graph Theory” (JAlg129, abstract); the proof rather makes essential use of the *plane* graphs, i.e., is rather an application of planarity than of what is typically seen as graph theory.

While “heavy use” is an overstatement in my opinion, this seems a nice example of *common ground* between category theory and graph theory. It apparently has not been made clear enough what is necessary for what.

I did not yet look closely into the question how much of the planarity is *indispensable* for Power’s proof to work out, and decided to ask first since this seems an obvious question and likely to have been asked answered before, but I do not find it.

The *obvious* question is of course: is there a non-planar relevant counterexample in the literature? I have been searching around for quite some time now.

It seems to me that, roughly speaking, one can *decide* to impose *additional non-planar axioms*, although one just happens not to need to do so in order to ensure coherence.

So, do you think Power and Yetter just *happened to tame higher-composition restricted to the plane*, using the plane as a convenient frame in which to carry out the induction-proof, or am I missing something essential because of which one can rest assured that no *non-planar* “pasting diagrams” (the latter in an informal sense) will be needed?

If not, the right formalization of pasting diagrams and nonambiguity of composition might perhaps not yet have been found.

]]>Is the category Hom of bicategories with homomorphisms as the morphisms, in the sense of

Ross Street, Fibrations in bicategories, Cahiers de topologie et géométrie différentielle catégoriques, tome 21, no 2 (1980), p. 111-160

already (recognizably) documented on the nLab ? (I did a sem-cursory search in this respect, but did not find it documented (in its own right, I mean, the article of Street appears.)

Should it be?

Should it have an article of its own?

To me it seems it should (my motivation is that I am using and documenting bicategories currently, and are studying Street’s 1980 paper as a sort of background reading to Garner–Shulman, Adv. Math. 289), but its traditional name Hom seems unfortunate, creating yet another meaning of Hom.

My suggestion would be to call it (and its article)

$BiCat$

]]>(reconsidered; no time at the moment to bring post into satisfactory form)

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