[New thread because, although it existed since 2012, pasting scheme appears not to have had a LatestChanges thread]

Started to expand pasting schemes. Intend to do more on this soon, in an integrated fashion with digraph and planar graph.

PLEASE note: ACCIDENTALLY a page pasting schemes was created too, as a result of some arcane issues with pluralized names of pages-still-empty. Please delete pasting schemes.

]]>Created digraph. Some background: this discussion. Created with permission, in the sense of

]]>If you really want to split off material that is pertinent to digraphs in the graph-theorist’s sense, then I myself would have no objection to a new article “digraph”.

**Changes-note**. Changed the already existing page 201707071634 to now contain a different svg illustration, planned to be used in an integrated way in pasting schemes soon.

**Metadata.** Like here, except that in 201707071634 symbols (arrows) indicating what is to be interpreted to 2-cells are given, in the same direction as in Power’s paper.

**Changes-note**. Changed the already existing page 201707071626 to now contain a different svg illustration, planned to be used in pasting schemes soon.

**Metadata.** What 201707051600 is: relevant material to create an nLab article on pasting schemes.
This is (a *labelling* of) the (plane diagram underlying the) pasting diagram A. J. Power gives as an example in his proof of his pasting theorem herein.

Unlike there, the 2-cells are not indicated in 201707051600.

Related concepts: pasting diagram, pasting scheme, digraph, planar graph, higher category theory.

]]>Changed 201707051620.

**Metadata.** by-and-large, cf. this thread.
A difference to 201707051600 is that here what A 2-Categorical Pasting Theorem, Journal of Algebra 129 (1990) calls a “boundary of the face F” is indicated by bold arrows.

EDIT: (proof of necessity of hypothesis in [A 2-Categorical Pasting Theorem, Journal of Algebra 129 (1990] and relevance to 201707051600 temporarily removed, to make it more uniform)

]]>EDIT:

**Changes note.** Changed the already existing page 201707040601 to contain an svg illustration relevant to pasting scheme and [this thread]

**Meta data.** cf. [this thread]; difference is that in 201707040601 a *face* $F$ of the plane digraph is named and one of the two orientations of the euclidean plane is indicated by a circular gray arrows. A connection to [Power’s proof] can be seen by letting $q_{-\infty}:=s$ (in Power’s sense), and $q_{\infty}=t$, and $F$ the “F” in Power’s paper.

OLD, bug-related discussion:

For some reason unknown to me, the “discussion” (actually, it is merely meant to be the obligatory “log what you do” entry), the discussion with name ‘201707040601’ that I started seems to have technical problems: the comment I entered is not displayed (to me). I would delete it, but apparently it is not possible to delete “discussions” one has started. Please do with it whatever seems most appropriate.

]]>**Changes-note**. Changed the already existing page 201707051600 I created, to now contain another svg illustration, planned to be used in pasting schemes soon. Sort-of-a-permission for this is

Power’s proof of (I guess you mean) his pasting theorem would probably be very handy to have discussed at the nLab. It would seem to fit at one of pasting diagram or pasting scheme, but less well at an article on some notion of graph I think. If you could even just write down the precise definitions of these various notions, that would also be very fine in my opinion.

**End of changes-note**

**Metadata.** What 201707051600 is: relevant material to create an nLab article on pasting schemes.
More specifically: to document A. J. Power’s proof of one of the rigorous formalizations of the notational practice of pasting diagrams.
201707051600 shows a plane digraph $G$.
Vertex $q_{-\infty}$ is an $\infty$-coking in $G$.
Vertex $q_{\infty}$ is an $\infty$-king in $G$.
Connection to A 2-Categorical Pasting Theorem, Journal of Algebra 129 (1990): therein, the author calls $q_{-\infty}$ a “source”, and $q_{\infty}$ a “sink”. This is fine but not in tune with contemporary (digraph-theoretic) terminology, whereas “king” and “coking” are.
These technical digraph-theoretic terms will be defined in digraph.

Related concepts: pasting diagram, pasting scheme, digraph, planar graph, higher category theory.

[ Some additional explanation: it was bad practice of me, partly excusable by the apparent LatestChanges-thread-starting-with-a-numeral-make-that-thread-invisible-forum-software-bug, to have created this page without notification and having it left unused for so long. Within reason, *every* illustration one publishes should be taken seriously, and documented. Much can be read on this of course, one useful reference for mathematicians is the TikZ&PGF manual, Version 3.0.0, Chapter 7, Guidelines on Graphics. My intentions were well-meant, in particular to improve the documentation of monoidal-enriched bicategories on the nLab. This is still work in progress, but to get the digraph/pasting scheme project under way is more urgent. Will re-use the 201707* named pages for this purpose, for tidiness. ]

[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.

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