One small question that has often occurred to me:

- in the three usual axioms specifying how the unit interacts with parenthesizing in a monoidal bicategory, is there any known reason for drawing one of the three diagrams as a square (as opposed to a triangle, like the other two) even though one of the 1-cells is the
*identity*id$\otimes$id, except for the (certainly important) aesthetical/visual/psychological reason that otherwise (if using the conventional notation) the tip of the arrow giving the 2-cell would point from a 1-cell to a 0-cell?

(Technical note: I chose the “Latest Changes” category, even though no change to monoidal bicategory was made yet, because monoidal bicategory appears to not have had a thread of its own yet, and it is not inconceivable that this page will evolve in the future and need a thread)

]]>Is there a way (or a convenient way, or a standard way) of writing string diagrams on the nLab? In particular I could use something for just usual symmetric monoidal categories.

]]>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 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.

]]>**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.

]]>Created 201707040601 for further use in some notes on icons in $\mathcal{V}$-enriched bicategories that I am writing.

]]>[[twosets_op_with_names20170618]] is identical to [[twosets_op_nonames20170618]] except for the morphisms having names. ]]>

Created twosets20170617. Contains an svg illustration of a full subcategory of $\mathsf{Set}$ consisting of a terminal object and a two-element set. Uses the convention that an identity arrow is labelled by its object. Intended for use in some graph-theoretical considerations from an nPOV. Sufficiently general to be possibly of use in some other nLab articles too.

]]>In a seminar I’m teaching, I wanted to share some of the relationships between some of the different sorts of objects that might be called “presentations of $\infty$-categories”. I ended up with this diagram. (Incidentally, I don’t know if “diagrams” is actually the correct category for this nforum discussion; if not, anyone should feel free to change it.)

But so, it occurred to me that this might be pedagogically useful to others. In particular, it might make sense to link on the nLab somewhere, though I don’t know where offhand. So I would appreciate any comments or suggestions on this front. (I’d be fine with hearing the opinion that it’s not worth putting on the nLab; I know it’s super rough, and I definitely wouldn’t be offended.)

Of course, it’d be better if this were a texed diagram instead of a picture, but I don’t have the time to do that right now. On the other hand, I think most aspects are more-or-less self-explanatory to someone who’s in the know, but definitely not everything. So I’ll at least make a few comments.

The lowercased objects such as $cat$, $relcat$, etc., are “strict” objects, i.e. their objects have

*sets*of objects. For example, $cat$ is reflective inside of $sSet$, whereas $Cat \subset Cat_\infty$ is a full ($\infty$-)subcategory.The “flag” hanging off to the left comes logically before the rest of the diagram. That is, the interpretation of the rest of the diagram is premised on the facts contained therein.

Things pretty much commute “as much as you would expect” (for a diagram involving a bunch of adjoints), with the following caveats.

The upper-left triangle (describing the two different ways of extracting an adjunction of $\infty$-categories from an enriched adjunction of simplicial model categories) isn’t (yet) known to commute.

The corresponding triangle for Quillen equivalences is known to commute. (I think? Actually I’m not sure how to prove this offhand…)

The pentagon with $modelcat^\delta_{sSet}$ at the top-left

*doesn’t*commute.However, the two ways of proceeding from $modelcat^\delta_{sSet}$ down to $Cat_\infty$ are naturally equivalent (although I guess “naturally equivalent” is kind of a weak assertion for a pair of functors off of a discrete category; maybe there’s something slightly better to say). This is a result of Dwyer–Kan.

This is all elaborated upon much more fully in the appendix to my “simplicial spaces” paper.

]]>Using the LaTeX macro package TikZ, I’ve redrawn most of the SVGs on the knots and links pages. I hope that I haven’t trodden on any toes in so doing! I may have missed a few diagrams as well.

I’ve shifted the actual SVGs to pages of their own. This makes it easier to edit the pages with them on - TikZ’s SVG export isn’t as compact as the inbuilt SVG editor - and easier to include on other pages. For example, I can imagine that the trefoil knot is going to appear again and again!

(Incidentally, are the two trefoils distinct? If so, which have I drawn at trefoil knot - SVG).

I’ve named the pages with `- SVG`

in their name, though for the moment I’ve also put in redirects to the name without the SVG. When actually including the diagram, one should always use the *canonical* name (ie *with* the `- SVG`

) since it may be that we actually write a page about the trefoil knot one day. But I thought that for the moment, a nice aspect of hyperlinks is that if we mention the trefoil knot in a page then we can put in a link to an actual picture.

Diagrams done so far:

- trefoil knot - SVG
- trefoil knot (2 bridge) - SVG
- Hopf link - SVG
- Whitehead link - SVG
- Borromean link - SVG (also redirects from Borromean rings)
- Reidemeister move 1 - SVG (it’s amazing how many ways it is possible to misspell Reidemeister)
- Reidemeister move 2 - SVG
- Reidemeister move 3 - SVG
- figure 8 knot - SVG

Pages with includes include: link, Reidemeister move, colorability, bridge number.

What would be fantastic here is if the “source” link took one to the actual LaTeX/TikZ source! I do intend to put that up on the nLab, but I need to clean it up a little as it depends on some customised style files that have a lot of crud in them.

]]>