# Is there anything even close to a “standard symbol” to indicate that a diagram is required to commute?

It is evident from contemporary mathematical practice that there does not exist any usual notation to denote that a diagram commutes. This is almost always stated in meta-language above or below a diagram.

Questions. Is there anything even close to a standard symbolic notation to indicate that a diagram commutes? Maybe in higher category theory?

Is there any noteworthy book or paper which makes use of a symbol, regardless of whether this is to be considered standard or advisable, instead of using surrounding words to impose the condition of commmutativity?

Remarks.

• Part of the motivation for this question is wishing not to have to make up a symbol if there is any noteworthy precedent in the literature, but, needless to say, I have never seen any such.

• Since there is e.g. a standard symbol to denote pullback diagrams, it seems somewhat surprising that the contemporary consensus seems to be to use surrounding words instead of a symbol.

• Of couse, the omission of certain composite morphisms in a diagram can be seen as some sort of signalling that a diagram is required to commute. The iconic example is a commutative square, in which, for good reason, the two composites are not drawn, and one could see this very omission to by indicative enough. Yet, to me, this seems not to yield a precise pictorial language, least of all when it comes to using diagrams as input to machine-computations.

• I seem to recall to have once read a paper, which I do not find, saying that in the early days of category theory the writing style in books and papers was more diagrammatical than nowadays, so, it at all, a notational convention such as the one I am looking for might be found in early category theoretical papers.
• Often a small circle (like the composition symbol) or a small clockwise arrow for smaller commutative diagrams – Osama Ghani Aug 3 '17 at 15:32
• @OsamaGhani: thanks for this hint. For me, personally, this seems not an option, since it interferes with the diagram proper too much: a small circle looks like the letter 'O', which I often use for objects, and a circled arrow looks, well, too much like an endomorphism of some object. I recognize that this answer probably simply does not have any answer relieving me of having to invent a symbol, or of sticking to a word near the diagram. Just asking since there is always something one does not know. – Peter Heinig Aug 3 '17 at 15:38
• @OsamaGhani: moreover, a circled arrow involves having to make the choice whether to use clockwise or anticlockwise order, and whatever choice one makes, it looks as if one wants to convey specific information by the orientation chosen. This is all the more an issue if (as I happen to do) works on a proof where having chosen an orientation of the Euclidean plane plays an essential role. Such orientations are often indicated by circled arrows. Thanks again. – Peter Heinig Aug 3 '17 at 15:40
• In lectures on a blackboard, I've seen people draw three marks in the middle of the diagram, like this : $'''$. – Arnaud D. Aug 3 '17 at 15:43
• Also in lectures, I've seen something like the # symbol rotated 45 degrees, in the middle of the diagram. But the circle-arrow seems fairly common: tex.stackexchange.com/questions/43395/…, tex.stackexchange.com/questions/80413/… – Hans Lundmark Aug 3 '17 at 15:46

The closest thing you'll get to a "standard" symbol is to insert an identity 2-cell. $$\require{AMScd} \begin{CD} A @>f>> B \\ @VhV\qquad\!\!\Downarrow\, id V @VVkV \\ C @>g>> D \end{CD}$$ This necessarily requires the two ends are equal just like the identity arrow requires the source and target objects are equal. This can be generalized to any dimension.

Usually diagrams are assumed to commute and no additional notation is necessary. Indeed, you occasionally see notation to explicitly indicate that a diagram doesn't commute.

• Thanks for this anwer. I was aware of 2-cell, yet in using them for this, there are other issues: (0) the 2-cell symbols are so 'directional' and force one to make the arbitrary choice where to have them point to exactly, and, more seriously for my purposes, (2) if you deal with diagrams more complex than the iconic commutative square, then the 2-cell arrows, to me, do not even seem correct (or at least they commit one to one of the existing pasting-scheme-conventions). Arrows are just not 'global-looking' enough, they do not convey 'the diagram in its entirety is commutative'. Thanks again. – Peter Heinig Aug 5 '17 at 5:12
• For example, already in example diagram I post in my summarizing answer in this thread, it seems that 2-cell arrows won't do, or, at least, their use brings about additional problems, which is the opposite of what I intend. – Peter Heinig Aug 5 '17 at 5:15

To get this notational question wrapped up, here is a summarizing answer.

Generalities.

Needless to say, marking out diagrams as commutative is unusual, and---if done badly--spoils them.

The beauty and usefulness of commutative diagrams lies in that they contains hardly anything unnecessary.

There can be reasons for a symbolic notation though, in particular if one in a proof works with multiple diagrams, as mathematical objects in their own right, and finds accompanying words lying about somewhere in their periphery inadequately precise.

This is why I, personally, opted not to use any symbol within the diagram, neither the $\circlearrowleft$ kindly suggested by Osama Ghani, nor the $'''$ suggested by Arnaud D., nor the rotated # kindly suggested by Hans Lundmark, in the comments to the OP, nor even the identity 2-cell kindly suggested by Derek Elkins in one of the answers (in the example below, one would need more than one higher cell, which brings about new problems, while the intent is just to mark the diagram in its entirety as commutative).

In my work, I use rectangular light-gray frames around the diagram, annotated with the word "commutes", always at the bottom of the rectangle (see below).

For certain purposes, it still seems to me that a standardized notation for a diagram being commutative is useful.

Please note that the below is unusual; I'm just adding it here to suggest to others one of the infinitely-many solutions one can find to this notational problem.

represents a set of axioms($=$ set of pairs of words over a signature of category theory).