The original question a friend asked me is what to call the coherence cells in a lax monoidal functor. After looking around, I was surprised to realize that when it comes to monoidal functors, pseudofunctors, morphisms of bicategories, or lax morphisms of algebras of a 2-monad more generally, the most common choice of terminology among various authors seems to be a steadfast refusal to refer to these cells in any way other than "coherence cell", or by some fixed Greek letter if more specificity is needed. In a very few places, e.g. the nlab, I've found "compositor" used. Are there other names I'm missing, or that just doesn't make their way into papers?

In principle it seems suboptimal to just say "coherence cell" given that, especially as one goes up in dimension, there can be a lot of different kinds of "coherence cells" around. But I can think of a couple reasons for not giving these cells a real name, and just using "coherence cell"

  1. There is such a wide variety of 2-monads, it might not make sense to choose a single name to use for their coherence cells every time. For example, "compositor" seems good for a morphism of bicategories in general, but for a monoidal functor maybe it is not so apt.

  2. Coherence cells typically satisfy coherence results saying that "all diagrams formed out of coherence cells commute". Hence as long as one doesn't work "too strictly", it's typically possible to tell which coherence cell is intended by looking at domains and codomains. So typically the only person who really needs to refer to the cell in a more specific way is the person proving the coherence result, and they can safely just use Greek letters.

Of course, not every coherence result is of the form mentioned in (2) -- the simplest counterexample is symmetric/braided monoidal categories where the symmetrizer/braiding can give rise to multiple canonical coherence maps between the same object. It seems to me that the coherence cells of a lax morphism satisfies a pretty simple coherence theorem which is of this form. But as soon as one's lax morphism is interacting with other sorts of categorical structure -- for example if one considers a lax monoidal functor between symmetric monoidal caetgories -- things may become more complicated, and more specificity might be needed.

  • $\begingroup$ In the case of lax-monoidal functor $F$, I would call the natural transformation $F(X) \otimes F(Y) \to F(X\otimes Y)$ the "lax-monoidality data", and the morphism $1 \to F(1)$ the "lax pointing". $\endgroup$ – Theo Johnson-Freyd Feb 25 '16 at 3:46
  • $\begingroup$ @Theo: I think you mean the "lax monoidor" and the "lax pointor," of course! $\endgroup$ – Qiaochu Yuan Feb 25 '16 at 5:02
  • $\begingroup$ Anyway, I think you want to give these cells names because there are situations where you want to point out that interesting information is hiding in them. For example, if you stare at the proof of Tannaka duality for finite groups, you crucially make use of the monoidor of the fiber functor in order to recover the group; without the monoidor, as explained at mathoverflow.net/questions/155743/…, you only recover the character table. $\endgroup$ – Qiaochu Yuan Feb 25 '16 at 5:07
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    $\begingroup$ I usually refer to these things as "comparison" cells, maps, structure etc. It seemed to me a fairly common terminology. $\endgroup$ – Dimitri Chikhladze Feb 25 '16 at 13:34
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    $\begingroup$ As a general term, I usually use "constraint" rather than "coherence", especially in the lax case. But either way, one can add add noun adjuncts to indicate the particular constraint: unit constraint, tensor product constraint, associativity constraint, etc. $\endgroup$ – Mike Shulman Feb 25 '16 at 18:00

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