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Recall Mac Lane's version of coherence for monoidal categories, which one can state informally as follows:

"Simple-minded" coherence for monoidal categories

Let $A$, $A^\prime$ be two bracketings on the monoidal product $A_{1} \otimes \ldots \otimes A_{n}$ (with possibly some additional occurances of the monoidal unit $I$). Let $i_{1}, i_{2}: A \rightarrow A^\prime$ be two isomorphisms built from associator and left/right unitors. Then $i_{1} = i_{2}$.

This theorem tells us that there is always canonical way to identify two products of $A_{1},\ldots, A_{n}$ with possibly different bracketings. Here I say "simple-minded" to mean that it can be presented as a statement of the form "some diagram commutes".

However, coherence for monoidal categories can also refer to the following result:

"Strictifying" coherence for monoidal categories

Any monoidal category is equivalent to a strict monoidal category.

The strictifying version of coherence is an important theorem on its own right, but it also implies the simple-minded version of coherence with the following argument.

Let $i_{1}, i_{2}: A \rightarrow A^\prime$ be two isomorphisms as above in a monoidal category $M$ and let $T: M \rightarrow N$ be an equivalence into a strict monoidal category. Then $T(i_{1}), T(i_{2})$ are both isomorphic to analogous composites of associators and left/right unitors in $N$, with the isomorphism given by constraint cells of $T$. Since $N$ is strict, the analogous composites in $N$ are both equal to the identity and we conclude that $T(i_{1}) = T(i_{2})$.

(The argument seems to be well known, although I learned it from these short notes of Tom Leinster.)

My question concerns known results about such "simple-minded" coherence for monoidal bicategories (ie. one object tricategories), which I had trouble finding in the literature.

Recall that coherence for tricategories as proved by Gordon, Street, Power has the following form:

"Strictifying" coherence for tricategories

Any tricategory is triequivalent to a Gray-category, ie. to a category enriched over the category of strict 2-categories equipped with the Gray tensor product. In particular, any monoidal bicategory is equivalent to a Gray monoid.

This is certainly an important and powerful result. However, what is not clear to me is how to extract from this some "simple-minded" corollaries, ie. statements about some diagrams (say, of 3-cells) commuting in any tricategory. I believe some argument similar in spirit to the one from notes of Tom Leinster should work, but triequivalences (or more generally, homomorphisms of tricategories) are such complicated objects that it is not quite obvious for me how to do this.

Is there any general framework for proving that some classes of diagrams commute in every tricategory? Has this been covered in the literature? What are the possible references?

For example, is the following naive generalization of Mac Lane coherence true?

"Naive" version of coherence for monoidal bicategories

Let $A$, $A^\prime$ be two bracketings on the monoidal product $A_{1} \otimes \ldots \otimes A_{n}$ (with possibly some additional occurances of the monoidal unit $I$) in a monoidal bicategory. Let $i_{1}, i_{2}: A \rightarrow A^\prime$ be two 1-cells built from associators, left/right unitors and their pseudoinverses. Moreover, let $j_{1}, j_{2}: i_{1} \sim i_{2}$ be two isomorphisms built from the constraint $2$-cells. Then $j_{1} = j_{2}$.

Here I am using the algebraic definition of a monoidal bicategory, ie. the associatiors and unitors of the monoidal structure come with a chosen pseudoinverse and unit/counit modifications that make them into adjoint equivalences. I believe in this these unit/counit 2-cells should also be considered "constraint", but I hope I can be forgiven for leaving this a little vague just to see what might be true and what isn't.

I stumbled upon this type of questions while studying possible definitions of a dual pair of objects in a monoidal bicategory. I frequently find it very problematic to prove any uniqueness results due to the relevant computations being difficult. Hence, I am looking for techniques that could simplify working with a general tricategory.

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  • $\begingroup$ You probably know this, but in a Gray-monoid all the coherence cells are identities apart from the interchange isomorphism: the one showing multiplication is functorial up to a iso., equivalently the one mediating between horiz. composition followed by vertical composition, and vertical composition followed by horiz. composition. $\endgroup$ – theHigherGeometer Jan 20 '14 at 23:59
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    $\begingroup$ There are probably relevant papers of Nick Gurski on ArXiv, and his thesis (much expanded) has now been published as ``Coherence in three-dimensional category theory. It has quite a lot beyond what is in Gordon, Street, Power, including making the theory of tricategories fully algebraic. $\endgroup$ – Peter May Jan 21 '14 at 2:27
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I am still certainly no expert on tricategories, but I believe I have found the needed result.

The question is answered in a paper of Nick Gurski, "An algebraic theory of tricategories" and probably also in his new book ("Coherence in Three-dimensional Category Theory"). I will reference the former, where Corollary 10.2.3. reads

Let $X$ be a 2-locally discrete category-enriched 2-graph. Then in the free tricategory on $X$, $FX$, every diagram of 3-cells commutes.

Translating it into the easier language of monoidal bicategories we obtain the following.

Recall a category-enriched graph $E$ is simply a set of objects $Ob(E)$ together with a category $E(a, b)$ for any pair $a, b \in Ob(E)$. In particular, any bicategory is a category-enriched graph. A category enriched graph is locally discrete if all the categories $E(-, -)$ are discrete, ie. do not contain non-identity morphisms.

Let $E$ be a locally discrete category-enriched graph. In the free monoidal bicategory $FE$ on $E$ any diagram of $2$-cells commutes.

Free monoidal bicategories (in the above sense) have universal properties with respect to strict homomorphisms so it is easy to deduce that in any monoidal bicategory, any diagram of 2-cells that can be presented as image of some diagram in $FX$ under a strict homomorphism, where $X$ is some locally discrete category-enriched graph, necessarily commutes.

In particular the "naive" version of coherence for monoidal bicategories I asked for above is true.

The situation is not ideal, however, as one could also want to obtain a similar result for free monoidal bicategories on more general generating data (say, allow generating 1-cells of the type $I \rightarrow A \otimes B$ or $C \otimes D \rightarrow E$). This generalization turns out to be false. This is related to the fact that a one-object monoidal bicategory is "morally the same" as a braided monoidal category (a result due to Gordon, Power, Street), with the braiding given by a clever composition of 2-cells. As braiding are in general not symmetries, some diagrams of constraint 2-cells in monoidal bicategories do not commute in general.

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