ScalingStacks

While we have made some progress in understanding monoidal 2-categories and braided monoidal 2-categories, it seems clear that a truly elegant, not to mention correct, treatment of these concepts requires a better understanding of 3-categories and 4-categories. In this spirit, we would like to conclude with a list of some issues that are not yet resolved.

1) We have not included in our definition of semistrict braided monoidal 2-category any axioms involving the unit object $I$ (other than those appearing in the definition of semistrict monoidal 2-category). In the case of a strict braided monoidal category, where $A\otimes I=A\otimes I=A$ for any object $A$, there are theorems saying that $R_{A,I}=R_{I,A}=1_{A}$. In the 2-categorical setting the proof for this theorem turns into an isomorphism: $R_{A,I}\cong R_{I,A}\cong 1_{A}$. If we assumed these isomorphisms were equations and in addition that $R_{f,I}=R_{I,f}=1_{f}$ for all 1-morphisms $f:I\->I$, then we could conclude that any braided monoidal category with one object gives a symmetric monoidal category, as expected.

2) In the center $\mathcal{Z}(\mathcal{C})$ as we have defined it, the 2-morphisms $\tilde{R}_{(A,B|C)}$ are all identity 2-morphisms, while the 2-morphisms $\tilde{R}_{(A|B,C)}$ are not. This points to a curious assymetry in our definition of the center. One could equally well have defined the center so that $\tilde{R}_{(A|B,C)}$ was always the identity, and not $\tilde{R}_{(A,B|C)}$, but the question is: why is any ‘symmetry-breaking’ required? It may be relevant that Gordon, Power and Street’s proof [17] that any tricategory is triequivalent to a semistrict 3-category involves a ‘symmetry-breaking’ maneuver. This occurs because the definition of semistrict 3-category has an inherent asymmetry, in that given $f\colon A\to A^{\prime}$ and $g\colon B\to B^{\prime}$, the 2-morphism $\otimes_{f,g}$ goes from $(A\otimes g)(f\otimes B^{\prime})$ to $(f\otimes B)(A^{\prime}\otimes g)$ rather than vice versa. Perhaps, therefore, the center construction involves no asymmetries at the level of weak $n$-categories, but an arbitrary symmetry breaking is needed to translate it into the framework of semistrict $n$-categories.

Because the strong braided monoidal 2-functor of Theorem 18 is injective on objects, morphisms and 2-morphisms, this result thus serves as a strictification theorem asserting that any semistrict braided monoidal 2-category $\mathcal{C}$ is equivalent (in a precise sense) to one for which $\tilde{R}_{(-,-|-)}$ is trivial. Indeed, one may prove this strictification in other ways as well. One can also, of course, show that any any semistrict braided monoidal 2-category $\mathcal{C}$ is equivalent in the same sense to one for which $\tilde{R}_{(-|-,-)}$ is trivial.