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It was J.F. Jardine established the so called supercoherence theory in Journal of Pure and Applied Algebra Volume 75, Issue 2, 18 October 1991, Pages 103–194. The result can be roughly stated as follows:

A pseudo-functor $\Delta^{op}\to \mathbf{Cat}$, from the opposite of simplex category to the 2-category of small categories, is fully determined by its value on objects, face maps and degeneracy maps, natural isomorphisms corresponding to the standard simplicial identities, and a small list of higher order coherence conditions (consisting 17 identities).

This supercoherence subsumes "all diagrams commute type" coherence for monoidal category, bicategory(?).

Now it's natural to replace the target strict 2-category $\mathbf{Cat}$ by a general bicategory $\mathbf{C}$. Is the naive analogue still true?

The naive approach is first looking at any strict 2-cateogry, then embedding a bicategory in a strict 2-category by the coherence theorem for bicategory. I do not know if this works actually.

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  • $\begingroup$ The coherence theorem for bicategories states that every bicategory is equivalent to a strict one. What do you mean by embedding into a strict 2-cat? $\endgroup$
    – Stephan Müller
    Commented Nov 15, 2013 at 12:45
  • $\begingroup$ @StephanMüller I am saying the same thing. One way to show the coherence theorem is to consider the Yoneda functor, which is an embedding. $\endgroup$
    – Ma Ming
    Commented Nov 15, 2013 at 15:15

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As far as I understand, the strictness is not very relevant here since the commutative diagrams are of certain 2-morphisms which are given as data. I don't think that in any place there you need to consider triple compositions of 1-morphisms, so the comparisons simply don't appear.

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