Mike- I have written up such definitions before, but nothing in publication as the completely general definition of, say, a sylleptic monoidal bicategory has not yet been more useful than the stricter definitions you could find in the work of Day and Street. I do have some work on braided monoidal bicategories under revision in which the most general definition will appear, and some ongoing joint work of mine with Mikhail Kapranov studies Picard 2-categories which are special kinds of symmetric monoidal bicategories. I even have a document which has lots of these definitions written down, but nothing else, not even sketches of how you might compose the higher cells. I could make that available if there is general interest. On the other hand, there is your question about chopping up the axioms in other ways to produce more symmetric looking pasting diagrams. The answer is yes, you can do it in any way you like, by the coherence theorem bicategories and in particular John Power's coherence result about pastings. The point is that any pasting diagram of 2-cells is actually just a composite of the component 2-cells vertically; any horizontal composition is taken care of by whiskering and then composing whiskered cells vertically. The coherence theorems above say that any two ways you can compose the same 2-cells give the same answer. So chopping up your diagram differently but retaining the same 0-cell source and target really just means composing both sides of an equation between 2-cells with some invertible 2-cells, and then probably cancelling some things on one side. If you want to change a 0-cell source or target, then it is likely that what you are doing is taking the original pasting, whiskering it by some equivalence 1-cell, composing both sides with the same invertible 2-cells, and then using some equations on one side. Since those 1-cells were equivalences, your new equality of pastings is logically equivalent to the old one, so even when you interpret the idea of "chopping up the diagram" quite liberally you get the same answer. I suppose the moral here is that if you have an equality of 2-cells in a bicategory (that is what these axioms are, after all), then you should feel free to alter it by either composing with invertible 2-cells or whiskering by equivalence 1-cells, and then demanding equality. Nick Mike Stay said the following on 12/01/2010 23:17:
I'm writing up the definitions of braided, sylleptic, and symmetric monoidal bicategories and would like to reorient some of the polyhedral coherence diagrams to make their symmetry more apparent. All the 2-morphisms are isomorphisms and all the 1-morphisms are equivalences. In such a case, it seems like any way of chopping up the polyhedron into two "sides" will commute as long as one way does. Is this common wisdom, or a folk theorem, or has someone proved it?
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