Commuting diagrams in a bicategory

Commuting diagrams in a bicategory

[Mike Stay]

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?
-- 
Mike Stay - metaweta@gmail.com
http://math.ucr.edu/~mike
http://reperiendi.wordpress.com


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Re: Commuting diagrams in a bicategory

[Nick Gurski]

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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