On Wed, Sep 1, 2010 at 2:42 PM, Greg Meredith <lgreg.meredith@biosimilarity.com> wrote:
i have always taken this situation
There are *several* definitions that are almost surely "right"
to be a pretty clear critique of category theory's basic formulation.
On the contrary, I think that one of the things category theory contributes to mathematics is a precise theory of many different levels at which two (or more) definitions can be "the same". It happens all over mathematics that different definitions give rise to "the same" object; category theory gives general contexts in which to discuss such things. Moreover it often happens that some subtler sorts of "sameness" are difficult to characterize and prove. Thus it is perhaps not really surprising that in category theory itself, one encounters some of the subtlest and most difficult sorts of "sameness" to deal with.
In the definition of a bicategory, one could replace the coherence axioms by the statement that all diagrams built from the canonical ismorphisms commute. Can it be generalized to n=3, ... , omega.
You could say that's the basic idea behind Batanin's definition.
Indeed -- but one has to be careful (hence the need for operads in Batanin's definition) because in general, it is not the case that "all diagrams built from the canonical isomorphisms commute" in the naive sense. What is true is that all diagrams built *in a canonincal way* from the canonical diagrams should commute -- but then one has to make precise how we are allowed to build canonical diagrams. Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]