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I don’t think anyone has written down a definition of ‘weak’ extranatural transformations for bicategories, so let’s do that. There are no surprises, but all we’ll be doing is defining one particular kind and showing that they compose with ordinary pseudonatural transformations, so there may well be unexpected wrinkles elsewhere.
Let be a (pseudo)functor and be an object of . An -indexed family of morphisms of is extranatural in if for each in there is an invertible 2-cell
satisfying some fairly obvious axioms corresponding to the usual ones for pseudonatural transformations:
The notion of a modification between such transformations is the obvious one, so we get a category .
Now let be another functor and be an ordinary pseudonatural transformation. We want to show that the family is again extranatural. The naturality (1) and unit (2) axioms are obvious from the diagrams; the only non-trivial part is axiom (3). To draw the diagrams here would be quite painful, so I’ll just point out that in , and that this gives you the equations you need between etc.
It’s easy to see that, for a bicategory , the morphisms are extranatural in . In fact, this is the universal extranatural transformation out of the terminal category , in the following sense.
Let be a pseudofunctor. Then there is an equivalence of categories , given in one direction by composition with .
An extranatural yields a pseudonatural with components (one could also choose the isomorphic ). The mediating are obtained from the compositors of , the unitors of , and . These are all suitably natural, so is indeed a pseudonatural transformation. For a natural , the mediating isomorphisms provide the components of an invertible modification , which is in fact natural in . The (natural) isomorphisms complete the equivalence.
In fact, because (the -weighted limit of ), what we have shown is that the 2-category admits bicategorical ends.
Last revised on October 31, 2012 at 03:22:19. See the history of this page for a list of all contributions to it.