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 if for each in there is an invertible 2-cell
satisfying some fairly obvious axioms corresponding to the usual ones for pseudonatural transformations:
The assignment is natural with respect to 2-cells .
is an identity, modulo the unitors of .
For , is equal to a suitable pasting composite of and . (It’s pretty obvious if you try to draw it: paste and along and complete the square (so to speak) using the bifunctoriality isomorphisms of .)
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.
Revision on May 25, 2011 at 22:26:04 by
Finn Lawler?.
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