Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
A transfor between pseudo/lax natural transformations is sometimes called a modification. Hence modifications are the 3-morphisms in a 3-category 2Cat of 2-categories and 2-functors between them.
Just as a natural transformation between functors is a collection of -cells indexed by -cells, a modification between transformations is an indexed collection of 2-cells.
To be most general, start with lax transformations . Given those, a modification assigns to each -cell a -cell in that commutes suitably with the -cell components of and (see Leinster for details).
If are strict transformations, then the complicated-looking modification condition becomes simply a naturality square with globs where the -cells would normally be.
When you get tired of thinking individually about -categories, functors, transformations, modifications, and so on, check out (n,k)-transformation.
(Some discussion from here has also been moved to there.)
Tom Leinster, Basic bicategories, arXiv.
Revised on February 25, 2013 15:31:41
by Urs Schreiber