nLab
modification

Modifications

Idea

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 11-cells indexed by 00-cells, a modification between transformations is an indexed collection of 2-cells.

Definitions

To be most general, start with lax transformations α,β:F˙G:CD\alpha, \beta : F \dot{\to} G : C \to D. Given those, a modification m:α¨βm : \alpha \ddot{\to} \beta assigns to each 00-cell ACA\in C a 22-cell m A:α Aβ Am_A : \alpha_A \Rightarrow \beta_A in DD that commutes suitably with the 22-cell components of α\alpha and β\beta (see Leinster for details).

If α,β\alpha,\beta are strict transformations, then the complicated-looking modification condition becomes simply a naturality square with globs m A:α Aβ Am_A : \alpha_A \Rightarrow \beta_A where the 11-cells α A\alpha_A would normally be.

Generalisation

When you get tired of thinking individually about nn-categories, functors, transformations, modifications, and so on, check out (n,k)-transformation.

(Some discussion from here has also been moved to there.)

References

Tom Leinster, Basic bicategories, arXiv.

Revised on February 25, 2013 15:31:41 by Urs Schreiber (89.204.139.45)