Definitions
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 $1$-cells indexed by $0$-cells, a modification between transformations is an indexed collection of 2-cells.
To be most general, start with lax transformations $\alpha, \beta : F \dot{\to} G : C \to D$. Given those, a modification $m : \alpha \ddot{\to} \beta$ assigns to each $0$-cell $A\in C$ a $2$-cell $m_A : \alpha_A \Rightarrow \beta_A$ in $D$ that commutes suitably with the $2$-cell components of $\alpha$ and $\beta$ (see Leinster for details).
If $\alpha,\beta$ are strict transformations, then the complicated-looking modification condition becomes simply a naturality square with globs $m_A : \alpha_A \Rightarrow \beta_A$ where the $1$-cells $\alpha_A$ would normally be.
When you get tired of thinking individually about $n$-categories, functors, transformations, modifications, and so on, check out (n,k)-transformation.
(Some discussion from here has also been moved to there.)
Niles Johnson, Donald Yau, Section 4.4 of: 2-Dimensional Categories, Oxford University Press 2021 [arXiv:2002.06055, doi:10.1093/oso/9780198871378.001.0001]
Tom Leinster, Basic bicategories [arXiv:math/9810017]
Last revised on September 26, 2023 at 16:33:15. See the history of this page for a list of all contributions to it.