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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 $1$-cells indexed by $0$-cells, a modification between transformations is an indexed collection of 2-cells.

Definitions

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.

Generalisation

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.)

Related concepts

• 2-functor 2-category

References

• Niles Johnson, Donald Yau, Section 4.4 of: 2-Dimensional Categories , Oxford University Press 2021 ( [[arXiv:2002.06055](http://arxiv.org/abs/2002.06055),arXiv:2002.06055, doi:10.1093/oso/9780198871378.001.0001 ) ]

• Tom Leinster, Basic bicategories , ( [[arXiv:math/9810017](http://arxiv.org/abs/math/9810017)]arXiv:math/9810017)