nLab modification

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

Given

• $\mathcal{X}, \mathcal{Y}$ a pair of 2-categories,

• $F,G \colon \mathcal{X} \longrightarrow \mathcal{Y}$ a pair of parallel 2-functors between these,

• a pair of parallel lax/pseudonatural transformations between those,

then a modification from $\eta$ to $\eta'$ is a function from objects of $\mathcal{X}$ to 2-morphisms of $\mathcal{Y}$ of the form

which satisfies the following equations for all 1-morphisms $x \xrightarrow{f} y$ in $\mathcal{X}$:

$$=$$

Related concepts

• 2-functor 2-category

• (n,k)-transformation

References

• John W. Gray, section I.4 of: Formal category theory: Adjointness for 2-categories, Lecture Notes in Mathematics 391, Springer (1974) [doi:10.1007/BFb0061280]

For review see most texts on 2-categories, such as:

• 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]