Modification (mathematics)

In mathematics, specifically category theory, a modification is an arrow between natural transformations. It is a 3-cell in the 3-category of 2-cells (where the 2-cells are natural transformations, the 1-cells are functors, and the 0-cells are categories). The notion is due to Bénabou.

Given two natural transformations $𝜶,𝜷:𝐅\to 𝐆$, there exists a modification $𝝁:𝜶\to 𝜷$ such that:

• ${𝝁}_{𝐚}:{𝜶}_{𝐚}\to {𝜷}_{𝐚}$,
• ${𝝁}_{𝐛}:{𝜶}_{𝐛}\to {𝜷}_{𝐛}$, and
• ${𝝁}_{𝐟}:{𝜶}_{𝐟}\to {𝜷}_{𝐟}$.

The following commutative diagram shows an example of a modification and its inner workings.

An example of a modification in category theory.