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 → G {\displaystyle {\boldsymbol {\alpha ,\,\beta }}:{\boldsymbol {\mathbf {F} }}\rightarrow {\boldsymbol {\mathbf {G} }}}, there exists a modification μ : α → β {\displaystyle {\boldsymbol {\mathbf {\mu } }}:{\boldsymbol {\mathbf {\alpha } }}\rightarrow {\boldsymbol {\mathbf {\beta } }}} such that:

  • μ a : α a → β a {\textstyle {\boldsymbol {\mathbf {\mu _{a}} }}:{\boldsymbol {\mathbf {\alpha _{a}} }}\rightarrow {\boldsymbol {\mathbf {\beta _{a}} }}},
  • μ b : α b → β b {\textstyle {\boldsymbol {\mathbf {\mu _{b}} }}:{\boldsymbol {\mathbf {\alpha _{b}} }}\rightarrow {\boldsymbol {\mathbf {\beta _{b}} }}}, and
  • μ f : α f → β f {\textstyle {\boldsymbol {\mathbf {\mu _{f}} }}:{\boldsymbol {\mathbf {\alpha _{f}} }}\rightarrow {\boldsymbol {\mathbf {\beta _{f}} }}}.

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

An example of a modification in category theory.
An example of a modification in category theory.
  • Kelly, G. M.; Street, Ross (1974). "Review of the elements of 2-categories". In Kelly, Gregory M. (ed.). Category Seminar: Proceedings of the Sydney Category Theory Seminar, 1972/1973. Lecture Notes in Mathematics. Vol. 420. Springer. pp. 75–103. doi:. ISBN 978-3-540-06966-9. MR .