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).^[1] The notion is due to Bénabou.^[2]

Given two natural transformations ${\displaystyle {\boldsymbol {\alpha ,\,\beta }}:{\boldsymbol {\mathbf {F} }}\rightarrow {\boldsymbol {\mathbf {G} }}}$ , there exists a modification ${\displaystyle {\boldsymbol {\mathbf {\mu } }}:{\boldsymbol {\mathbf {\alpha } }}\rightarrow {\boldsymbol {\mathbf {\beta } }}}$ such that:

• ${\textstyle {\boldsymbol {\mathbf {\mu _{a}} }}:{\boldsymbol {\mathbf {\alpha _{a}} }}\rightarrow {\boldsymbol {\mathbf {\beta _{a}} }}}$ ,
• ${\textstyle {\boldsymbol {\mathbf {\mu _{b}} }}:{\boldsymbol {\mathbf {\alpha _{b}} }}\rightarrow {\boldsymbol {\mathbf {\beta _{b}} }}}$ , and
• ${\textstyle {\boldsymbol {\mathbf {\mu _{f}} }}:{\boldsymbol {\mathbf {\alpha _{f}} }}\rightarrow {\boldsymbol {\mathbf {\beta _{f}} }}}$ .^[1]

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

An example of a modification in category theory.

References

1. Mac Lane, Saunders (2010). Categories for the working mathematician. Graduate texts in mathematics (2nd. ed., Softcover version of original hardcover edition 1998 ed.). New York, NY: Springer. p. 278. ISBN 978-1-4419-3123-8.
2. Kelly & Street 1974, § 1.4.

• 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:10.1007/BFb0063101. ISBN 978-3-540-06966-9. MR 0357542.