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 ${\boldsymbol {\alpha ,\,\beta }}:{\boldsymbol {\mathbf {F} }}\rightarrow {\boldsymbol {\mathbf {G} }}$ , there exists a modification ${\boldsymbol {\mathbf {\mu } }}:{\boldsymbol {\mathbf {\alpha } }}\rightarrow {\boldsymbol {\mathbf {\beta } }}$ 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. source ↗

References

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.
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.

[:0-1] 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.

1

2