In mathematics, specifically

category theory Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...

, a modification is an arrow between

natural transformation In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...

s. 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

\boldsymbol : \boldsymbol \rightarrow \boldsymbol

, there exists a modification

\boldsymbol : \boldsymbol \rightarrow \boldsymbol

such that: *

\boldsymbol : \boldsymbol \rightarrow \boldsymbol

, *

\boldsymbol : \boldsymbol \rightarrow \boldsymbol

, and *

\boldsymbol : \boldsymbol \rightarrow \boldsymbol

. The following commutative diagram shows an example of a modification and its inner workings. Modification in category theory

References

*{{cite conference , author1-link=Max Kelly , author2-link=Ross Street , last1 = Kelly , first1 = G. M. , last2 = Street , first2 = Ross , editor-last = Kelly , editor-first = Gregory M. , contribution = Review of the elements of 2-categories , doi = 10.1007/BFb0063101 , isbn = 978-3-540-06966-9 , mr = 357542 , pages = 75–103 , publisher = Springer , series = Lecture Notes in Mathematics , title = Category Seminar: Proceedings of the Sydney Category Theory Seminar, 1972/1973 , volume = 420 , year = 1974 Category theory Functors