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]

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.