Sections, retractions, and idempotent operations

A section is a function for which another function exists that can bring you back to the original argument after having applied the section. This other function is called its retraction. For example, the function that maps a month to the first day of the month expressed as an integer between 1 and 356 is a section. Its retraction is a function that maps each day of the year to its corresponding month.

Suppose that, instead of starting with a month, we start with some day between 1 and 356, apply the retraction, and then the section. We will not, in general, get the same day. But the day won’t change if we do the same thing again. “Applying a section after its retraction” is an idempotent operation.

These slides from this course contain a good explanation of right and left inverses of functions. Sections are right inverses, retractions are left inverses.

The notions of section and retraction go beyond functions and sets. This is a good series of posts explaining the concepts in the more general setting of Category Theory.