Beck modules for a monoid

Abstract:

Given an object *X* of a category *C*, a Beck module for *X* is an
abelian group object in the slice category *C/X*. For a ring *R*, it is a
2-sided *R*-module; for a group *G*, it is a right *G*-module; for a
commutative ring *R*, it is just an *R*-module; for a Lie algebra *L* it is
the usual notion of *L*-module. These depend on the ambient category of
course. But when a student in Paris named Leonard Guetta asked me
what was a Beck module for a monoid, I assumed it was an ordinary
module. But I was wrong. I will describe what it actually is.