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.