It’s common to think of monads as generalized algebraic theories; the most familiar examples, such as the monads on encoding groups, rings, and so forth, have this flavor. However, this intuition is really only appropriate for certain monads (e.g. finitary monads on , which are the same thing as Lawvere theories).
It’s also common to think of monads as generalized monoids; previously we discussed why this was a reasonable thing to do.
Today we’ll discuss a different intuition: monads are (loosely) categorifications of idempotents.