M Theory Lesson 7
So let's go back to the relation $T^2 = T$. The reason for the capital $T$ (besides our swanky new latex capabilities) is that rather than sources or targets for arrows in a category, we would now like to weaken the relation and talk about monads.
A monad is a functor $T: C \rightarrow C$ with natural transformations $\mu: TT \rightarrow T$ and $\eta: 1 \rightarrow T$. Think of these as multiplication and unit. They satisfy an associativity and unit law. The square that represents associativity may have its vertices labelled by signs --, -+, +- and ++ where the source -- is the composition TTT before bracketing. Such parity cubes appear naturally in higher categorical contexts.