nLab Mnd

Idea

The 2-category $Mnd$ is the free-standing monad (sometimes called the ‘walking monad’).

A 2-functor $Mnd \to K$ is a monad in the 2-category $K$. These 2-functors form the 2-category of monads of $K$.

Definition

$Mnd$ is the 2-category composed of a single object $\ast$, such that $Hom(\ast, \ast)$ is the augmented simplex category.

It is a one object sub-2-category of Adj, the free-standing adjunction.

References

• John Baez, Lecture 8: The walking monad, the augmented simplex category and the simplex category, Category Theory Seminar – Fall 2018.