$\require{AMScd}$Let $\cal C$ be a 2-category. Every 2-monad $T$ on $\cal C$ induces a Kleisli 2-category where
- 0-cells are the objects of $\cal C$
- a 1-cell $X\looparrowright Y$ consists of a 1-cell $X\to TY$ in $\cal C$
- a 2-cell $\alpha : f\Rightarrow g$ is a 2-cell between the 1-cells $f,g : X\to TY$.
Composition of 1-cells is defined as in 1-dimensional Kleisli categories: $$ g\bullet f := \mu_Z\circ Tg\circ f $$ for $f :X\to TY, g : Y\to TZ$ (and identity of $X$ is the unit $\eta_X$ of the monad).
Vertical composition of 2-cells is obvious to define; horizontal composition takes into account how 1-cells compose, but it's equally rather easy and employs horizontal composition $\boxminus$ of 2-cells in $\cal C$: $$ \beta \boxminus_\text{Kleisli}\alpha := \mu_Z * (T\beta \boxminus \alpha) $$ for every pair of 2-cells $\alpha : u \Rightarrow v : A\looparrowright B$ and $ \beta : s\Rightarrow t : B\looparrowright C$.
Now,
I would like to know what is a monad in this 2-category, and if it is something already known under another name.
Such a monad amounts to
- a 0-cell $A$ in $Kl(T)$;
- an endo-1-cell $s : A\looparrowright A$, i.e. a 1-cell $s : A\to TA$ in $\cal C$
- a pair of 2-cells $\sigma : s\bullet s \Rightarrow s$ and $\nu : \eta_A\Rightarrow s$ such that associativity[1] and unit[2.1, 2.2] axioms hold: $$ \begin{CD} \mu_A\cdot Ts\cdot \mu_A\cdot Ts\cdot s @>\mu_A * T\sigma * s>> \mu_A \cdot Ts \cdot s\\ @V\mu_ATs * \sigma VV [1] @VV\sigma V \\ \mu_A \cdot Ts\cdot s @>>\sigma > s \end{CD} \qquad \begin{CD} \mu_A\cdot T\eta_A\cdot s @>\mu_A * T\nu * s >> \mu_A \cdot Ts\cdot s @<\mu_A Ts * \nu<< \mu_A \cdot Ts\cdot \eta_A\\ @| [2.1] @V\sigma VV [2.2] @|\\ s @= s @= s \end{CD} $$ I have no idea what such a thing could be.