Definition of strict 2-monad on Cat

Question

The category $\text{Cat}$ can be thought of as a $2$-category. I was hoping somebody could help by telling me the explicit definition of a strict 2-monad $(T, \eta, \mu)$ on $\text{Cat}.$ In particular, I would like to know how the 2-monad is supposed to work on the $0,1$ and $2$ cells of $\text{Cat},$ and which laws are supposed to be satisfied. It would be great if answers could be as explicit as possible.

Answer 1

I'm going to write $\mathfrak{Cat}$ for the $2$-category of categories and $\mathbf{Cat}$ for the $1$-category of categories. A strict $2$-monad on $\mathfrak{Cat}$ is essentially the same data as a $1$-monad on $\mathbf{Cat}$; the point is that the strictness makes all coherence transformations and identities collapse to the identity operations/transformations. Let's see this in practice!

A strict $2$-monad on $\mathfrak{Cat}$ is given by a triple $(T,\eta,\mu)$ where:

• $T:\mathfrak{Cat} \to \mathfrak{Cat}$ is a strict $2$-functor (so $T(g \circ f) = T(g) \circ T(f)$ for any composable pair of functors $\mathscr{C} \xrightarrow{f} \mathscr{D} \xrightarrow{g} \mathscr{E}$, $T(\beta \circ \alpha) = T(\beta) \circ T(\alpha)$ for vertical composition of natural transformations, and $T(\gamma \ast \alpha) = T(\gamma) \ast T(\alpha)$ for horizontal composition of natural transformations). Strict $2$-functors are essentially functors between $2$-categories which preserve all categorical information strictly on the nose.
• $\eta:\operatorname{id}_{\mathfrak{Cat}} \Rightarrow T$ is a strict pseudonatural transformation (so all the coherence conditions of pseudonatural transformations must be satisfied on the nose, i.e., each coherence transformation must be the identity).
• $\mu:T^{2} \Rightarrow T$ is also a strict pseudonatural transformation (again all the coherence transformations must be the identity).
• The $1$-monad laws described below must hold strictly in $\mathfrak{Cat}$, i.e., the diagrams corresponding to the identities written below must hold on the nose:
  • $\mu \circ (T \ast \eta) = \iota_{T} = \mu \circ (\eta \ast T)$, where $T \ast \eta$ and $\eta \ast T$ denote the right and left whiskering of $T$ by $\eta$, respectively, and $\iota_{T}$ is the identity transformation on $T$;
  • $\mu \circ (T \ast \mu) = \mu \circ (\mu \ast T)$, where $T \ast \mu$ and $\mu \ast T$ denote the right and left whiskering of $T$ by $\mu$, respectively;

In this way a strict $2$-monad on $\mathfrak{Cat}$ simply ports the usual $1$-monad identities and diagrams over and asks that they hold on the nose.