# The formal theory of monads on $\bf PDer$

I have two questions on the 2-category of prederivators $\bf PDer$:

1. Does $\bf PDer$ "admit the construction of algebras" (ATCOA) in the sense of Street's Formal theory of monads? I recall that by ATCOA for a 2-category $\cal K$ I mean that the inclusion functor ${\cal K} \to {\bf Mnd}({\cal K})$ has a right 2-adjoint. One can show by hand (see below) that there are well-behaved objects of algebras for a monad $T : \mathbb D \to \mathbb D$, but I wouldn't be able to guess if something goes wrong when trying to glue these objects to a 2-adjoint ${\bf Mnd}({\bf PDer})\to \bf PDer$...

2. Regarding a monad as a lax functor $1\to \cal K$ its "object of EM-algebras" is its lax limit (dually, the Kleisli object is its lax colimit). Is it known that given a monad $T$ on a prederivator $\mathbb D$ there are adjunctions $$\text{Alg}_T(\mathbb D) \leftrightarrows \mathbb D \quad\text{and}\quad \text{Kl}_T(\mathbb D) \leftrightarrows \mathbb D$$ obtained gluing the various categories of algebras and free algebras $\text{Alg}_{T_J}(\mathbb D(J))$ and $\text{Kl}_{T_J}(\mathbb D(J))$ on the monads $T_J : \mathbb D(J) \to \mathbb D(J)$ induced by the components of $T$, in order to form the prederivator of EM-algebras and of free algebras.

Are these two prederivators still characterized by the universal property of the lax co/limit of $T : 1\to \bf PDer$?

The 2-category of prederivators is just the functor 2-category $[\mathrm{Dia}^{op},\mathrm{Cat}]_p$ with pseudonatural transformations as morphisms. As long as the domain 2-category $\mathrm{Dia}$ is small with respect to the codomain $\mathrm{Cat}$, this is of the form $T$-$\mathrm{Alg}_p$, i.e. the 2-category of strict $T$-algebras and pseudo $T$-morphisms, for a suitable 2-monad on the 2-category $\mathrm{Cat}^{\mathrm{ob}(\mathrm{Dia})}$. Blackwell-Kelly-Power "Two-dimensional monad theory" showed that for a good 2-monad $T$, the 2-category $T$-$\mathrm{Alg}_p$ has pie-limits lifted from the underlying 2-category, which include lax limits such as EM-objects, and also has bicolimits by a more involved construction, which include lax bicolimits such as "weak Kleisli objects".
Edit: Alexander Campbell points out that actually strict Kleisli objects exist as well, since $[\mathrm{Dia}^{op},\mathrm{Cat}]_p$ is also 2-comonadic.
• One also has that for a small 2-category $A$, $[A,\textbf{Cat}]_p^{\text{op}}$ is of the form $S\text{-Alg}_p$ for a cocontinuous 2-monad $S$ on $[\text{ob}A,\textbf{Cat}]^\text{op}$. (The corresponding 2-comonad on $[\text{ob}A,\textbf{Cat}]$ is given by right Kan extension followed by restriction along $\text{ob}A \to A$.) So by the same two-dimensional monad theory result, "strict" Kleisli objects exist in $[A,\textbf{Cat}]_p$ and are calculated objectwise. – Alexander Campbell Jun 9 '17 at 2:13