I have a monad on an ind-category (specifically, my ind-category has a monoidal structure and I have an algebra object, so the monad is tensoring with it). It would be very useful in my work if the Eilenberg-Moore category of the monad it itself an ind-category. Is there a known criterion for the monad for this to be true? Further, is there a handy description for a generating sub-category of modules? I'm also happy to use the dual case of a comonad on a pro-category if there's a break in symmetry that makes that an easier question.



When is the Eilenberg-Moore category of a monad on an ind-category itself an ind-category?
Let $T$ be a monad on an [accessible category](https://ncatlab.org/nlab/show/accessible+category) (i.e. an $\mathbf{Ind}$-category). If the underlying endofunctor of $T$ is [finitary](https://ncatlab.org/nlab/show/finitary+monad) (i.e. preserves filtered colimits), then the Eilenberg–Moore category of $T$ is also accessible. This is proven (in slightly more generality) in Theorem 2.78 of Adámek–Rosický's [Locally presentable and accessible categories](https://www.cambridge.org/core/books/locally-presentable-and-accessible-categories/94CB48295B6AF097FC232313A57BDE17), for instance.


