Do the algebras for a $\infty$-monad form a stable $\infty$-category? I'm wondering if a monad $T$ on a stable $\infty$-category $\cal C$ has a stable $\infty$-category of algebras, provided $T$ preserves finite limits/colimits.
Is this true?
Edit: Is something similar true in the triangulated world?
 A: The statement about stable $\infty$-categories is true. Just as in ordinary categories the basic facts about limits and colimits in the category of algebras are that the forgetful functor $\mathrm{Alg}(T) \to \mathcal{C}$


*

*creates any limits that $\mathcal{C}$ admits, without any assumption on $T$, and

*creates colimits of any shape $X$ such that $\mathcal{C}$ admits $X$-shaped colimits and the underlying functor of $T$ preserves them.
These are Theorem 5.7 and Corollary 5.5 in Emily Riehl and Dominic Verity's paper Completeness results for quasi-categories of algebras, homotopy limits, and related general constructions.
If $\mathcal{C}$ is stable and the underlying functor of $T$ is exact, 1 and 2 tell you that a pushout square of algebras is a pushout square in $\mathcal{C}$ is a pullback square in $\mathcal{C}$ is a pullback square of algebras; so the $\infty$-category of algebras of $T$ is also stable.
EDIT: About triangulated categories I don't know, but take a look at Paul Balmer's Separability and triangulated categories, which seems to contain a similar result for something called separable monads.
