Maybe an update on the literature on homotopical refinements of monadicity

The article

• Kathryn Hess, A general framework for homotopic descent and codescent, (arXiv:1001.1556)

discusses homotopical monadicity in terms of simplicial model categories.

The article

• Emily Riehl, Dominic Verity, Homotopy coherent adjunctions and the formal theory of monads (arXiv:1310.8279)

discusses it in terms of quasi-categories. Finally, as mentioned in the comments above

• Jacob Lurie, section 6.2 of Higher Algebra

discusses it more abstractly in $\infty$-category theory.

Maybe as a caveat, in Hess's nice article the monads are ordinary (if maybe simplicially enriched) monads on the underlying categories, so that I suppose that there should be some extra discussion of "rectification", namely discussion of under which conditions this presents an $\infty$-monad with all its higher coherence data. See the comments on the nLab at infinity-Monad -- Properties -- Homotopy coherence.

Maybe an update on the literature on homotopical refinements of monadicity:

The article

• Kathryn Hess, A general framework for homotopic descent and codescent, (arXiv:1001.1556)

discusses homotopical monadicity in terms of simplicial model categories.

The article

• Emily Riehl, Dominic Verity, Homotopy coherent adjunctions and the formal theory of monads (arXiv:1310.8279)

discusses it in terms of quasi-categories. 

Finally, as mentioned in the comments above

• Jacob Lurie, section 6.2 of Higher Algebra

discusses it more abstractly in $\infty$-category theory.

Maybe as a caveat, in Hess's nice article the monads are ordinary (if maybe simplicially enriched) monads on the underlying categories, so that I suppose that there should be some extra discussion of "rectification", namely discussion of under which conditions this presents an $\infty$-monad with all its higher coherence data. See the comments on the nLab at infinity-Monad -- Properties -- Homotopy coherence.