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](http://arxiv.org/abs/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](http://arxiv.org/abs/1310.8279))

discusses it in terms of quasi-categories. 

Finally, as mentioned in the comments above

* Jacob Lurie, section 6.2 of _[Higher Algebra](http://ncatlab.org/nlab/show/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](http://ncatlab.org/nlab/show/%28infinity%2C1%29-monad#HomotopyCoherence)_.