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Let $\mathbf{C}$ be a cofibrantly generated model category (assume for simplicity that all objects are fibrant) and $\mathbf{C}^{\mathrm{T}}$ the category of $\mathrm{T}$-algebras with the induced model structure (same weak equivalences and fibrations as in the underlying model category $\mathbf{C}$). By definition, the adjuction $\mathrm{T}:\mathbf{C}\rightleftharpoons\mathbf{C}^{\mathrm{T}}: \mathrm{U}$ is monadic. How about the homotopical version, i.e, $\mathbb{L}\mathrm{T}:Ho\mathbf{C}\rightleftharpoons Ho(\mathbf{C}^{\mathrm{T}}): \mathbb{R}\mathrm{U}$

is there any result about the "homotopical" monadicity theorem, which compares $Ho(\mathbf{C}^{\mathrm{T}})$ and $Ho(\mathbf{C})^{\mathbb{L}\mathrm{T}}$.

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    $\begingroup$ As Mike already said, $Ho(C^T)$ is almost never monadic over $Ho(C)$. However if you are willing to use a homotopy coherent version of a monad and an algebra, Jacob Lurie has an $\infty$-category version of the Barr-Beck/monadicity theorem in his book on higher algebra. $\endgroup$
    – Justin Noel
    Commented Feb 13, 2012 at 22:32

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$Ho(C^T)$ is almost never monadic over $Ho(C)$. The objects of $Ho(C^T)$ are $T$-algebras in $C$, which means in particular that their $T$-algebra structure commutes strictly, whereas the algebras for the induced monad on $Ho(C)$ will only have algebra structure commuting up to (non-specified, non-coherent) isomorphism.

For instance, if $T$ is the monad derived from an $E_\infty$-operad, then $T$-algebras are $E_\infty$-spaces, whereas $Ho(T)$-algebras are "$H_\infty$-spaces". These have an obstruction theory specifying when they can be $E_\infty$-ized.

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    $\begingroup$ @Mike: what you said is true if $T$ is the monad associated to an $E_\infty$ operad. In the $A_\infty$ case $Ho(T)$-algebras are homotopy associative H-spaces. Niles Johnson and I have worked out the obstruction theory in lifting a map of $Ho(T)$-algebras to a homotopy class of $T$-algebra maps (under appropriate restrictions) in a forthcoming paper. $\endgroup$
    – Justin Noel
    Commented Feb 13, 2012 at 22:22
  • $\begingroup$ Mike, Noel, Thank you for your answers! $\endgroup$
    – Ilias A.
    Commented Feb 14, 2012 at 9:25
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    $\begingroup$ Thanks, Justin! I've corrected the answer. I actually just heard Niles talk about your result (which I think is awesome by the way) at the JMM last month; I should have remembered it and mentioned it. $\endgroup$
    – Mike Shulman
    Commented Feb 15, 2012 at 5:15
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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

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.

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