# Monadicity theorem in homotopy theory.

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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Are you aware of the paper "Algebras and Modules in Monoidal Model Categories" by Schwede and Shipley? They use the language of monads there, but call them "triples." I think Lemma 2.3 and Remark 4.5 might be of use to you. Here is a link: homepages.math.uic.edu/~bshipley/monoidal.pdf –  David White Feb 13 '12 at 17:14
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. –  Justin Noel Feb 13 '12 at 22:32
$Ho(C^T)$ 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.
@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. –  Justin Noel Feb 13 '12 at 22:22