nLab
model structure on algebras over a monad

Idea

For $C$ a monoidal model category and $T : C \to C$ a monad on $C$, there is under mild conditions a natural model category structure on the category of algebras over a monad over $T$.

Definition

Let $C$ be a cofibrantly generated model category and $T : C \to C$ a monad on $C$.

Then under mild conditions there exists the transferred model structure on the category of algebras over a monad, transferred along the free functor/forgetful functor adjunction

$$(F \dashv U) : Alg T \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} C \,.$$

See (SchwedeShipley, lemma 2.3).

Related concepts

• model structure on monoids in a monoidal model category

• model structure on algebras over an operad

• model structure on algebras over a monad

• model structure on simplicial T-algebras

References

• Stefan Schwede, Brooke Shipley, Algebras and modules in monoidal model categories Proc. London Math. Soc. (2000) 80(2): 491-511 (pdf)