nLab
model structure on algebras over a monad

Idea

For $\mathcal{C}$ a cofibrantly generated model category and $T \colon \mathcal{C} \longrightarrow \mathcal{C}$ a monad on $\mathcal{C}$, there is under mild conditions a natural model category structure on the category of algebras over a monad over $T$.

Definition

Let $\mathcal{C}$ be a cofibrantly generated model category and $T \colon \mathcal{C} \longrightarrow \mathcal{C}$ a monad on $\mathcal{C}$.

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

$$(F \dashv U) \;\colon\; Alg_T(\mathcal{C}) \stackrel{\overset{F}{\longleftarrow}}{\underset{U}{\longrightarrow}} \mathcal{C} \,.$$

See (Schwede-Shipley 00, 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 coalgebras over a comonad

• 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)