This extra structure supplies more control over constructions in the model category. For instance its choice induces a weak factorization system also in every diagramcategory of the given model category.

Definition

An algebraic model structure on a homotopical category$(M,W)$ consists of a pair of algebraic weak factorization systems $(C_t, F)$, $(C,F_t)$ together with a morphism of algebraic weak factorization systems

$(C_t,F) \to (C,F_t)$

such that the underlying weak factorization systems form a model structure on $M$ with weak equivalences $W$.

A morphism of algebraic weak factorization systems consists of a natural transformation

Any algebraic model category has a fibrant replacement monad $R$ and a cofibrant replacement comonad $Q$. There is also a canonical distributive law$RQ \to QR$ comparing the two canonical bifibrant replacement functors.

References

The notion was introduced in

Emily Riehl, Algebraic model structures, New York J. Math. 17 (2011) 173-231 (journa, arXiv)