related by the Dold-Kan correspondence
The structure of an algebraic model category is a refinement of that of a model category.
Where a bare model category structure is a category with weak equivalences refined by two weak factorization systems ( and ) in an algebraic model structure these are refined further to algebraic weak factorization systems plus a bit more.
This extra structure supplies more control over constructions in the model category. For instance its choice induces a weak factorization system also in every diagram category of the given model category.
An algebraic model structure on a homotopical category consists of a pair of algebraic weak factorization systems , together with a morphism of algebraic weak factorization systems
such that the underlying weak factorization systems form a model structure on with weak equivalences .
A morphism of algebraic weak factorization systems consists of a natural transformation
comparing the two functorial factorizations of a map that defines a colax comonad morphism and a lax monad morphism .
Every cofibrantly generated model category structure can be lifted to that of an algebraic model category.
Any algebraic model category has a fibrant replacement monad and a cofibrant replacement comonad . There is also a canonical distributive law comparing the two canonical bifibrant replacement functors.
The notion was introduced in
The algebraic analog of monoidal model categories is discussed in