model category, model $\infty$-category
on chain complexes/model structure on cosimplicial abelian groups
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 ((cofibrations, acyclic fibrations) and (acyclic cofibrations, fibrations)) 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 $(M,W)$ consists of a pair of algebraic weak factorization systems $(C_t, F)$, $(C,F_t)$ together with a morphism of algebraic weak factorization systems
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
comparing the two functorial factorizations of a map $f$ that defines a colax comonad morphism $C_t \to C$ and a lax monad morphism $F_t \to F$.
Every cofibrantly generated model category structure can be lifted to that of an algebraic model category. It is not clear whether or not this is true for any accessible model category.
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.
Algebraic model structures: Quillen model structures, mainly on locally presentable categories, and their constituent categories with weak equivalences and weak factorization systems, that can be equipped with further algebraic structure and “freely generated” by small data.
structure | small-set-generated | small-category-generated | algebraicized |
---|---|---|---|
weak factorization system | combinatorial wfs | accessible wfs | algebraic wfs |
model category | combinatorial model category | accessible model category | algebraic model category |
construction method | small object argument | same as $\to$ | algebraic small object argument |
The notion was introduced in
The algebraic analog of monoidal model categories is discussed in