nLab algebraic model category



Model category theory

model category, model \infty -category



Universal constructions


Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general \infty-algebras

specific \infty-algebras

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks



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)(M,W) consists of a pair of algebraic weak factorization systems (C t,F)(C_t, F), (C,F t)(C,F_t) together with a morphism of algebraic weak factorization systems

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

such that the underlying weak factorization systems form a model structure on MM with weak equivalences WW.

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

domf C tf Cf Rf ξ f Qf Ff F tf codf \array{ & \text{dom} f & \\ {}^{C_{t}f}\swarrow & & \searrow {}^{{C}{f}} \\ Rf & \stackrel{\xi_f}{\to} & Qf \\ {}_{{F}{f}}\searrow & & \swarrow {}_{F_{t}f} \\ & \text{cod} f & }

comparing the two functorial factorizations of a map ff that defines a colax comonad morphism C tCC_t \to C and a lax monad morphism F tFF_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 RR and a cofibrant replacement comonad QQ. There is also a canonical distributive law RQQRRQ \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.

weak factorization systemcombinatorial wfsaccessible wfsalgebraic wfs
model categorycombinatorial model categoryaccessible model categoryalgebraic model category
construction methodsmall object argumentsame as \toalgebraic small object argument


The notion was introduced in:

The algebraic analog of monoidal model categories is discussed in


See also:

  • Patrick M. Schultz, Algebraic Weak Factorization Systems in Double Categories, PhD thesis, University of Orego (2014) [hdl:1794/18429, pdf]

  • Gabriel Bainbridge, Some Constructions of Algebraic Model Categories, PhD thesis, Ohio State University (2021) [pdf, pdf]

Last revised on May 10, 2023 at 07:56:21. See the history of this page for a list of all contributions to it.