nLab algebraic model category

Idea

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.

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

$$\array{ & \text{dom} f & \\ {}^{C_{t}f}\swarrow & & \searrow {}^{{C}{f}} \\ Rf & \stackrel{\xi_f}{\to} & Qf \\ {}_{{F}{f}}\searrow & & \swarrow {}_{F_{t}f} \\ & \text{cod} f & }$$

comparing the two functorial factorizations of a map $f$ that defines a colax comonad morphism $C_t \to C$ and a lax monad morphism $F_t \to F$.

Properties

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.

Related pages

• algebraic weak factorization system
• accessible model category

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

References

The notion was introduced in:

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

The algebraic analog of monoidal model categories is discussed in

• Emily Riehl, Monoidal algebraic model structures (arXiv:1109.2883)

Review:

• Emily Riehl, §12.9 in: Categorical Homotopy Theory, Cambridge University Press (2014) [doi:10.1017/CBO9781107261457, pdf]

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]