nLab algebraic model category

Contents

Context

Model category theory

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for $\infty$-groupoids

for ∞-groupoids

for equivariant $\infty$-groupoids

for rational $\infty$-groupoids

for rational equivariant $\infty$-groupoids

for $n$-groupoids

for $\infty$-groups

for $\infty$-algebras

general $\infty$-algebras

specific $\infty$-algebras

for stable/spectrum objects

for $(\infty,1)$-categories

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

for $(\infty,1)$-operads

for $(n,r)$-categories

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

Contents

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.

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.

structuresmall-set-generatedsmall-category-generatedalgebraicized
weak factorization systemcombinatorial wfsaccessible wfsalgebraic wfs
model categorycombinatorial model categoryaccessible model categoryalgebraic model category
construction methodsmall object argumentsame as $\to$algebraic small object argument

References

The notion was introduced in:

The algebraic analog of monoidal model categories is discussed in

Review: