nLab
algebraic model category

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Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (∞,1)-categories

Model structures

for ∞-groupoids

for ∞-groupoids

for n-groupoids

for ∞-groups

for ∞-algebras

general

specific

for stable/spectrum objects

for (∞,1)-categories

for stable (∞,1)-categories

for (∞,1)-operads

for (n,r)-categories

for (∞,1)-sheaves / ∞-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,acyclicfibrations) and (acycliccofibrations,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)→(C,F t)(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

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 f that defines a colax comonad morphism C t→C and a lax monad morphism F t→F.

Properties

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 R and a cofibrant replacement comonad Q. There is also a canonical distributive law RQ→QR comparing the two canonical bifibrant replacement functors.

References

The notion was introduced in

The algebraic analog of monoidal model categories is discussed in

Revised on January 9, 2012 22:13:32 by Emily Riehl (74.104.37.214)
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