model category

for ∞-groupoids

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.

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.

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)