nLab weak model category

Idea

The concept of weak model categories is a relaxation of that of model categories, even weaker than the concept of semimodel categories, but such that it still allows for a rich theory largely analogous to that of actual model categories:

The weak analogue of the construction of the homotopy category of a model category still exists, as do notions of Quillen adjunction and Quillen equivalence.

Also, for example, an analogue of left or right Bousfield localization of model categories still makes sense for weak model categories; and, as a bonus in contrast to the usual case, it does not require the assumption of left or right properness.

Definition

A weak model category is a premodel category that satisfies the following two axioms:

1. Cylinder axiom: Every cofibration $A\to X$ from a cofibrant object to a fibrant object admits a relative strong cylinder object

   $$X\sqcup_A X\to I_A X\to X,$$

   where the left map is a cofibration and its first component $X\to I_A X$ is an acyclic cofibration.

2. Path object axiom: Every fibration $A\to X$ from a cofibrant object to a fibrant object admits a relative strong path object

   $$A\to P_X A\to A\times_X A,$$

   where the right map is a fibration and its first component $P_X A\to A$ is an acyclic fibration.

Properties

Relation to premodel categories

A premodel category can be upgraded to a weak model category as follows.

Theorem

If a premodel category admits a weak Quillen cylinder, then it is a weak model category.

Definition

A weak Quillen cylinder on a premodel category $C$ is a pair of left adjoint functors $I,D\colon C\to C$ together with the following commutative square of natural transformations of functors $C\to C$:

$$\begin{matrix} id_C\sqcup \id_C&\mathop{\longrightarrow}\limits^i&I\cr \downarrow\nabla&&\downarrow e\cr id_C&\mathop{\longrightarrow}\limits_j&D,\cr \end{matrix}$$

where $\nabla$ is the codiagonal, $i$ is a cofibration, $j$ is a trivial cofibration, and the first component of $i$ is a trivial cofibration.

Here a natural transformation $\lambda\colon F\to G$ of functors $C\to C$ is a cofibration if for any (trivial) cofibration $X\to Y$ the map $F(Y)\sqcup_{F(X)}G(X)\to G(Y)$ is a (trivial) cofibration. Likewise, $\lambda$ is a trivial cofibration if for any cofibration $X\to Y$ the above map is a trivial cofibration.

Reference: Henry 20, Section 6.

This is essentially a reformulation of Cisinski-Olschok theory?.

Relation to model categories

Model categories can be singled out from weak model categories

by adding the following properties:

1. Every fibrant object admits a strong path object and every cofibrant object admits a strong cylinder object.

2. All acyclic cofibrations are trivial cofibrations and all acyclic fibrations are trivial fibrations. (Trivial maps are given as data for a premodel category, whereas acyclic (co)fibrations are defined as (co)fibrations that satisfy a right (left) lifting property with respect to the class of cofibrations with cofibrant source (respectively fibrations with fibrant target.)

3. The two classes of weak equivalence corresponding to the left and right induced semimodel structures coincide.

Relation to combinatorial model categories

Every combinatorial weak model category can be connected to a combinatorial model category by a zigzag of Quillen equivalences.

Examples

Example

(weak model structure on semi-simplicial sets)
There is a weak model structure on semi-simplicial sets which is Quillen equivalen to that on simplicial sets. (Henry 18, Thm. 5.5.6).

Related concepts

• model category

• premodel category

References

• Simon Henry, Weak model categories in classical and constructive mathematics, Theory and Applications of Categories, Vol. 35, 2020, No. 24, pp 875-958. (arXiv:1807.02650, tac:35-24)

• Simon Henry, Combinatorial and accessible weak model categories (arXiv:2005.02360)