nLab premodel category

Idea

The notion of a premodel category is a relaxation of the notion of a model category. Combinatorial premodel categories form a 2-category that has all (small) limits and colimits and has representing objects for Quillen bifunctors.

The 2-category of combinatorial $V$-enriched premodel categories is the category of modules over a monoid $V$ in this 2-category. It inherits the same set of properties and additionally admits a model 2-category structure. In this model structure, a left Quillen functor is a weak equivalence if and only if it is a Quillen equivalence.

Definition

A premodel category is a bicomplete category equipped with a pair of weak factorization systems $(C,AF)$ and $(AC,F)$ such that $AC\subset C$ (equivalently, $AF\subset F$).

The members of AC are called anodyne cofibrations and the members of AF are called anodyne fibrations (as in anodyne morphism).

Criteria for model categories

Model categories can be singled out among premodel categories by imposing the additional requirement that the class

$$W \coloneqq AF \circ AC \,,$$

obtained by composing elements of $AC$ with those of $AF$, is closed under the 2-out-of-3 property. To check this condition, it suffices to check certain specialized cases of 2-out-of-3 (see Cavallo & Sattler 2022, Theorem 3.8):

Proposition

In a premodel category, $W$ satisfies the 2-out-of-3 property if and only it the following conditions are satisfied:

1. If $g,f$ are cofibrations such that $g f$ and $g$ are anodyne cofibrations, then $f$ is an anodyne cofibration. If $g,f$ are fibrations such that $g f$ and $f$ are anodyne fibrations, then $g$ is an anodyne fibration.

2. Any (cofibration, anodyne fibration) factorization or (anodyne cofibration, fibration) factorization of a map in $W$ is an (anodyne cofibration, anodyne fibration) factorization.

3. Any composite of an anodyne fibration followed by an anodyne cofibration is in $W$.

Anodyne and trivial (co)fibrations

The notion of premodel category doesn’t come with a good general notion of weak equivalence. But if a particular premodel category has a good notion of weak equivalence, such as one of Barton‘s relaxed premodel categories, one needs to distinguish between two types of cofibrations (and analogously between two types of fibrations):

• An anodyne cofibration is a member of the class AC
• A trivial cofibration is a cofibration that is also a weak equivalence

In principle one must also distinguish a third class of cofibrations that have the left lifting property with respect to fibrations between fibrant objects. However, in a relaxed premodel category, these are trivial cofibrations. (Barton, Prop 3.5.2)

Cylindrical premodel categories

A cylindrical premodel category is one equipped with well-behaved cylinder and cocylinder functors. In the following, we assume that $\mathcal{E}$ is a finitely complete and finitely cocomplete category.

Definition

An adjoint functorial cylinder on $\mathcal{E}$ is a cylinder functor with a right adjoint: a pair of adjoint functors $\mathcal{E} : C \dashv P : \mathcal{E}$ together with natural transformations

Notation

Given a cylinder functor $C \colon \mathcal{E} \to \mathcal{E}$, write $\partial \colon Id_{\mathcal{E}} + \Id_{\mathcal{E}} \to C$ for the copairing $\partial = [\delta_0,\delta_1]$.

Definition

Let $C \dashv P$ be an adjoint functorial cylinder on $\mathcal{E}$. A weak factorization system $(L,R)$ on $\mathcal{E}$ is cylindrical if $L$ is closed under the pushout application $\widehat{ev}(\partial,-) \colon \mathcal{E}^{\mathbf{2}} \to \mathcal{E}^{\mathbf{2}}$.

Definition

Let $C \dashv P$ be an adjoint functorial cylinder on $\mathcal{E}$. A premodel category (C,AF) and (AC,F) is cylindrical when (C,AF) and (AC,F) are cylindrical and the pushout applications $\widehat{ev}(\delta_0,-), \widehat{ev}(\delta_1,-) \colon \mathcal{E}^{\mathbf{2}} \to \mathcal{E}^{\mathbf{2}}$ send cofibrations to anodyne cofibrations.

Note

This notion is distinct from Richard Williamson's cylindrical model structures.

In a cylindrical premodel structure, the first condition of Proposition implies the second (Cavallo & Sattler 2021, Lemma 3.22), yielding a simplified recognition theorem:

Proposition

In a cylindrical premodel category, $W$ satisfies the 2-out-of-3 property if and only if the following conditions are satisfied:

1. If $g,f$ are cofibrations such that $g f$ and $g$ are anodyne cofibrations, then $f$ is an anodyne cofibration. If $g,f$ are fibrations such that $g f$ and $f$ are anodyne fibrations, then $g$ is an anodyne fibration.

2. Any composite of an anodyne fibration followed by an anodyne cofibration is in $W$.

Related concepts

• model category

• weak model category

References

• Reid William Barton, A model 2-category of enriched combinatorial premodel categories. Doctoral dissertation. (arXiv:2004.12937)

• Evan Cavallo and Christian Sattler, Relative elegance and cartesian cubes with one connection, 2022. (arXiv:2211.14801)