Contents

model category

for ∞-groupoids

# Contents

## 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, $AF\subset F$).

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.

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

## 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)