premodel category



Model category theory

model category



Universal constructions


Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for rational \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras



for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks



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 VV-enriched premodel categories is the category of modules over a monoid VV 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.


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

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

WAFAC, W \coloneqq AF \circ AC \,,

obtained by composing elements of ACAC with those of AFAF, 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)


Last revised on May 9, 2020 at 14:38:11. See the history of this page for a list of all contributions to it.