model category, model -category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of -categories
Model structures
for -groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant -groupoids
for rational -groupoids
for rational equivariant -groupoids
for -groupoids
for -groups
for -algebras
general -algebras
specific -algebras
for stable/spectrum objects
for -categories
for stable -categories
for -operads
for -categories
for -sheaves / -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 -enriched premodel categories is the category of modules over a monoid 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 and such that (equivalently, ).
Model categories can be singled out among premodel categories by imposing the additional requirement that the class
obtained by composing elements of with those of , 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).
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):
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 18:38:11. See the history of this page for a list of all contributions to it.