model category, model $\infty$-category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of $(\infty,1)$-categories
Model structures
for $\infty$-groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant $\infty$-groupoids
for rational $\infty$-groupoids
for rational equivariant $\infty$-groupoids
for $n$-groupoids
for $\infty$-groups
for $\infty$-algebras
general $\infty$-algebras
specific $\infty$-algebras
for stable/spectrum objects
for $(\infty,1)$-categories
for stable $(\infty,1)$-categories
for $(\infty,1)$-operads
for $(n,r)$-categories
for $(\infty,1)$-sheaves / $\infty$-stacks
Introduced by Hovey 1998, the notion of semimodel categories s a relaxation of that of model categories which allows for a largely similar theory.
The notion of a weak model category and premodel category relaxes the definition even further.
(See Hovey 98, Theorem 3.3.)
A left semimodel category is a relative category equipped with a class of cofibrations and fibrations such that weak equivalences are closed under retracts and the 2-out-of-3 property, cofibrations have a left lifting property with respect to trivial fibrations, trivial cofibrations with cofibrant source have a left lifting property with respect to fibrations, and morphisms with cofibrant source can be factored as a cofibration followed by a fibration, either one of which can be further made trivial.
A right semimodel category is defined by passing to the opposite category.
(semimodel structure on semisimplicial sets) There exists a right semi-model structure on the category of semi-simplicial sets (Rooduijn 2018).
The definition is due to:
The example of the semimodel structure on semisimplicial sets:
Last revised on June 26, 2021 at 12:00:52. See the history of this page for a list of all contributions to it.