# nLab semimodel category

Contents

### Context

#### Model category theory

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for $\infty$-groupoids

for ∞-groupoids

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

# Contents

## Idea

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.

## Definition

(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.

## Examples

###### Example

(semimodel structure on semisimplicial sets) There exists a right semi-model structure on the category of semi-simplicial sets (Rooduijn 2018).

## References

The definition is due to:

The example of the semimodel structure on semisimplicial sets:

• Jan Rooduijn, A right semimodel structure on semisimplicial sets, Amsterdam 2018 (pdf, mol:4787)

Last revised on June 26, 2021 at 12:00:52. See the history of this page for a list of all contributions to it.