#
nLab

semimodel category

Contents
### Context

#### Model category theory

**model category**

## 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

#### specific

### 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

## 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 08:00:52.
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