#
nLab

pointed model 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 rational $\infty$-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

## Definition

A model category is **pointed** if its underlying category is a pointed category, i.e., if the unique morphism from the initial object to the terminal object is an isomorphism, in which case both of them are denoted by $0$ (the zero object).

In any pointed category, one has a canonical **zero morphism** between any objects $A$ and $B$, given by the composition $A\to 0\to B$.

The homotopy equalizer of $f\colon A\to B$ and $0\colon A\to B$ is known as the homotopy fiber of $f$.

The homotopy coequalizer of $f\colon A\to B$ and $0\colon A\to B$ is known as the homotopy cofiber of $f$.

In particuar there is the homotopy (co)-fiber of the zero object with itself, the loop space object- and reduced suspension-operation. Asking these operations to be equivalences in a suitable sense leads to the concept of *linear model categories* and *stable model categories*.

## Examples

Model categories which are pointed without being linear or even stable:

Pointed model categories which are stable:

## References

Last revised on February 10, 2021 at 07:54:50.
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