# nLab pointed model 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

## 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 pair of 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:

###### Example

(model categories of pointed objects)
Given any model category, its model category of pointed objects is a pointed model category.

In the case of the classical model structure on topological spaces this is the classical model structure on pointed topological spaces.

Pointed model categories which are stable: