#
nLab

classical model structure on pointed topological spaces

### Context

#### Model category theory

**model category**, model $\infty$-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

The *classical model category structure on pointed topological spaces* $Top^{\ast/}_{Quillen}$ is the model structure on pointed objects of the classical model structure on topological spaces $Top_{Quillen}$ under the point (a pointed model category).

Equipped with the smash product this is a monoidal model category.

## Properties

### Cofibrant generation

Recall that the generatic cofibrations of the classical model structure on topological spaces are

$I_{Top}
\coloneqq
\left\{
S^{n-1} \overset{\iota_n}{\longrightarrow} D^n
\right\}_{n \in \mathbb{N}}$

and the generating acylic cofibrations are

$J_{Top}
\coloneqq
\left\{
D^n \overset{(id,\delta_0)}{\longrightarrow} D^n \times I
\right\}_{n \in \mathbb{N}}
\,.$

Write

$(-)_+
\;\colon\;
Top \longrightarrow Top^{\ast/}$

for the operation of freely adjoining a basepoint.

###### Proposition

The coslice model structure $(Top_{Quillen})^{\ast/}$ is itself cofibrantly generated, with generating cofibrations

$I_{Top^{\ast/}}
=
\left\{
S^{n-1}_+ \overset{(\iota_n)_+}{\longrightarrow} D^n_+
\right\}$

and generating acyclic cofibrations

$J_{Top^{\ast/}}
=
\left\{
D^n_+ \overset{(id, \delta_0)_+}{\longrightarrow} (D^n \times I)_+
\right\}
\,.$

This is a special case of a general statement about cofibrant generation of coslice model structures, see this proposition.

## References

Textbook accounts: