for ∞-groupoids

# 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: