nLab classical model structure on pointed topological spaces

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.

Related concepts

• model structure on topological spaces

  • classical model structure on topological spaces

    • classical model structure on pointed topological spaces

  • Strøm model structure

  • model structure on Delta-generated topological spaces

References

Textbook accounts:

• Mark Hovey, Cor. 2.4.20 in: Model Categories, Mathematical Surveys and Monographs, Volume 63, AMS (1999) (ISBN:978-0-8218-4361-1, doi:10.1090/surv/063, pdf, Google books)