nLab classical model structure on pointed topological spaces

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

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:

Last revised on July 20, 2021 at 10:34:56. See the history of this page for a list of all contributions to it.