#
nLab

classical model structure on pointed topological spaces

### Context

#### Model category theory

**model category**

## Definitions

category with weak equivalences

weak factorization system

homotopy

small object argument

resolution

## Morphisms

Quillen adjunction

## Universal constructions

homotopy Kan extension

homotopy limit/homotopy colimit

Bousfield-Kan map

## Refinements

monoidal model category

enriched model category

simplicial model category

cofibrantly generated model category

algebraic model category

compactly generated model category

proper model category

cartesian closed model category, locally cartesian closed model category

stable model category

## Producing new model structures

on functor categories (global)

on overcategories

Bousfield localization

transferred model structure

Grothendieck construction for model categories

## Presentation of $(\infty,1)$-categories

(∞,1)-category

simplicial localization

(∞,1)-categorical hom-space

presentable (∞,1)-category

## Model structures

Cisinski model structure### for $\infty$-groupoids

for ∞-groupoids

on topological spaces

Thomason model structure

model structure on presheaves over a test category

on simplicial sets, on semi-simplicial sets

model structure on simplicial groupoids

on cubical sets

on strict ∞-groupoids, on groupoids

on chain complexes/model structure on cosimplicial abelian groups

related by the Dold-Kan correspondence

model structure on cosimplicial simplicial sets

### for $n$-groupoids

for n-groupoids/for n-types

for 1-groupoids

### for $\infty$-groups

model structure on simplicial groups

model structure on reduced simplicial sets

### for $\infty$-algebras

#### general

on monoids

on simplicial T-algebras, on homotopy T-algebras

on algebas over a monad

on algebras over an operad,

on modules over an algebra over an operad

#### specific

model structure on differential-graded commutative algebras

model structure on differential graded-commutative superalgebras

on dg-algebras over an operad

model structure on dg-modules

### for stable/spectrum objects

model structure on spectra

model structure on ring spectra

model structure on presheaves of spectra

### for $(\infty,1)$-categories

on categories with weak equivalences

Joyal model for quasi-categories

on sSet-categories

for complete Segal spaces

for Cartesian fibrations

### for stable $(\infty,1)$-categories

on dg-categories### for $(\infty,1)$-operads

on operads, for Segal operads

on algebras over an operad,

on modules over an algebra over an operad

on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations

### for $(n,r)$-categories

for (n,r)-categories as ∞-spaces

for weak ∞-categories as weak complicial sets

on cellular sets

on higher categories in general

on strict ∞-categories

### for $(\infty,1)$-sheaves / $\infty$-stacks

on homotopical presheaves

model structure for (2,1)-sheaves/for stacks

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# Contents

## Idea

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

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

- Mark Hovey, around corollary 2.4.20 of
*Model categories*

Last revised on April 14, 2016 at 13:33:20.
See the history of this page for a list of all contributions to it.