Contents

model category

for ∞-groupoids

# Contents

## Idea

The classical model structure on topological spaces $Top_{Qu}$ restricts to compactly generated topological spaces and further to weak Hausdorff spaces among these, to yield Quillen equivalent model structures $k Top_{Qu}$ and $h k Top_{Qu}$there. Since these model structures on k-spaces are Cartesian monoidal closed as model categories (Prop. below) while $Top_{QU}$ is not, they provide a more convenient foundations for much of homotopy theory in terms of model categories.

## Statement

Recall (from here) the sequence of adjoint functors

(1)$h k Top \underoverset {\underset{}{\hookrightarrow}} {\overset{ h }{\longleftarrow}} {\;\;\;\; \bot \;\;\;\; } k Top \underoverset {\underset{k}{\longleftarrow}} {\overset{}{\hookrightarrow}} {\;\;\;\; \bot \;\;\;\;} Top$

exhibiting the coreflective subcategory inside all of Top of compactly generated topological spaces and further the reflective subcategory of weak Hausdorff spaces among these.

### On k-Spaces

###### Proposition

(classical model structure on compactly generated topological spaces)
The classical model structure on topological spaces $Top_{Qu}$ restricts along $k Top \xhookrightarrow{\;} Top$ (1) to a cofibrantly generated model category structure $k Top_{Qu}$ on compactly generated topological spaces, and the coreflection becomes a Quillen equivalence:

$k Top \underoverset {\underset{k}{\longleftarrow}} {\overset{}{\hookrightarrow}} {\;\;\;\; \simeq_{\mathrlap{Qu}} \;\;\;\;} Top$

(e.g. Hovey 1999, Thm. 2.4.23) A proof is spelled out with this Thm. at classical model structure on topological spaces (which is this Thm. at Introduction to Homotopy Theory).

###### Proposition

The model structure on compactly generated topological spaces from Prop. is a cartesian monoidal model category;

(e.g. Hovey 1999, Prop. 4.2.11) A proof is given with this Prop. at classical model structure on topological spaces (which is this Prop. at Introduction to Homotopy Theory).

### On weakly Hausdorff k-Spaces

Similarly:

###### Proposition

(classical model structure on compactly generated weak Hausdorff spaces)
The model structure on compactly generated topological spaces $k Top_{Qu}$ from Prop. restricts along $h k top \xhookrightarrow{\;} k Top$ (1) to a model category structure on weakly Hausdorff k-spaces $h k Top_{Qu}$, and the reflection is a Quillen equivalence:

$h k Top_{Qu} \underoverset {\underset{}{\hookrightarrow}} {\overset{ h }{\longleftarrow}} {\;\;\;\; \simeq_{\mathrlap{Qu}} \;\;\;\; } k Top_{Qu}$

(e.g. Hovey 1999, Thm. 2.4.25)

Again, this is a Cartesian monoidal model category (e.g. Hovey 1999, Thm. 2.4.23).

### On Delta-generated spaces

In the same way as above (e.g. Gaucher 2007, p. 7, Haraguchi 2013):

the model structure on compactly generated topological spaces restricts further along the inclusion of Delta-generated topological spaces $D Top \hookrightarrow k Top$, to give a Quillen equivalent model structure on Delta-generated topological spaces:

$Top_{Qu} \underoverset { \underset{ k }{\longrightarrow} } { {\hookleftarrow} } { \;\;\;\;\;\;\simeq_{\mathrlap{Qu}}\;\;\;\;\;\; } k Top_{Qu} \underoverset { \underset{ D }{\longrightarrow} } { {\hookleftarrow} } { \;\;\;\;\;\;\simeq_{\mathrlap{Qu}}\;\;\;\;\;\; } D Top_{Qu} \,.$

## References

Textbook accounts:

Lecture notes:

See also:

• Philippe Gaucher, p. 7 of: Homotopical interpretation of globular complex by multipointed d-space, Theory and Applications of Categories, vol. 22, number 22, 588-621, 2009 (arXiv:0710.3553)

Last revised on September 30, 2021 at 03:19:08. See the history of this page for a list of all contributions to it.