nLab model structure on compactly generated 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 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: