nLab
model structure on compactly generated topological spaces

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Model category theory

model category

Definitions

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Universal constructions

Refinements

Producing new model structures

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

Model structures

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Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

The classical model structure on topological spaces Top QuTop_{Qu} restricts to compactly generated topological spaces and further to weak Hausdorff spaces among these, to yield Quillen equivalent model structures kTop Quk Top_{Qu} and hkTop Quh k Top_{Qu} there. Since these model structures on k-spaces are Cartesian monoidal closed as model categories (Prop. below) while Top QUTop_{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)hkTophkTopkTop 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 QuTop_{Qu} restricts along kTopTopk Top \xhookrightarrow{\;} Top (1) to a cofibrantly generated model category structure kTop Quk Top_{Qu} on compactly generated topological spaces, and the coreflection becomes a Quillen equivalence:

kTop QukTop 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 kTop Quk Top_{Qu} from Prop. restricts along hktopkToph k top \xhookrightarrow{\;} k Top (1) to a model category structure on weakly Hausdorff k-spaces hkTop Quh k Top_{Qu}, and the reflection is a Quillen equivalence:

hkTop Qu QuhkTop Qu 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 DTopkTopD Top \hookrightarrow k Top, to give a Quillen equivalent model structure on Delta-generated topological spaces:

Top Qu QukkTop Qu QuDDTop Qu. 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.