nLab model structure on Delta-generated topological spaces

Statement

Proposition

The category of Delta-generated topological spaces carries the structure of a cofibrantly generated model category with the same generating (acyclic) cofibrations as for the classical model structure on topological spaces and such that the coreflection into all topological spaces (this Prop.) is a Quillen equivalence with the classical model structure on topological spaces:

$$Top_{Qu} \underoverset {\underset{Cdfflg}{\longrightarrow}} {\hookleftarrow} {\;\;\;\;\;\simeq_{\mathrlap{Qu}}\;\;\;\;\;} D Top_{Qu}$$

(Haraguchi 13, Theorem 3.3)

By the same kind of argument (Gaucher 2007), this factors by a Quillen equivalence through the model structure on compactly generated topological spaces:

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

Related concepts

• model structure on topological spaces

  • classical model structure on topological spaces (on pointed spaces)

    • model structure on compactly-generated topological spaces

    • model structure on Delta-generated topological spaces

  • Strøm model structure

• model structure on diffeological spaces

References

• 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)

• Tadayuki Haraguchi, On model structure for coreflective subcategories of a model category, Math. J. Okayama Univ. 57 (2015), 79–84 (arXiv:1304.3622, doi:10.18926/mjou/53040, MR3289294, Zbl 1311.55027)

See also:

• Sterling Ebel, Chris Kapulkin, Thm. 4.9 in: Synthetic approach to the Quillen model structure on topological spaces [arXiv:2310.14235]