nLab model structure on Delta-generated topological spaces

Contents

Context

Model category theory

model category, model \infty -category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general \infty-algebras

specific \infty-algebras

for stable/spectrum objects

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

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

for (,1)(\infty,1)-operads

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

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

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

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 QuCdfflgDTop Qu 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 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

Last revised on September 30, 2021 at 03:21:01. See the history of this page for a list of all contributions to it.