nLab
classical model structure on pointed topological spaces

Contents

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

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

Contents

Idea

The classical model category structure on pointed topological spaces Top Quillen */Top^{\ast/}_{Quillen} is the model structure on an undercategory of the classical model structure on topological spaces Top QuillenTop_{Quillen} under the point.

With the smash product this is a monoidal model category.

Properties

Cofibrant generation

Recall that the generatic cofibrations of the classical model structure on topological spaces are

I Top{S n1ι nD n} n I_{Top} \coloneqq \left\{ S^{n-1} \overset{\iota_n}{\longrightarrow} D^n \right\}_{n \in \mathbb{N}}

and the generating acylic cofibrations are

J Top{D n(id,δ 0)D n×I} n. J_{Top} \coloneqq \left\{ D^n \overset{(id,\delta_0)}{\longrightarrow} D^n \times I \right\}_{n \in \mathbb{N}} \,.

Write

() +:TopTop */ (-)_+ \;\colon\; Top \longrightarrow Top^{\ast/}

for the operation of freely adjoining a basepoint.

Proposition

The coslice model structure (Top Quillen) */(Top_{Quillen})^{\ast/} is itself cofibrantly generated, with generating cofibrations

I Top */={S + n1(ι n) +D + n} I_{Top^{\ast/}} = \left\{ S^{n-1}_+ \overset{(\iota_n)_+}{\longrightarrow} D^n_+ \right\}

and generating acyclic cofibrations

J Top */={D + n(id,δ 0) +(D n×I) +}. J_{Top^{\ast/}} = \left\{ D^n_+ \overset{(id, \delta_0)_+}{\longrightarrow} (D^n \times I)_+ \right\} \,.

This is a special case of a general statement about cofibrant generation of coslice model structures, see this proposition.

References

  • Mark Hovey, around corollary 2.4.20 of Model categories

Last revised on April 14, 2016 at 13:33:20. See the history of this page for a list of all contributions to it.