nLab model structure on topological spaces

Idea

Philosophically, model structures allow one to localize a category at a particular collection of weak equivalences, which one would like to formally invert.

For topological spaces, there are two natural candidates for the collection $W$ of weak equivalences:

1. the weak homotopy equivalences

2. and the homotopy equivalences.

Both of these have accompanying model structures. Interestingly, these two model structures can also be combined to form what’s known as the mixed model structure.

All of these model structures exist not only on the category of all topological spaces, but also on most convenient categories of topological spaces. Using a nice category instead is sometimes important, such as if we want the model structure to be monoidal.

Definition

Classical Quillen Model Structure

The first, and most prevalent, model category structure on Top, called the classical model structure on topological spaces (Quillen67, II.3) has

• weak equivalences are the weak homotopy equivalences;

• fibrations are the Serre fibrations, maps which have the right lifting property with respect to all inclusions of the form $i_0 : D^n \hookrightarrow D^n \times I$ that include the $n$-disk as $D^n \times \{0\}$.

• cofibrations are the “retracts of relative cell complexes”.

This is a cofibrantly generated model category. The cofibrations $C$ are generated by the set of boundary inclusions $S^{n-1} \hookrightarrow D^n$ for all $n \in \mathbb{N}$ in the sense that they are the smallest saturated class containing these morphisms. As a consequence, of Quillen’s small object argument, all cofibrations have the form described above, where a relative cell complex is a transfinite composite of pushouts of coproducts of these generating maps.

This model structure is sometimes called the Quillen model structure on topological spaces or $q$-model structure on Top.

Hurewicz (or Strøm) Model Structure

A second model structure – the Strøm model structure – has

• weak equivalences are the homotopy equivalences;

• fibrations are the Hurewicz fibrations, which are defined to be maps that have the right lifting property with respect to all inclusions $i_0 : A \hookrightarrow A \times I$ for any topological space $A$.

• cofibrations are determined by these classes and are called the closed Hurewicz cofibrations.

This model structure is sometimes called the Hurewicz model structure, since it uses Hurewicz fibrations and cofibrations, or also the $h$-model structure, where $h$ can stand for either “Hurewicz” or “homotopy equivalence.” However, it is also sometimes called the Strøm model structure, since it was first proven to exist by Arne Strøm.

Mixed Model Structure

From the definitions, Hurewicz fibrations are necessarily Serre fibrations. It is well-known that homotopy equivalences are weak homotopy equivalences. If we write $(C_1, F_1, W_1)$ for the classes of the first model structure and $(C_2, F_2, W_2)$ for the classes of the second, we have $W_2 \subset W_1$ and $F_2 \subset F_1$.

In general given two model structures with these inclusions, we get a third mixed model structure $(C_m, F_2, W_1)$ where the cofibrations $C_m$ are determined by the other two classes.

On topological spaces, this model structure has

• weak equivalences the weak homotopy equivalences

• fibrations the Hurewicz fibrations;

• cofibrant spaces (the m-cofibrant spaces) are precisely those spaces that are homotopy equivalent to CW complexes.

Properties

Restriction to convenient categories of topological spaces

For the discussion of the homotopy theory given by the model structure on topological spaces, it is necessary or at least useful to pass to convenient categories of topological spaces.

Definition

Write

• $k Top \hookrightarrow Top$ for the full subcategory of k-spaces;

• $CGTop \hookrightarrow Top$ for the full subcategory of compactly generated spaces.

Proposition

(model structure on compactly generated topological spaces)

There is a model category structure $kTop_{Qu}$ on $k Top$ in which a morphism is a cofibration, fibration or weak equivalence, respectively, precisely if it is so under the inclusion $k Top \hookrightarrow Top$ in the classical model structure on topological spaces. This inclusion is the left adjoint in a Quillen equivalence

$$k Top \underoverset {\underset{k}{\longleftarrow}} {\overset{}{\hookrightarrow}} {\;\;\;\; \simeq_{\mathrlap{Qu}} \;\;\;\;} Top$$

This appears for instance as (Hovey, theorem 2.4.23)

Proposition

There is a model category structure $CGTop_{Quillen}$ on $CGTop$ in which a morphism is a cofibration, fibration or weak equivalence, respectively, precisely if it is so under the inclusion $CGTop \hookrightarrow Top$. And this inclusion is the right adjoint in a Quillen equivalence

$$CGTop_{Qu} \underoverset {\underset{}{\hookrightarrow}} {\overset{ h }{\longleftarrow}} {\;\;\;\; \simeq_{\mathrlap{Qu}} \;\;\;\; } k Top_{Qu} \mathrlap{\,.}$$

(e.g. Hovey 1999, Thm. 2.4.25)

To appreciate the following, notice that $Top_{Qu}$ is not a monoidal model category, because Top itself is not (cartesian) closed.

Proposition

Both $k Top_{Qu}$ and $CGTop_{Qu}$ are symmetric monoidal model categories.

This appears as (Hovey, prop. 4.2.11).

Proposition

In fact $CGTop_{Quillen}$ is a cartesian closed model category.

(This is briefly mentioned in Berger-Moerdijk 2003, Sec. 3.3.2.)

Relation between $Top_{Quillen}$ and $sSet_{Quillen}$

The Quillen model structure $Top_{Qullen}$ is Quillen equivalent to the standard (Quillen) model structure on simplicial sets via the total singular complex and geometric realization functors.

$$(\vert-\vert \dashv Sing) \colon Top_{Quillen} \stackrel{\overset{|-|}{\leftarrow}}{\underset{Sing}{\to}} sSet_{Quillen} \,.$$

Since the standard model structure on simplicial sets is a presentation of the (∞,1)-category ∞Grpd of ∞-groupoids realized as Kan complexes, this identifies topological spaces with ∞-groupoids in an (∞,1)-categorical sense. Notably it says that every $\infty$-groupoid is, up to equivalence, the fundamental ∞-groupoid of some topological space.

This statement is called the homotopy hypothesis (which here is a theorem). See there for more details.

Relation between $Top_{Quillen}$ and $Top_{Strom}$

The identity functor constitutes a Quillen adjunction

$$(Id \dashv Id) : Top_{Strom} \stackrel{\leftarrow}{\to} Top_{Quillen}$$

between the Quillen model structure and the Strom model structure on $Top$. Here $Top_{Strom} \to Top_{Quillen}$ is the right Quillen functor.

Related concepts

• homotopical structure on C*-algebras

• model structure on topological spaces

  • classical model structure on topological spaces

    • classical model structure on pointed topological spaces

  • Strøm model structure

  • model structure on Delta-generated topological spaces

References

The original “Quillen” or “q-” model structure is due to

• Daniel Quillen, Ch. II, Sec. 3 of Homotopical Algebra, Lecture Notes in Mathematics 43, Springer 1967(doi:10.1007/BFb0097438)

An expository, concise and comprehensive writeup is in

• Philip Hirschhorn, The Quillen model category of topological spaces, 2015 (arXiv:1508.01942)

Standard textbooks references include

• Mark Hovey, Model Categories, Mathematical Surveys and Monographs, Volume 63, AMS (1999) (ISBN:978-0-8218-4361-1, doi:10.1090/surv/063, pdf, Google books)

• Philip Hirschhorn, Model Categories and Their Localizations, AMS Math. Survey and Monographs Vol 99 (2002) (ISBN:978-0-8218-4917-0, pdf toc, pdf)

• Kate Ponto, Peter May, section 17 of More concise algebraic topology (pdf)

For the “Hurewicz,” “h-” or “Strøm model structure”:

• Arne Strøm, The homotopy category is a homotopy category, Archiv der Mathematik 23 (1972) (pdf, pdf)

For the “m-” or “mixed model structure”:

• Michael Cole, Mixing model structures, Topology Appl. 153 no. 7 (2006) doi.

The generalization to the model structure on topological operads is due to

• Clemens Berger, Ieke Moerdijk, Axiomatic homotopy theory for operads, Comment. Math. Helv. Vol. 78 (2003), no. 4 (arXiv:math/0206094)