nLab
model structure on topological spaces

Many Model Structures

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: the weak homotopy equivalences 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 nice categories of spaces. Using a nice category instead is sometimes important, such as if we want the model structure to be monoidal.

Quillen Model Structure

The first, and most prevalent, model structure has $W$ the class of weak homotopy equivalences. The fibrations $F$ 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 $D^n\times \{0\}$.

The cofibrations have a nice description as well as “retracts of relative cell complexes.” In fact, this situation is quite general. 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 or $q$-model structure on Top.

This model structure is Quillen equivalent to the standard model structure on simplicial sets via the total singular complex and geometric realization functors.

$${\mathrm{Top}}_{\mathrm{Quillen}}\stackrel{\stackrel{\mid -\mid }{\leftarrow }}{\underset{\mathrm{Sing}}{\to }}{\mathrm{sSet}}_{\mathrm{Quillen}}.$$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 $\mathrm{\infty }$-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.

Hurewicz (or Strøm) Model Structure

A second model structure has $W$ the class of homotopy equivalences and $F$ the class of 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 $A$. The 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 homotopy equivalences for weak equivalences and Hurewicz fibrations for fibrations. The cofibrant spaces (the m-cofibrant spaces) are precisely those spaces that are homotopy equivalent to CW complexes.

References

For the original “Quillen” or “q-” model structure:

• D. G. Quillen, Homotopical algebra, Lecture Notes in Mathematics 43, Springer-Verlag 1967, iv+156 pp.

• Any book on model categories, such as Hovey’s Model categories or Hirschhorn’s Model categories and their localizations.

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

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

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

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