related by the Dold-Kan correspondence
For topological spaces, there are two natural candidates for the collection of weak 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.
weak equivalences are the weak homotopy equivalences;
This is a cofibrantly generated model category. The cofibrations are generated by the set of boundary inclusions for all 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.
A second 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 for any topological space .
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 -model structure, where 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.
From the definitions, Hurewicz fibrations are necessarily Serre fibrations. It is well-known that homotopy equivalences are weak homotopy equivalences. If we write for the classes of the first model structure and for the classes of the second, we have and .
In general given two model structures with these inclusions, we get a third mixed model structure where the cofibrations are determined by the other two classes.
On topological spaces, this model structure has
weak equivalences the weak homotopy equivalences
fibrations the Hurewicz fibrations;
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].]
There is a model category structure on in which a morphism is a cofibration, fibration or weak equivalence, respectively, precisely if it is so under the inclusion . And this inclusion is the left adjoint in a Quillen equivalence
This appears for instance as (Hovey, theorem 2.4.23)
There is a model category structure on in which a morphism is a cofibration, fibration or weak equivalence, respectively, precisely if it is so under the inclusion . And this inclusion is the right adjoint in a Quillen equivalence
This appears as (Hovey, prop. 4.2.11).
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 -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.
The identity functor constitutes a Quillen adjunction
The original “Quillen” or “q-” model structure is due to
An expository, concise and comprehensive writeup is in
Standard textbooks references include
Mark Hovey Model categories
Hirschhorn Model categories and their localizations.
For the “Hurewicz,” “Strøm,” or “h-” model structure:
For the “mixed” or “m-” model structure:
The generalization to the model structure on topological operads is due to