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Ilya Nikokoshev
Ilya Nikokoshev
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  • simplicial sets,
  • CW-complexes (Serre fibrations),
  • topological spaces (Hurewicz fibrations),
  • simplicial objects in other categories, like abelian groups, rings, etc,
  • diagrams of simplicial sets, especially,
  • cosimplicial simplicial sets, and
  • presheaves of simplicial sets,
  • quasi-categories (a category whose objects are a special type of simplicial set),
  • the category of small categories (see Thomason),
  • A^1A1-homotopy invariant Nisnevich presheaves with transfers. It is important to emphasize that these categories often carry several different model structures giving non-equivalent homotopy categories. Another point is that most of these are in some sense built out of simplicial sets. Grothendieck in letters (to Brown?) suggested the idea of a test category, categories which could play the same role as the category Delta. In addition these may come with extra structure. I believe this is the case for cubical objects and similar constructions.

It is important to emphasize that these categories often carry several different model structures giving non-equivalent homotopy categories. Another point is that most of these are in some sense built out of simplicial sets. Grothendieck in letters (to Brown?) suggested the idea of a test category, categories which could play the same role as the category Delta. In addition these may come with extra structure. I believe this is the case for cubical objects and similar constructions.

There are also two common operations to do on such model categories: localization and stabilization. The first produces a new model structure on a fixed model category, while the second produces a new category with a model structure.

The idea of localization is to allow us to make certain types of morphisms into isomorphisms in the homotopy category. For instance, in the category of simplicial sets, we may want to consider p-local equivalence. There is a model category structure on simplicial sets such that X->YX -> Y is an isomorphism in the homotopy category if and only if it induces an isomorphism on p-localized homotopy groups. Here, p-localized usually means p-adic. This was first systematically laid out in the book Homotopy limits, completions, and localizations by Bousfield and Kan. A modern account is in Hirschhorn's book Model categories and their localizations.

  • simplicial sets,
  • CW-complexes (Serre fibrations),
  • topological spaces (Hurewicz fibrations),
  • simplicial objects in other categories, like abelian groups, rings, etc,
  • diagrams of simplicial sets, especially,
  • cosimplicial simplicial sets, and
  • presheaves of simplicial sets,
  • quasi-categories (a category whose objects are a special type of simplicial set),
  • the category of small categories (see Thomason),
  • A^1-homotopy invariant Nisnevich presheaves with transfers. It is important to emphasize that these categories often carry several different model structures giving non-equivalent homotopy categories. Another point is that most of these are in some sense built out of simplicial sets. Grothendieck in letters (to Brown?) suggested the idea of a test category, categories which could play the same role as the category Delta. In addition these may come with extra structure. I believe this is the case for cubical objects and similar constructions.

There are also two common operations to do on such model categories: localization and stabilization. The first produces a new model structure on a fixed model category, while the second produces a new category with a model structure.

The idea of localization is to allow us to make certain types of morphisms into isomorphisms in the homotopy category. For instance, in the category of simplicial sets, we may want to consider p-local equivalence. There is a model category structure on simplicial sets such that X->Y is an isomorphism in the homotopy category if and only if it induces an isomorphism on p-localized homotopy groups. Here, p-localized usually means p-adic. This was first systematically laid out in the book Homotopy limits, completions, and localizations by Bousfield and Kan. A modern account is in Hirschhorn's book Model categories and their localizations.

  • simplicial sets,
  • CW-complexes (Serre fibrations),
  • topological spaces (Hurewicz fibrations),
  • simplicial objects in other categories, like abelian groups, rings, etc,
  • diagrams of simplicial sets, especially,
  • cosimplicial simplicial sets, and
  • presheaves of simplicial sets,
  • quasi-categories (a category whose objects are a special type of simplicial set),
  • the category of small categories (see Thomason),
  • A1-homotopy invariant Nisnevich presheaves with transfers.

It is important to emphasize that these categories often carry several different model structures giving non-equivalent homotopy categories. Another point is that most of these are in some sense built out of simplicial sets. Grothendieck in letters (to Brown?) suggested the idea of a test category, categories which could play the same role as the category Delta. In addition these may come with extra structure. I believe this is the case for cubical objects and similar constructions.

There are also two common operations to do on such model categories: localization and stabilization. The first produces a new model structure on a fixed model category, while the second produces a new category with a model structure.

The idea of localization is to allow us to make certain types of morphisms into isomorphisms in the homotopy category. For instance, in the category of simplicial sets, we may want to consider p-local equivalence. There is a model category structure on simplicial sets such that X -> Y is an isomorphism in the homotopy category if and only if it induces an isomorphism on p-localized homotopy groups. Here, p-localized usually means p-adic. This was first systematically laid out in the book Homotopy limits, completions, and localizations by Bousfield and Kan. A modern account is in Hirschhorn's book Model categories and their localizations.

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Benjamin Antieau
Benjamin Antieau
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Besides the comments above, it might be useful to have a list of categories that have natural model structures:

  • simplicial sets,
  • CW-complexes (Serre fibrations),
  • topological spaces (Hurewicz fibrations),
  • simplicial objects in other categories, like abelian groups, rings, etc,
  • diagrams of simplicial sets, especially,
  • cosimplicial simplicial sets, and
  • presheaves of simplicial sets,
  • quasi-categories (a category whose objects are a special type of simplicial set),
  • the category of small categories (see Thomason),
  • A^1-homotopy invariant Nisnevich presheaves with transfers. It is important to emphasize that these categories often carry several different model structures giving non-equivalent homotopy categories. Another point is that most of these are in some sense built out of simplicial sets. Grothendieck in letters (to Brown?) suggested the idea of a test category, categories which could play the same role as the category Delta. In addition these may come with extra structure. I believe this is the case for cubical objects and similar constructions.

There are also two common operations to do on such model categories: localization and stabilization. The first produces a new model structure on a fixed model category, while the second produces a new category with a model structure.

The idea of localization is to allow us to make certain types of morphisms into isomorphisms in the homotopy category. For instance, in the category of simplicial sets, we may want to consider p-local equivalence. There is a model category structure on simplicial sets such that X->Y is an isomorphism in the homotopy category if and only if it induces an isomorphism on p-localized homotopy groups. Here, p-localized usually means p-adic. This was first systematically laid out in the book Homotopy limits, completions, and localizations by Bousfield and Kan. A modern account is in Hirschhorn's book Model categories and their localizations.

As for stabilization, this is the process of making the loop space functor invertible. In all of the examples above there exist small homotopy limits, and we can thus form the loop space of any object X as the homotopy fiber product of two maps sending a point to the base-point of X. In this setting we can apply stabilization. For instance, using this on simplicial sets gives us the category of spectra (or, pre-spectra), and the fibrant objects in this category are the Omega-spectra. This category has the incredibly important feature that fiber and cofiber sequences coincide. See the paper Homotopy theory of Gamma-spaces, spectra, and bisimplicial sets by Bousfield and Friedlander.

The more modern versions of spectra are S-modules and symmetric spectra. For the latter, see the paper by Hovey, Shipley, and Smith. These new versions address the following problem. On the normal category of spectra, there is a natural commutative tensor product (the smash product) on the homotopy category of spectra. But, this does not 'lift' to a commutative tensor structure on spectra. The category of symmetric spectra solves this problem by rigidifying, taking into account the action of the symmetric groups on tensors.

Finally, let me note that you don't get anything for free by using model categories. Specifically, while it is often easy to understand what fibrant objects are, the process of fibrant replacement is very very difficult to understand. This is entirely analogous to the point of view of chain complexes: injective objects are easy to understand. But, the injective resolution of some given object is very difficult.