nLab
model category

Idea

A model category (sometimes called a Quillen model category or a closed model category) is a context for doing homotopy theory. Quillen developed the definition of a model category to formalize the similarities between homotopy theory and homological algebra: the key examples which motivated his definition were the category of topological spaces, the category of simplicial sets, and the category of chain complexes.

So, what is a model category? For starters, it is a category equipped with three classes of morphisms, each closed under composition: weak equivalences, fibrations and cofibrations:

• The weak equivalences play the role of ‘homotopy equivalences’ or something a bit more general. Already in the case of topological spaces, it is useful to say that two spaces have the same homotopy type if there is a map from one to the other that induces isomorphisms on homotopy groups for any choice of basepoint in the first space. These maps are more general than homotopy equivalences, so they are called ‘weak equivalences’.

• The fibrations play the role of ‘nice surjections’. For example, in the category of topological spaces with its usual model structure, a locally trivial fiber bundle is typically a fibration.

• The cofibrations play the role of ‘nice inclusions’. For example, in the category of topological spaces with its usual model structure, an NDR pair is typically a cofibration.

A bit more technically: we can define an (∞,1)-category starting from any category with weak equivalences. The idea is that this (∞,1)-category keeps track of objects in our original category, morphisms between objects, homotopies between maps, homotopies between homotopies, and so on, ad infinitum. However, the extra structure of a model category makes it easier to work with this (∞,1)-category. We can obtain this (∞,1)-category in various ways, including simplicial localization if we want to obtain a simplicially enriched category as a variant of (∞,1)-category. Alternatively, to obtain a quasicategory, given a model category $M$, the simplicial nerve $N_{\Delta }(M_{\mathrm{cf}})$ of the subcategory $M_{\mathrm{cf}}\subset M$ of cofibrant-fibrant objects is a quasicategory. We say this (∞,1)-category is presented (or modeled) by the model category, and that the objects of the model category are models for the objects of this $(\mathrm{\infty }\mathrm{,1})$-category. Not every (∞,1)-category is obtained in this way (otherways it would necessarily have homotopy limits and colimits).

In this sense model categories are ‘models for homotopy theory’ or ‘categories of models for homotopy theory’. (The latter sense was the one intended by Quillen, but the former is also a useful way to think.)

Recall that the idea of categories with weak equivalences is to work just with 1-morphisms instead of with $n$-morphisms for all $n$, but to carry around extra information to remember which 1-morphisms are really equivalences in the full (∞,1)-category, i.e. isomorphisms in the corresponding homotopy category.

In a model category the data of weak equivalences is accompanied by further auxiliary data that helps to compute the (∞,1)-categorical hom-space, the homotopy category and derived functors. See homotopy theory for more on that.

If the model category happens to be a combinatorial simplicial model category $A$ it presents the category $A^{\circ }$ in the form of a simplicially enriched category given by the full SSet-enriched subcategory on objects that are both fibrant and cofibrant.

Definition

A model structure on a category $K$ consists of three distinguished classes of morphisms - the cofibrations $C$, the fibrations $F$, and the weak equivalences $W$ - satisfying the following two properties.

(i) $W$ makes $K$ into a category with weak equivalences, meaning that it is closed under 2-out-of-3: given a composable pair of morphisms $f,g$, if two out of the three morphisms $f,g,gf$ are in $W$, so is the third.

(ii) $(C,F\cap W)$ and $(C\cap W,F)$ are two weak factorization systems on $K$.

A model category is a complete and cocomplete category $K$ with a model structure $(C,F,W)$. (Quillen’s original definition required only finite limits and colimits, which are enough for the basic constructions. Colimits of larger cardinality are sometimes required for the small object argument, however.)

One says that an object is cofibrant if the map $0\to X$ from the initial object is a cofibration, and fibrant if the map $X\to 1$ to the terminal object is a fibration. Often, the fibrant and cofibrant objects are the ones one is “really” interested in, but the category consisting only of these is not well-behaved (as a 1-category). The factorizations supply fibrant and cofibrant replacement functors which allow us to treat any object of the model category as a ‘model’ for its fibrant-cofibrant replacement.

Variants

There are several notions of category with weak equivalences with similar but less structure than a full model category.

• A category of fibrant objects has a notion of just weak equivalences and fibrations, none of cofibrations. As the name implies, all of its objects are fibrant; the canonical example is the subcategory of fibrant objects in a model category.

• A Waldhausen category dually has a notion of weak equivalences and cofibrations, and all of its objects are cofibrant.

There is also a slight variant of the full notion of model category by Thomason that is designed to make the global model structure on functors more naturally accessible: this is the notion of Thomason model category.

Notes

• Some authors, notably Mark Hovey, require that the factorizations given by (ii) are actually functorial. In practice, Quillen’s small object argument means that many model categories can be made to have functorial factorizations.

• As a consequence of the above definition, the classes $C,F,$ and $W$ are all closed under retracts and composition and contain the isomorphisms of $K$. This is least obvious in the case of $W$. In the presence of functorial factorizations, it is easy to show that closure under retracts follows from (i) and (ii); with a bit of cleverness, this can also be done without functoriality.

Examples

Classical model structures

The archetypical model structures are the

• Quillen model structure on topological spaces and the
• Quillen model structure on simplicial sets.

These model categories are Quillen equivalent and encapsulate much of “classical” homotopy theory. From a higher-categorical viewpoint, they can be regarded as models for ∞-groupoids (in terms of CW complexes or Kan complexes, respectively).

Other classical topological objects of study, such as spectra, equivariant spaces, and parametrized spaces?, also form model categories.

In addition, there are also

• model structures on chain complexes and similarly
• model structures on dg-algebras,

which encapsulate classical homological algebra, and are related to the model structure on simplicial sets by the Dold-Kan correspondence. In fact, Quillen’s original definition of model categories was motivated by the analogy between homotopy theory and homological algebra.

Categorical model structures

Of interest to category theorists is that many notions of higher categories come equipped with model structures, witnessing the fact that when retaining only invertible transfors between $n$-categories they should form an $(\mathrm{\infty }\mathrm{,1})$-category. Many of these are called

• folk model structures, including “categorical” model structures for

  • categories
  • (strict or weak) 2-categories
  • strict ω-categories, and
  • strict ω-groupoids.

Model categories have successfully been used to compare many different notions of (∞,1)-category. The following definitions of $(\mathrm{\infty }\mathrm{,1})$-category all form Quillen equivalent model categories:

• simplicially enriched categories
• quasi-categories (via the Joyal model structure on simplicial sets)
• Segal categories
• complete Segal spaces

Other “higher categorical structures” can also be expected to form model categories, such as the

• model structure on dendroidal sets

which generalizes the Joyal model structure from (∞,1)-categories to (∞,1)-operads.

There is also another class of model structures on categorical structures, often called Thomason model structures (not to be confused with the notion of “Thomason model category”). In the “categorical” or “folk” model structures, the weak equivalences are the categorical equivalences, but in the Thomason model structures, the weak equivalences are those that induce weak homotopy equivalences of nerves. Thomason model structures are known to exist on 1-categories and 2-categories, at least, and are generally Quillen equivalent to the Quillen model structures on topological spaces and simplicial sets (via the nerve construction).

Parametrized model structures

The parameterized version of the model structure on simplicial sets is a

• model structure on simplicial presheaves or
• model structure on simplicial sheaves or

which serves as a model for ∞-stack (∞,1)-toposes (for hypercomplete (∞,1)-toposes, more precisely).

Functor and localized model structures

Many model structures, including those for complete Segal spaces, simplicial presheaves, and diagram spectra, are constructed by starting with a model structure on a functor category, such as a

• projective model structure,
• injective model structure, or
• Reedy model structure,

and applying a general technique called Bousfield localization which forces a certain class of morphisms to become weak equivalences. It can also be thought of as forcing a certain class of objects to become fibrant.

References

An introductory survey of some key concepts is in the set of slides

• Peter May, Model Category Primer

There is an unpublished manuscript of Chris Reedy from around 1974 that’s been circulating as an increasingly faded photocopy. It’s been typed into LaTeX, and the author has given permission for it to be posted on the net:

• Chris Reedy, Homotopy theory of model categories (pdf)

Recent review:

• Paul Goerss, Kristen Schemmerhorn, Model Categories and Simplicial Methods (arXiv)

A nice first introduction:

• Dwyer, Spalinski, Homotopy theories and model categories (pdf)

Monographs:

• Philip S. Hirschhorn, Model Categories and Their Localizations (AMS)
• Mark Hovey, Model Categories (Google books)

See

• P. Hirschhorn, personal website: Mathematics

for errata and more.

For yet another introduction to model categories, with an eye towards their use as presentations of $(\mathrm{\infty }\mathrm{,1})$-categories see Appendix A.2 of

• Jacob Lurie, Higher Topos Theory