related by the Dold-Kan correspondence
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.
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 Top of topological spaces with its usual model structure on topological spaces, a locally trivial fiber bundle is typically a fibration.
The cofibrations play the role of ‘nice inclusions’. For example, in the category Top of topological spaces with its usual model structure on topological spaces, 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 , the simplicial nerve of the subcategory 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 -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 -morphisms for all , 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 it presents the category in the form of a simplicially enriched category given by the full SSet-enriched subcategory on objects that are both fibrant and cofibrant.
(i) makes into a category with weak equivalences, meaning that it is closed under 2-out-of-3: given a composable pair of morphisms , if two out of the three morphisms are in , so is the third.
(ii) and are two weak factorization systems on .
The morphisms in (the fibrations that are also weak equivalences) are called trivial fibrations or acyclic fibrations
The morphisms in (the cofibrations that are also weak equivalences) are called trivial cofibrations or acyclic cofibrations.
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.
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.
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.
There are several extra conditions that strengthen the notion of a model category:
A left/right proper model category is one where the weak equivalences are stable under pushforward along cofibrations / pullback along fibrations
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.
This is least obvious in the case of . In the presence of functorial factorizations, it is easy to show that closure under retracts follows from axioms (i) and (ii); with a bit of cleverness, this can also be done without functoriality.
It is clear that given a model category structure, any two of the three classes of special morphisms (cofibrations, fibrations, weak equivalences) determine the third:
given and , we have ;
given and , we have ;
given and , we find as the class of morphisms which factor into a morphism in followed by a morphism in .
But, in fact, already the cofibrations and the fibrant objects determine the model structure.
A model structure on a category is determined by its class of cofibrations and its class of fibrant objects.
This statement appears for instance as (Joyal, prop. E.1.10)
So let with be a model category.
By the above remark it is sufficient to show that the cofibrations and the fibrant objects determine the class of weak equivalences. Moreover, these are already determined by the weak equivalences between cofibrant objects, because for any morphism, functorial cofibrant replacement and with 2-out-of-3 implies that is a weak equivalence precisely if is.
is an isomorphism, namely a bijection of sets. The equivalence relation that defines may be taken to be given by left homotopy induced by cylinder objects, which in turn are obtained by factoring codiagonals into cofibrations followed by acyclic fibrations. So all this is determined already by the class of cofibrations, and hence weak equivalences are determined by the cofibrations and the fibrant objects.
its weak equivalences are those morphisms whose dual was a weak equivalence in , its fibrations are those morphisms that were cofibrations in and similarly for its cofibrations.
The archetypical model structures are the
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).
In addition, there are also
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.
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 -categories they should form an -category. Many of these are called
Model categories have successfully been used to compare many different notions of (∞,1)-category. The following definitions of -category all form Quillen equivalent model categories:
There are related model structures for enriched higher categories:
Other “higher categorical structures” can also be expected to form model categories, such as the
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 “canonical” 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).
The parameterized version of the model structure on simplicial sets is a
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
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.
An account is in
and appendix E of
An introductory survey of some key concepts is in the set of slides
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:
A nice first introduction:
for errata and more.