Because of the above example regarding homology, the study of closed model categories is sometimes thought of as homotopical algebra.
In practice the distinction has not proven significant and most recent authors (e.g., Mark Hovey and Philip Hirschhorn) work with closed model categories and simply drop the adjective 'closed'.
The category of topological spaces, Top, admits a standard model category structure with the usual (Serre) fibrations and with weak equivalences as weak homotopy equivalences.
The cofibrations are not the usual notion found here, but rather the narrower class of maps that have the left lifting property with respect to the acyclic Serre fibrations.
Equivalently, they are the retracts of the relative cell complexes, as explained for example in Hovey's Model Categories.
The category of (nonnegatively graded) chain complexes of R-modules carries at least two model structures, which both feature prominently in homological algebra: or This explains why Ext-groups of R-modules can be computed by either resolving the source projectively or the target injectively.
In fact, there are always two candidates for distinct model structures: in one, the so-called projective model structure, fibrations and weak equivalences are those maps of functors which are fibrations and weak equivalences when evaluated at each object of C. Dually, the injective model structure is similar with cofibrations and weak equivalences instead.
If C is a model category, then so is the category Pro(C) of pro-objects in C. However, a model structure on Pro(C) can also be constructed by imposing a weaker set of axioms to C.[5] Every closed model category has a terminal object by completeness and an initial object by cocompleteness, since these objects are the limit and colimit, respectively, of the empty diagram.
Cofibrations can be characterized as the maps which have the left lifting property with respect to acyclic fibrations, and acyclic cofibrations are characterized as the maps which have the left lifting property with respect to fibrations.
Similarly, fibrations can be characterized as the maps which have the right lifting property with respect to acyclic cofibrations, and acyclic fibrations are characterized as the maps which have the right lifting property with respect to cofibrations.
A pair of adjoint functors between two model categories C and D is called a Quillen adjunction if F preserves cofibrations and acyclic cofibrations or, equivalently by the closed model axioms, such that G preserves fibrations and acyclic fibrations.
A typical example is the standard adjunction between simplicial sets and topological spaces: involving the geometric realization of a simplicial set and the singular chains in some topological space.
Therefore, simplicial sets are often used as models for topological spaces because of this equivalence of homotopy categories.