Model Category - Examples

Examples

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. This structure is not unique; in general there can be many model category structures on a given category. For the category of topological spaces, another such structure is given by Hurewicz fibrations and standard cofibrations, and the weak equivalences are the (strong) homotopy equivalences.

The category of (nonnegatively graded) chain complexes of R-modules carries at least two model structures, which both feature prominently in homological algebra:

• weak equivalences are maps that induce isomorphisms in homology;
• cofibrations are maps that are monomorphisms in each degree with projective cokernel; and
• fibrations are maps that are epimorphisms in each nonzero degree

or

• weak equivalences are maps that induce isomorphisms in homology;
• fibrations are maps that are epimorphisms in each degree with injective kernel; and
• cofibrations are maps that are monomorphisms in each nonzero degree.

This explains why Ext-groups of R-modules can be computed by either resolving the source projectively or the target injectively. These are cofibrant or fibrant replacements in the respective model structures.

The category of arbitrary chain-complexes of R-modules has a model structure that is defined by

• weak equivalences are chain homotopy equivalences of chain-complexes;
• cofibrations are monomorphisms that are split as morphisms of underlying R-modules; and
• fibrations are epimorphisms that are split as morphisms of underlying R-modules.

Other examples of categories admitting model structures include the category of all small categories, the category of simplicial sets or simplicial presheaves on any small Grothendieck site, the category of topological spectra, and the categories of simplicial spectra or presheaves of simplicial spectra on a small Grothendieck site.

Simplicial objects in a category are a frequent source of model categories; for instance, simplicial commutative rings or simplicial R-modules admit natural model structures. This follows because there is an adjunction between simplicial sets and simplicial commutative rings (given by the forgetful and free functors), and in nice cases one can lift model structures under an adjunction.

Denis-Charles Cisinski has developed a general theory of model structures on presheaf categories (generalizing simplicial sets, which are presheaves on the simplex category).