Әлсіз теңдестіктер, фибрациялар және кофибрациялар бар математикалық категориялар

Mathematical category with weak equivalences, fibrations and cofibrations
In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows') called 'weak equivalences', 'fibrations' and 'cofibrations' satisfying certain axioms relating them. These abstract from the category of topological spaces or of chain complexes (derived category theory). The concept was introduced by
In recent decades, the language of model categories has been used in some parts of algebraic K theory and algebraic geometry, where homotopy theoretic approaches led to deep results.

Model categories can provide a natural setting for homotopy theory: the category of topological spaces is a model category, with the homotopy corresponding to the usual theory. Similarly, objects that are thought of as spaces often admit a model category structure, such as the category of simplicial sets. Another model category is the category of chain complexes of R modules for a commutative ring R. Homotopy theory in this context is homological algebra. Homology can then be viewed as a type of homotopy, allowing generalizations of homology to other objects, such as groups and R algebras, one of the first major applications of the theory. Because of the above example regarding homology, the study of closed model categories is sometimes thought of as homotopical algebra.

The axioms imply that any two of the three classes of maps determine the third (e. g., cofibrations and weak equivalences determine fibrations). Also, the definition is self dual: if C is a model category, then its opposite category also admits a model structure so that weak equivalences correspond to their opposites, fibrations opposites of cofibrations and cofibrations opposites of fibrations.

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.

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. A simplicial model category is a simplicial category with a model structure that is compatible with the simplicial structure. Given any category C and a model category M, under certain extra hypothesis the category of functors Fun (C, M) (also called C diagrams in M) is also a model category. 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. In both cases the third class of morphisms is given by a lifting condition (see below). In some cases, when the category C is a Reedy category, there is a third model structure lying in between the projective and injective. The process of forcing certain maps to become weak equivalences in a new model category structure on the same underlying category is known as Bousfield localization. For example, the category of simplicial sheaves can be obtained as a Bousfield localization of the model category of simplicial presheaves. Denis Charles Cisinski has developed a general theory of model structures on presheaf categories (generalizing simplicial sets, which are presheaves on the simplex category). 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.

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. Given an object X in the model category, if the unique map from the initial object to X is a cofibration, then X is said to be cofibrant. Analogously, if the unique map from X to the terminal object is a fibration then X is said to be fibrant. If Z and X are objects of a model category such that Z is cofibrant and there is a weak equivalence from Z to X then Z is said to be a cofibrant replacement for X. Similarly, if Z is fibrant and there is a weak equivalence from X to Z then Z is said to be a fibrant replacement for X. In general, not all objects are fibrant or cofibrant, though this is sometimes the case. For example, all objects are cofibrant in the standard model category of simplicial sets and all objects are fibrant for the standard model category structure given above for topological spaces. Left homotopy is defined with respect to cylinder objects and right homotopy is defined with respect to path space objects. These notions coincide when the domain is cofibrant and the codomain is fibrant. In that case, homotopy defines an equivalence relation on the hom sets in the model category giving rise to homotopy classes.

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.

The homotopy category of a model category C is the localization of C with respect to the class of weak equivalences. This definition of homotopy category does not depend on the choice of fibrations and cofibrations. However, the classes of fibrations and cofibrations are useful in describing the homotopy category in a different way and in particular avoiding set theoretic issues arising in general localizations of categories. More precisely, the "fundamental theorem of model categories" states that the homotopy category of C is equivalent to the category whose objects are the objects of C which are both fibrant and cofibrant, and whose morphisms are left homotopy classes of maps (equivalently, right homotopy classes of maps) as defined above. (See for instance Model Categories by Hovey, Thm 1.2.10)

Applying this to the category of topological spaces with the model structure given above, the resulting homotopy category is equivalent to the category of CW complexes and homotopy classes of continuous maps, whence the name.