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MOTIVATION Model categories can provide a natural setting for s. Another model category is the category of Chain Complex es 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 . FORMAL DEFINITION The definition given initially by Quillen was that of a closed model category, the assumptions of which seemed strong at the time, motivating others to weaken some of the assumptions to define a model category. In practice the distinction has not proven significant and most recent authors work with closed model categories and simply drop the adjective 'closed'. The definition has been separated to that of a model structure on a category and then further categorical conditions on that category, the necessity of which may seem unmotivated at first but becomes important later. The following definition follows that given by Hovey. A model structure on a category ''C'' consists of three distinguished classes of morphisms (equivalently subcategories): Weak Equivalence s, Fibration s, and Cofibration s, and two functorial factorizations and subject to the following axioms. Note that a fibration that is also a weak equivalence is called an '''acyclic fibration''' and a cofibration that is also a weak equivalence is called an '''acyclic cofibration'''. ;Axioms: # ''Retracts'': each of the distinguished classes is closed under retracts. # ''2 of 3'': if ''f'' and ''g'' are maps in ''C'' such that ''f'', ''g'', and ''gf'' are defined and any two of these are weak equivalences then so is the third. # ''Lifting'': acylic cofibrations have the Left Lifting Property with respect to fibrations and cofibrations have the left lifting property with respect to fibrations. # ''Factorization'': for every morphism morphism in ''C'', is a cofibration, is an acyclic fibration, is an acyclic cofibration, and is a fibration. A model category is a category that has a model structure and all (small) limits and colimits, i.e. a complete and cocomplete category with a model structure. Note that acyclic fibrations are sometimes called trivial fibrations, and analagously for cofibrations, but this terminology is ambiguous. EXAMPLE: TOPOLOGICAL SPACES The Category Of Topological Spaces , Top, admits a model category structure with the usual (Serre) fibrations and cofibrations and with weak equivalences as weak homotopy equivalences. 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 cofibrations. EXAMPLE: CHAIN COMPLEXES OF ''R''-MODULES The category of (nonnegatively graded) chain complexes of ''R''-modules can be realized as a model category by defining the distinguished classes as follows:
SOME CONSTRUCTIONS Every closed model category has a Terminal Object by the completeness axiom and an Initial Object by the cocompleteness axiom 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''. HOMOTOPY AND THE HOMOTOPY CATEGORY Given a model category, one can then define an associated homotopy category by Localizing with respect to the class of weak equivalences. This suggests that the information regarding homotopy is contained in the class of weak equivalences whereas the classes of fibrations and cofibrations are useful in making constructions within the category. Left homotopy is defined with respect to cylinder objects and right homotopy is defined with respect to path 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. REFERENCES
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