Model category

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In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows'): 'weak equivalences', 'fibrations' and 'cofibrations'. These abstract from a conventional homotopy category, of topological spaces or of chain complexes (derived category theory). This concept was introduced in 1967 by Daniel G. Quillen.

Contents 1 Motivation 2 Formal definition 3 Example: Topological spaces 4 Example: Bounded from below chain complexes of R-modules 5 Example: Arbitrary chain-complexes of R-modules 6 Some constructions 7 Homotopy and the homotopy category 8 References

[edit] Motivation

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.

[edit] 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 equivalences, fibrations, and cofibrations, and two functorial factorizations (α,β) and (γ,δ) 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

1. Retracts: each of the distinguished classes is closed under retracts.
2. 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.
3. Lifting: acylic cofibrations have the left lifting property with respect to fibrations and cofibrations have the left left lifting property lifting property with respect to acyclic fibrations.
4. Factorization: for every morphism f in C, α(f) is a cofibration, β(f) is an acyclic fibration, γ(f) is an acyclic cofibration, and δ(f) 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 analogously for cofibrations, but many readers find this terminology ambiguous.

[edit] 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.

[edit] Example: Bounded from below 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:

• weak equivalences are maps that induce isomorphism 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.

[edit] Example: Arbitrary chain-complexes of R-modules

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

1. weak equivalences are chain-homotopy equivalences of chain-complexes, i.e., a chain-map is a weak-equivalence if and only if there exists a chain-map and chain-homotopies and where:

• 

• 

here and are the differentials of C and D, respectively.

2. Cofibrations are monomorphisms that are split as morphisms of underlying R-modules

3. Fibrations are epimorphisms that are split as morphisms of underlying R-modules

[edit] 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.

[edit] 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.

[edit] References

• W. G. Dwyer and J. Spalinski: Homotopy Theories and model categories, 1995. [1]

• Mark Hovey: Model Categories, 1999, ISBN 0-8218-1359-5.