Model category

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 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 (e.g., Mark Hovey and Philip Hirschhorn) 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 ${\displaystyle (\alpha ,\beta )}$ and ${\displaystyle (\gamma ,\delta )}$ subject to the following axioms. A fibration that is also a weak equivalence is called an acyclic (or trivial) fibration[1] and a cofibration that is also a weak equivalence is called an acyclic (or trivial) cofibration (or sometimes called an anodyne morphism).

Axioms

1. Retracts: if g is a morphism belonging to one of the distinguished classes, and f is a retract of g (as objects in the arrow category ${\displaystyle C^{2}}$, where 2 is the 2-element ordered set), then f belongs to the same distinguished class. Explicitly, the requirement that f is a retract of g means that there exist i, j, r, and s, such that the following diagram commutes:

   

2. 2 of 3: if f and g are maps in C such that gf is defined and any two of these are weak equivalences then so is the third.
3. Lifting: acyclic cofibrations have the left lifting property with respect to fibrations, and cofibrations have the left lifting property with respect to acyclic fibrations. Explicitly, if the outer square of the following diagram commutes, where i is a cofibration and p is a fibration, and i or p is acyclic, then there exists h completing the diagram.

   

4. Factorization:
   • every morphism f in C can be written as ${\displaystyle p\circ i}$ for a fibration p and an acyclic cofibration i;
   • every morphism f in C can be written as ${\displaystyle p\circ i}$ for an acyclic fibration p and a cofibration i.

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.

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.

• (∞,1)-category
• Cocycle category
• Stable model category

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• Dwyer, William G.; Spaliński, Jan (1995), "Homotopy theories and model categories" (PDF), Handbook of algebraic topology, Amsterdam: North-Holland, pp. 73–126, doi:10.1016/B978-044481779-2/50003-1, MR 1361887
• Philip S. Hirschhorn: Model Categories and Their Localizations, 2003, ISBN 0-8218-3279-4.
• Mark Hovey: Model Categories, 1999, ISBN 0-8218-1359-5.
• Klaus Heiner Kamps and Timothy Porter: Abstract homotopy and simple homotopy theory, 1997, World Scientific, ISBN 981-02-1602-5.
• Georges Maltsiniotis: La théorie de l'homotopie de Grothendieck. Astérisque, (301) 2005, vi+140 pp.

• Riehl, Emily (2014), Categorical homotopy theory, Cambridge University Press, doi:10.1017/CBO9781107261457, ISBN 978-1-107-04845-4, MR 3221774

• Quillen, Daniel G. (1967), Homotopical algebra, Lecture Notes in Mathematics, No. 43, 43, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0097438, MR 0223432

• "Do we still need model categories?"
• "(infinity,1)-categories directly from model categories"
• Paul Goerss and Kristen Schemmerhorn, Model Categories and Simplicial Methods

• Model category in nLab
• Model category in Joyal's catlab