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 Daniel G. Quillen (1967). 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.
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.
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 (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 ( α , β ) {\displaystyle (\alpha ,\beta )} and ( γ , δ ) {\displaystyle (\gamma ,\delta )} subject to the following axioms. A fibration that is also a weak equivalence is called an acyclic (or trivial) fibration and a cofibration that is also a weak equivalence is called an acyclic (or trivial) cofibration (or sometimes called an anodyne morphism).
Axioms
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 C 2 {\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 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. 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.
Factorization: every morphism f in C can be written as p ∘ i {\displaystyle p\circ i} for a fibration p and an acyclic cofibration i; every morphism f in C can be written as p ∘ i {\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.
Definition via weak factorization systems The above definition can be succinctly phrased by the following equivalent definition: a model category is a category C and three classes of (so-called) weak equivalences W, fibrations F and cofibrations C so that
C has all limits and colimits,
( C ∩ W , F ) {\displaystyle (C\cap W,F)} is a weak factorization system,
( C , F ∩ W ) {\displaystyle (C,F\cap W)} is a weak factorization system
W {\displaystyle W} satisfies the 2 of 3 property.
First consequences of the definition 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 C o p {\displaystyle {\mathcal {C}}^{op}} also admits a model structure so that weak equivalences correspond to their opposites, fibrations opposites of cofibrations and cofibrations opposites of fibrations.
Examples
Topological spaces 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.
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