User:Alodyne/Model category

In mathematics, model categories are the foundation of homotopy theory. Roughly speaking, they are categories with extra structure on certain distinguished morphism sets, described below. The framework of model categories provides a concrete model for a categorical localization while avoiding set-theoretic difficulties.

Two motivations
Model categories may be motivated in two general ways. The first is the avoidance of set-theoretic problems, as described above, that arise in the localization of a category remember to fix the latter article. Localization is the process of inverting a selected class of morphisms. For example, in the category of chain complexes of modules over a ring, it is sometimes desirable to invert the quasi-isomorphisms (the morphisms of complexes that induce isomorphisms on homology); in the category of CW complexes (or more generally, compactly generated spaces) we may wish to invert weak homotopy equivalences. The idea in each case is to pass to a new category, with the same objects as the old, but with the selected morphisms now isomorphisms. This may seem opaque, but there is a clear reason for doing so. In homological algebra we are interested in information that is detected at the level of homology. If two complexes are isomorphic "at the level of homology," then perhaps we would be better off not preserving the distinction between the two. The difficulties mentioned arise because in most cases one loses control of the morphisms; in the localized category, there is no reason why Hom(X,Y) should be a set (rather than a proper class). This in turn comes from the fact that a weak homotopy equivalence, for example, need not have any kind of reasonable inverse (whereas homotopy equivalences do indeed have homotopy inverses). The same is true of quasi-isomorphisms as compared to chain homotopy equivalences. The model category framework provides both a recipe for constructing the localized category (called the homotopy category of the model category) and assurance that the morphism sets are well-behaved.

On the other hand, in both of the examples mentioned above (homological algebra and classical homotopy theory), there are many cases in which extremely desirable constructions can only be made before passage to homotopy, even though one is primarily concerned with the homotopy categories (in these cases, the derived category and the homotopy category of topological spaces). All known interesting examples of homotopical categories (i.e., triangulated categories with certain properties) arise as the homotopy category of some model category. Thus we are at liberty to make our constructions at the model category level, then push the resulting objects down into the homotopy category.

Examples

 * Let Ch(R) be the category of chain complexes of R-modules for some ring R, with chain maps as morphisms. Then Ch(R) is an abelian category, so projective resolutions exist. If M and N are quasi-isomorphic complexes (so that they are isomorphic in the derived category), then they have quasi-isomorphic projective resolutions P and Q. Passage to the derived category identifies M with N and P with Q, so we in practice do not distinguish between them. Yet these resolutions are most easily constructed at the level of complexes rather than at the level of the derived category; moreover, even though the resolution is unique at the derived level, there are many representatives of it at the chain level.
 * Let T be the category of based compactly generated topological spaces, with based continuous maps as morphisms, and let H be the category of based CW complexes, with based homotopy classes of continuous maps as morphisms. The CW approximation theorem tells us that for every object X of T, there is an object QX of H and a map
 * q : QX &rarr; X

such that q is a weak homotopy equivalence. From the point of view of the derived category, we should think of q as an isomorphism (in a suitably constructed localization of T) and of QX as a projective resolution. In the same way as in the above example, any two choices of q are equivalent from this localized viewpoint. Nevertheless, the construction of q and QX takes place at the level of the spaces themselves, and as above, there are many choices for them at the level of spaces.

Definition
There are several slightly varying definitions in use. All follow the initial exposition of Daniel Quillen to great extent. We follow the treatment of Hovey. The greatest difference between Quillen's definition and the one given below concerns the factorizations. We assume they are functorial; Quillen did not do so. Yet in all the examples of interest, they occur functorially, and the assumption perhaps eases comprehension of so much abstraction all at once. We need several preliminary definitions first. For convenience, we use juxtaposition to denote composition of morphisms in all that follows.

Recall that if C is any category, we may form the category Map C, whose objects are morphisms of C and whose morphisms are commutative squares. Then we say that a morphism f in C is a retract of a morphism g in C if f is a retract of g as objects of Map C. This is the case if and only if there is a commuting diagram of the following form: where each horizontal composite is an identity morphism. We also say that a functorial factorization is an ordered pair (&alpha;,&beta;) of functors Map C &rarr; Map C such that f = &beta;(f)&alpha;(f) for all objects f of Map C. This construction is related to the more familiar case where a morphism is factored into two morphisms: one from the source to a third object, and a second from the third object to the target, such that the composition equals the original morphism. Finally, we must discuss lifting properties. This is perhaps the heart of the definition of model category. Let i : A &rarr; B and p : X &rarr; Y be morphisms in our category C. Suppose moreover that whenever these maps fit into a diagram of the following form, there is a lift h : B &rarr; X satisfying hi = f and ph = g. That is, given a diagram consisting of the solid arrows, we can always fill in the dashed arrow to make the entire diagram commute. Then we say that i has the left lifting property with respect to p, and that p has the right lifting property with respect to i.

A model structure on a category C is three classes of morphisms of C, called weak equivalences, cofibrations, and fibrations, together with two functorial factorizations (&alpha;,&beta;) and (&gamma;,&delta;) satisfying the following axioms. Call a morphism a trivial cofibration if it is both a cofibration and a weak equivalence; similarly for trivial fibration.
 * (2-of-3) If f and g are morphisms of C such that gf is defined and any two of f, g, and gf are weak equivalences, so is the third.
 * (retracts) If f and g are morphisms of C such that f is a retract of g and g is a weak equivalence, cofibration, or fibration, then so is f.
 * (lifting) Trivial cofibrations have the left lifting property with respect to fibrations, and cofibrations have the left lifting property with respect to trivial fibrations.
 * (factorization) For any morphism f, &alpha;(f) is a cofibration, &beta;(f) is a trivial fibration, &gamma;(f) is a trivial cofibration, and &delta;(f) is a fibration.

A model category is a category C with all small limits and colimits, together with a model structure on C.

Resolutions; or, Fibrant and cofibrant replacement

 * (Some context and motivation here)

Since C has limits and colimits it has, in particular, an initial object i and a terminal object t. An object X of C is called fibrant if the unique map from X to t is a fibration, and cofibrant if the unique map from i to X is a cofibration. According to the factorization axioms we may find, for each X, a cofibrant object QX and a trivial fibration QX &rarr; X. We may also find a fibrant object RX and a trivial cofibration X &rarr; RX. In particular, the maps QX &rarr; X and X &rarr; RX are weak equivalences; we refer to QX as a "cofibrant replacement" and RX as a "fibrant replacement" for X.

The homotopy category of a model category
The point of the foregoing mountain of abstraction is that we can use it to construct a category that is equivalent to Ho C, the localization of C with respect to the weak equivalences. This neatly sidesteps the set-theoretic issues outlines above, for it will be apparent that the morphisms form a set in the category we construct. Moreover, the nature of the morphisms themselves is much clearer than in the localized category.

Paths, cylinders, and Eckmann-Hilton duality

 * Define path and cylinder objects with references to topology and algebra. Anticipate the definitions of homotopy and give them. Explain about dual model categories, lifting properties, and EH. Functoriality of cylinder/path objects. When is homotopy an equivalence relation?

To "do homotopy theory" inside a model category, one needs an internal notion of "homotopy". In topology, if f and g are continuous maps between spaces X and Y, a homotopy is usually understood to be a continuous map H:I x X &rarr; Y. Here I x X is the cylinder over X, and the map H is required to agree with f on one side and g on the other side of this cylinder. Left homotopies in a model category are defined to be maps from a cylinder object with similar properties.

In more detail, if X is an object in a model category C, a cylinder over X is an object IX, together with a factorization X + X &rarr; IX &rarr; X of the "fold" map from X + X to X. Here X + X denotes the disjoint union of two copies of X, the map X + X &rarr; IX is required to be a cofibration, and the map IX &rarr; Y is required to be both an acyclic fibration, as in the factorization axiom above. If f and g are two maps X &rarr; Y, a left homotopy between f and g is a map H from IX to Y, such that H composed with X + X &rarr; IX coincides with the map f+g : X + X &rarr; Y.

There is another approach to homotopies which is maybe less vivid geometrically, but equally valid from the point of view of category theory: one could view a homotopy between f and g as a map X &rarr; YI from X to the space of paths in Y. Thus, right homotopies in a model category are defined to be maps to a path object. That is, if Y is an object in a model category C, a path object for Y is an object YI together with a factorization Y &rarr; YI &rarr; Y x Y of the "diagonal" map from Y to Y x Y, such that Y &rarr; YI is an acyclic cofibration, and YI is a fibration. If f and g are two map X &rarr; Y, a right homotopy between f and g is a map H from X to YI, such that YI &rarr; Y x Y composed with H coincides with the map (f,g): X &rarr; Y x Y.

If f and g are two maps between objects X and Y of a model category C, we shall say they are "left homotopic" if there is a left homotopy between them, and that they are "right homotopic" if there is a right homotopy between them. We shall refer to the two relations on Hom(X,Y) as the "left homotopy relation" and the "right homotopy relation." Neither relation is an equivalence relation in general, nor do the two relations agree. However, we have the following theorem:


 * Theorem Let X and Y be objects in a model category C.  If X is cofibrant, then the left homotopy relation is an equivalence relation on Hom(X,Y).  Similarly, if Y is fibrant, the right homotopy relation is an equivalence relation on Hom(X,Y).  Moreover, if X is cofibrant and Y is fibrant, the two relations coincide.

Examples of model categories
Chain complexes, simplicial sets, topological spaces.