(This page discusses the Theory of Homotopy Theories, as derived from the homotopical algebra approach to the subject. For other topics within the general area of Homotopy Theory, look at the list of related entries. You will probably find what you want. If you don’t why not write a page on the topic.)
Historically, the motivating example had been category Top of (suitably well-behaved…) topological spaces: objects are topological spaces, morphisms are continuous maps between these, 2-morphisms are homotopies of such maps, and k-morphisms are higher order homotopies of homotopies. If “suitably well-behaved” means at least of the homotopy type of CW-complexes, then this -category is equivalent to ∞ Grpd. One may think of the notion of ∞-groupoid as being the intrinsic notion, which has various realizations useful in computations, such as topological spaces or simplicial sets (see simplicial homotopy theory).
A convenient, powerful, and traditional way to deal with (∞,1)-categories is to “present” them by 1-categories with specified classes of morphisms called weak equivalences : a category with weak equivalences or homotopical category. The idea is as follows. Given a category with a class of weak equivalences, we can form its homotopy category or category of fractions by adjoining formal inverses to all the morphisms in . The (∞,1)-category presented by “ can be thought of as the result of regarding as an -category with only identity -cells for , then adjoining formal inverses to morphisms in in the -categorical sense; that is, making them into equivalences rather than isomorphisms. It is remarkable that most -categories that arise in mathematics can be presented in this way.
As with presentations of groups and other algebraic structures, very different presentations can give rise to equivalent -categories. For example, several different presentations of the -category of -groupoids are:
The latter three can hence be regarded as providing “combinatorial models” for the homotopy theory of topological spaces.
The value of working with presentations of -categories rather than the -categories themselves is that the presentations are ordinary 1-categories, and thus much simpler to work with. For instance, ordinary limits and colimits are easy to construct in the category of topological spaces, or of simplicial sets, and we can then use these to get a handle on -categorical limits and colimits in the -category of -groupoids. However, we always have to make sure that we use only 1-categorical constructions that are homotopically meaningful, which essentially means that they induce -categorical meaningful constructions in the presented -category. In particular, they must be invariant under weak equivalence.
Most presentations of -categories come with additional classes of morphisms, called fibrations and cofibrations, that are very useful in performing constructions in a homotopically meaningful way. Quillen defined a model category to be a 1-category together with classes of morphisms called weak equivalences, cofibrations, and fibrations that fit together in a very precise way (the term is meant to suggest “a category of models for a homotopy theory”). Many, perhaps most, presentations of -categories are model categories. Moreover, even when we do not have a model category, we often have classes of cofibrations and fibrations with many of the properties possessed by cofibrations and fibrations in a model category, and even when we do have a model category, there may be classes of cofibrations and fibrations, different from those in the model structure, that are useful for some purposes.
Unlike the weak equivalences, which determine the “homotopy theory” and the -category that it presents, fibrations and cofibrations should be regarded as technical tools which make working directly with the presentation easier (or possible). Whether a morphism is a fibration or cofibration has no meaning after we pass to the presented -category. In fact, every morphism is weakly equivalent to a fibration and to a cofibration. In particular, despite the common use of double-headed arrows for fibrations and hooked arrows for cofibrations, they do not correspond to -categorical epimorphisms and monomorphisms.
In a model category, a morphism which is both a fibration and a weak equivalence is called an acyclic fibration or a trivial fibration. Dually we have acyclic or trivial cofibrations. An object is called cofibrant if the map from the initial object to is a cofibration, and fibrant if the map to the terminal object is a fibration. The axioms of a model category ensure that for every object there is an acyclic fibration where is cofibrant and an acyclic cofibration where is fibrant.
For a (higher) category theorist, the following examples of model categories are perhaps the most useful to keep in mind. * sets, isomorphisms. All morphisms are both fibrations and cofibrations. The -category presented is again the 1-category . * categories, equivalences of categories. The cofibrations are the functors which are injective on objects, and the fibrations are the isofibrations. The acyclic fibrations are the equivalences of categories which are literally surjective on objects. Every object is both fibrant and cofibrant. The -category presented is the 2-category . This is often called the folk model structure. * (strict) 2-categories and (strict) 2-functors, 2-functors which are equivalences of bicategories. The fibrations are the 2-functors which are isofibrations on hom-categories and have an equivalence-lifting property. Every object is fibrant; the cofibrant 2-categories are those whose underlying 1-category is freely generated by some directed graph. The -category presented is the (weak) 3-category . This model structure is due to Steve Lack.
The morphisms from to in the -category presented by are zigzags ; these are sometimes called generalized morphisms. Many presentations (including every model category) have the property that any such morphism is equivalent to one with a single zag, as in . In a model category, a canonical form for such a zigzag is where is cofibrant and is fibrant. In this case we can moreover take to be an acyclic fibration and to be an acyclic cofibration.
Often it suffices to consider even shorter zigzags of the form or . In particular, this is the case if every object is fibrant or every object is cofibrant. For example:
If is cofibrant and is fibrant, then every generalized morphism from to is equivalent to an ordinary morphism. For example, if is a cofibrant 2-category, then every pseudofunctor is equivalent to a strict 2-functor
Quillen also introduced a highly structured notion of equivalence between model categories, now called a Quillen equivalence, which among other things ensures that they present the same -category. Quillen equivalences are now being used to compare different definitions of higher categories.
“The ultimate aim of algebraic homotopy is to construct a purely algebraic theory, which is equivalent to homotopy theory in the same sort of way that ‘analytic’ is equivalent to ‘pure’ projective geometry.”
This theme has been taken up more recently by Baues, using a type of abstract homotopy theory closely related to Ken Brown’s categories of fibrant objects. Whitehead’s own work was extended by Ronnie Brown and Phil Higgins, see nonabelian algebraic topology.
The original axiomatization of homotopy theory by model categories is due to
The similar axiomatization involving the weaker structure of a calculus of fractions is due to
A standard account of the modern form of simplicial homotopy theory is in
Formulation of abstract homotopy theory as the theory of (∞,1)-toposes is due to
H. J. Baues, Homotopy types, in Handbook of Algebraic Topology, (edited by I.M. James), North Holland, 1995.
William Dwyer, P .Hirschhorn, Daniel Kan, Jeff Smith, Homotopy Limit Functors on Model Categories and Homotopical Categories, volume 113 of Mathematical Surveys and Monographs, American Mathematical Society (2004) (there exists this pdf copy of what seems to be a preliminary version of this book)