In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topology, the theory has also been used in other areas of mathematics such as algebraic geometry (e.g., A1 homotopy theory) and category theory (specifically the study of higher categories).
In homotopy theory and algebraic topology, the word "space" denotes a topological space. In order to avoid pathologies, one rarely works with arbitrary spaces; instead, one requires spaces to meet extra constraints, such as being compactly generated, or Hausdorff, or a CW complex.
In the same vein as above, a "map" is a continuous function, possibly with some extra constraints.
Often, one works with a pointed space -- that is, a space with a "distinguished point", called a basepoint. A pointed map is then a map which preserves basepoints; that is, it sends the basepoint of the domain to that of the codomain. In contrast, a free map is one which needn't preserve basepoints.
Main article: Homotopy
Let I denote the unit interval. A family of maps indexed by I, is called a homotopy from to if is a map (e.g., it must be a continuous function). When X, Y are pointed spaces, the are required to preserve the basepoints. A homotopy can be shown to be an equivalence relation. Given a pointed space X and an integer , let be the homotopy classes of based maps from a (pointed) n-sphere to X. As it turns out, are groups; in particular, is called the fundamental group of X.
If one prefers to work with a space instead of a pointed space, there is the notion of a fundamental groupoid (and higher variants): by definition, the fundamental groupoid of a space X is the category where the objects are the points of X and the morphisms are paths.
A map is called a cofibration if given (1) a map and (2) a homotopy , there exists a homotopy that extends and such that . To some loose sense, it is an analog of the defining diagram of an injective module in abstract algebra. The most basic example is a CW pair ; since many work only with CW complexes, the notion of a cofibration is often implicit.
A fibration in the sense of Serre is the dual notion of a cofibration: that is, a map is a fibration if given (1) a map and (2) a homotopy , there exists a homotopy such that is the given one and . A basic example is a covering map (in fact, a fibration is a generalization of a covering map). If is a principal G-bundle, that is, a space with a free and transitive (topological) group action of a (topological) group, then the projection map is an example of a fibration.
Given a topological group G, the classifying space for principal G-bundles ("the" up to equivalence) is a space such that, for each space X,
Brown's representability theorem guarantees the existence of classifying spaces.
The idea that a classifying space classifies principal bundles can be pushed further. For example, one might try to classify cohomology classes: given an abelian group A (such as ),
where is the Eilenberg–MacLane space. The above equation leads to the notion of a generalized cohomology theory; i.e., a contravariant functor from the category of spaces to the category of abelian groups that satisfies the axioms generalizing ordinary cohomology theory. As it turns out, such a functor may not be representable by a space but it can always be represented by a sequence of (pointed) spaces with structure maps called a spectrum. In other words, to give a generalized cohomology theory is to give a spectrum.
A basic example of a spectrum is a sphere spectrum:
See also: Characteristic class, Postnikov tower, Whitehead torsion
Main article: Localization of a topological space
There are several specific theories
Main article: Homotopy hypothesis
One of the basic questions in the foundations of homotopy theory is the nature of a space. The homotopy hypothesis asks whether a space is something fundamentally algebraic.
Main article: Model category