In mathematics,[] homotopy theory is a systematic study of situations in which maps come with homotopies between them. It originated as a topic in algebraic topology but nowadays it is studied as an independent discipline. Besides algebraic topology, the theory has also been in used in other areas of mathematics such as algebraic geometry and category theory.
Concepts
Spaces
In homotopy theory, one typically does not work with an arbitrary topological space to avoid pathologies in point-set topology. Instead, one assumes a space is a reasonable space; the meaning depends on authors but it can mean that a space is compactly generatedHausdorff space or is a CW complex. Frequently, one works with a space X with some chosen point * in the space; such a space is called a based space. A map between based spaces are then required to preserve the base points. For example, if is the unit interval and 0 is the base point, then a map is a path from the base point to the point. The adjective “free” is used to indicate freeness of choice of base points; for example, a free path would be an arbitrary map that does not necessarily preserve the base point. A map between based spaces is also often called a based map, to emphasize that it is not a free map.
Homotopy
Let I denote the unit interval. A family of maps indexed by I, is called a homotopy from to if is a map. When X, Y are based spaces, then are required to preserve the base points. A homotopy can be shown to be an equivalence relation. Given a based space X and an integer, let be the homotopy classes of based maps from a 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 based space, there is the notion of a fundamental groupoid : by definition, the fundamental groupoid of a space X is the category where the objects are the points of X and the morphisms paths.
A map is called a cofibration if given a map and 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 a map and a homotopy, there exists a homotopy such that is the given one and. A basic example is a covering map. If is a principal G-bundle, that is, a space with a free and transitive groupaction of a group, then the projection map is an example of a fibration.
Classifying spaces and homotopy operations
Given a topological group G, the classifying space for principal G-bundles is a space such that, for each space X, where
the left-hand side is the set of homotopy classes of maps,
~ refers isomorphism of bundles, and
= is given by pulling-back the distinguished bundle on along a map.
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 groupA, 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 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:
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.
