Homotopy theory

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 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).
Concepts.
Spaces and maps.
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.
Homotopy.
Let "I" denote the unit interval. A family of maps indexed by "I", formula_1 is called a homotopy from formula_2 to formula_3 if formula_4 is a map (e.g., it must be a continuous function). When "X", "Y" are pointed spaces, the formula_5 are required to preserve the basepoints. A homotopy can be shown to be an equivalence relation. Given a pointed space "X" and an integer formula_6, let formula_7 be the homotopy classes of based maps formula_8 from a (pointed) "n"-sphere formula_9 to "X". As it turns out, formula_10 are groups; in particular, formula_11 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.
Cofibration and fibration.
A map formula_12 is called a cofibration if given (1) a map formula_13 and (2) a homotopy formula_14, there exists a homotopy formula_15 that extends formula_2 and such that formula_17. 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 formula_18; 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 formula_19 is a fibration if given (1) a map formula_20 and (2) a homotopy formula_21, there exists a homotopy formula_22 such that formula_2 is the given one and formula_24. A basic example is a covering map (in fact, a fibration is a generalization of a covering map). If formula_25 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 formula_26 is an example of a fibration.
Classifying spaces and homotopy operations.
Given a topological group "G", the classifying space for principal "G"-bundles ("the" up to equivalence) is a space formula_27 such that, for each space "X",
where
Brown's representability theorem guarantees the existence of classifying spaces.
Spectrum and generalized cohomology.
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 formula_34),
where formula_36 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: formula_37
Obstruction theory and characteristic class.
See also: Characteristic class, Postnikov tower, Whitehead torsion
Specific theories.
There are several specific theories
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.