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Concept# Homotopy theory

Summary

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 (specifically the study of ).
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, 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 where the 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.

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