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A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. [3] In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra.
For example, the category of (reasonable) topological spaces has a structure of a model category where a weak equivalence is a weak homotopy equivalence, a cofibration a certain retract and a fibration a Serre fibration. [20] Another example is the category of non-negatively graded chain complexes over a fixed base ring. [21
The homotopy groups are fundamental to homotopy theory, which in turn stimulated the development of model categories. It is possible to define abstract homotopy groups for simplicial sets. Homology groups are similar to homotopy groups in that they can represent "holes" in a topological space. However, homotopy groups are often very complex and ...
The older definition of the homotopy category hTop, called the naive homotopy category [1] for clarity in this article, has the same objects, and a morphism is a homotopy class of continuous maps. That is, two continuous maps f : X → Y are considered the same in the naive homotopy category if one can be continuously deformed to the other.
In this sense, ANRs avoid all the homotopy-theoretic pathologies of arbitrary topological spaces. For example, the Whitehead theorem holds for ANRs: a map of ANRs that induces an isomorphism on homotopy groups (for all choices of base point) is a homotopy equivalence. Since ANRs include topological manifolds, Hilbert cube manifolds, Banach ...
A typical example is a surface such as the one depicted at the right. Moreover, is a point in X called the base-point. (As is explained below, its role is rather auxiliary.) The idea of the definition of the homotopy group is to measure how many (broadly speaking) curves on X can be deformed into each other. The precise definition depends on ...
An equivalent definition of homotopical connectivity is based on the homotopy groups of the space. A space is n-connected (or n-simple connected) if its first n homotopy groups are trivial. Homotopical connectivity is defined for maps, too. A map is n-connected if it is an isomorphism "up to dimension n, in homotopy".
Similarly, one can define a colimit as the left adjoint to the diagonal functor Δ 0 given above. To define a homotopy colimit, we must modify Δ 0 in a different way. A homotopy colimit can be defined as the left adjoint to a functor Δ : Spaces → Spaces I where Δ(X)(i) = Hom Spaces (| N(I op /i) |, X), where I op is the opposite category of I.