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The classification of a space up to homotopy equivalence, retaining the properties invariant under continuous deformations with homotopy inverses rather than its exact point-set form.
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A homotopy equivalence is the relation-generating data used to classify spaces by homotopy type; the construction retains invariants such as homotopy and homology groups.
A mathematical concept supplies part of the formal language, state space, representation, or analytic machinery used by a scientific or mathematical-physics concept. The source is the mathematical predecessor; this does not claim that physical content follows from mathematics alone.
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Homotopy type treats spaces as equivalent when maps in both directions compose to maps homotopic to the corresponding identities, rather than when the spaces are homeomorphic.
Identify elements by a stated equivalence relation and equip the quotient with the induced structure. The edge detail must state the representatives and equivalence relation.
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