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Given two topological spaces X and Y, a homotopy equivalence between X and Y is a pair of continuous maps f : X → Y and g : Y → X, such that g ∘ f is homotopic to the identity map id X and f ∘ g is homotopic to id Y. If such a pair exists, then X and Y are said to be homotopy equivalent, or of the same homotopy type.
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 learned as an independent discipline.
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group , denoted π 1 ( X ) , {\displaystyle \pi _{1}(X),} which records information about loops in a space .
For example, a map is a homotopy equivalence if and only if it is an isomorphism in the homotopy category. homotopy colimit A homotopy colimit is a homotopically-correct version of colimit. homotopy over a space B A homotopy h t such that for each fixed t, h t is a map over B. homotopy equivalence 1.
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.
Several logical symbols are widely used in all mathematics, and are listed here. For symbols that are used only in mathematical logic, or are rarely used, see List of logic symbols. ¬ Denotes logical negation, and is read as "not".
Regular homotopy for immersions is similar to isotopy of embeddings: they are both restricted types of homotopies. Stated another way, two continuous functions f , g : M → N {\displaystyle f,g:M\to N} are homotopic if they represent points in the same path-components of the mapping space C ( M , N ) {\displaystyle C(M,N)} , given the compact ...
The homotopy pullback of along the identity is nothing but the mapping path space of . The universal property of a homotopy pullback yields the natural map , a special case of a natural map from a limit to a homotopy limit. In the case of a homotopy fiber, this map is an inclusion of a fiber to a homotopy fiber.