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A homotopy between two embeddings of the torus into : as "the surface of a doughnut" and as "the surface of a coffee mug".This is also an example of an isotopy.. Formally, a homotopy between two continuous functions f and g from a topological space X to a topological space Y is defined to be a continuous function: [,] from the product of the space X with the unit interval [0, 1] to Y such that ...
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, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different (but related) categories, as discussed below.
The nth homotopy group of a topological space is the group of homotopy classes of basepoint-preserving maps from the -sphere to , under the group operation of concatenation. The most fundamental homotopy group is the fundamental group π 1 ( X ) {\displaystyle \pi _{1}(X)} .
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 ...
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 .
A homotopy pullback (or homotopy fiber-product) is the dual concept of a homotopy pushout. It satisfies the universal property of a pullback up to homotopy. [ citation needed ] Concretely, given f : X → Z {\displaystyle f:X\to Z} and g : Y → Z {\displaystyle g:Y\to Z} , it can be constructed as
Cover of Homotopy Type Theory: Univalent Foundations of Mathematics.. In mathematical logic and computer science, homotopy type theory (HoTT) refers to various lines of development of intuitionistic type theory, based on the interpretation of types as objects to which the intuition of (abstract) homotopy theory applies.