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Homotopy

★★★★★Graduate+

📖Definition

Homotopy is a continuous deformation between two continuous functions. Homotopy equivalent spaces have the 'same shape' topologically.

H: X × [0,1] → Y, H(x,0) = f(x), H(x,1) = g(x)

Definition of homotopy

f ∼eq g

f and g are homotopic

X ∼eq Y ⇔ ∃ f: X → Y, g: Y → X : g ∘ f ∼eq id_X, f ∘ g ∼eq id_Y

Homotopy equivalence

예제 1

Explain why disk D² is homotopy equivalent to a point.

예제 2

Why are coffee cup and donut homotopy equivalent?

Space classification, invariants

Homotopy algebra

Homotopy type theory

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