📂Topological Data Analysis

Homotopy Type

Definition ¹

For two topological spaces $X, Y$, if there exist continuous functions $f : X \to Y$, $g: Y \to X$ that satisfy the following, then $X, Y$ is said to have the same Homotopy Type and $X, Y$ or $f, g$ is also referred to as Homotopy Equivalent. $$ \begin{align*} g \circ f \simeq& \text{id}_{X} \\ f \circ g \simeq& \text{id}_{Y} \end{align*} $$ Here, $\text{id}_{\cdot}$ is the identity function, and $f \simeq g$ means that $f,g$ is homotopic.

Explanation

The reason for considering homotopy equivalence, i.e., having the same homotopy type, is obviously to talk about a relaxed ‘sameness’ stepping back slightly from topological isomorphism. What is compromised for this purpose is the condition that $f$ and $g$ are inverse functions to each other in the definition, and it is sufficient if returning to the original space from the opposite side just returns to the original point.