Holes in Topological Spaces: Homotopy and Weak Homotopy Equivalence

This article explores the concept of 'holes' in topological spaces and introduces two equivalence relations: homotopy equivalence and weak homotopy equivalence. Homotopy equivalence allows spaces to be deformed while preserving the number of 'holes,' such as a coffee cup and a torus being homotopy equivalent. Weak homotopy equivalence is more relaxed, requiring only that spaces have the same homotopy groups, even if they differ in local structure. The article delves into the concept of homotopy groups and illustrates how to identify 'holes' in spaces using homotopy groups with the example of a torus. Finally, it mentions Grothendieck's conjecture that the infinity groupoid captures all information about a topological space up to weak homotopy equivalence, which is closely related to weak factorization systems and Quillen model categories.