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Homotopy Equivalences
Previously: Fibrations and Cofibrations. In topology, we say that two shapes are the same if there is a homeomorphism– an invertible continuous map– between them. Continuity means that …
In general, paths in a topological space , i.e. continuous mappings , naturally split into equivalence classes with respect to homotopy. So there is essentially a single fundamental group for the whole space, as long as is path-connected, i.e., it doesn’t split into multiple disconnected chunks. Grothendieck conjectured that the infinity groupoid captures all information about a topological space up to weak homotopy equivalence.
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