Homotopy theory MOC

Homotopy of paths

Let 𝛼,𝛽 :𝕀 →𝑋 be paths with common endpoints, i.e. 𝛼(0) =𝛽(0) and 𝛼(1) =𝛽(1). Then a homotopy of paths 𝐻 :𝛼 ≃𝛽 is a Homotopy of continuous maps with the additional constraint that the endpoints are the same for all 𝑡, #m/def/homotopy i.e. 𝐺 :[0,1] ×[0,1] →𝑋 with

This is equivalent to homotopy relative {0,1}. Homotopy classes of paths are the morphisms of the Fundamental groupoid.

Properties

• Path traversal lemma

#state/tidy | #lang/en | #SemBr