31. Abstract homotopy theory and motivic homotopy theory are also outside the scope. 32. Continuous deformation is required for homotopy , but not homeomorphism. 33. Grothendieck's conjectural theory of motivic homotopy theory, and motivic integration. 34. This is a " homeomorphism reduces to homotopy reduces to algebra " result. 35. One interpretation of the theorem is that it computes homotopy 1-types. 36. Homotopy maps re-enter the picture by defining homotopically equivalent chain maps.37. This leads to the idea of using multiple groupoid objects in homotopy theory. 38. The history in relation to homotopy groups is interesting. 39. If is contractible then and are homotopy equivalent spaces. 40. Analogous arguments using a homotopy of homotopies shows that this isomorphism is canonical.
