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