| 41. | For a path connected base, all the different fibers are homotopy equivalent.
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| 42. | Each of these constructions yield the same homotopy category.
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| 43. | This can be formulated in terms of homotopy theory.
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| 44. | If all these vanish, then the map f is a homotopy equivalence.
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| 45. | The cellular approximation theorem can be used to immediately calculate some homotopy groups.
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| 46. | This is a current research area in Homotopy Type Theory ( HoTT ).
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| 47. | Then a homotopy between the two systems is considered.
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| 48. | Homotopy Lie algebras and their morphisms define a category.
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| 49. | Now consider the homotopy class of continuous paths from to, within the space.
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| 50. | An active area of research is the univalent foundations arising from homotopy type theory.
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