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Jean-Pierre
Marquis (2006) "A path to Epistemology of Mathematics: Homotopy theory", pages 239 to 260 in
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local lifting of paths to a given sheet; the uniqueness is because the fibers of a covering map are
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corresponds to a dotted arrow making the diagram commute. This diagram is dual to that of the
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The outer square (without the dotted arrow) commutes if and only if the hypotheses of the
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Assume all maps are continuous functions between topological spaces. Given a map
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1033:{\displaystyle T\mathrel {:=} (X\times \{0\})\cup (Y\times )\subseteq X\times }
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493:{\displaystyle f_{\bullet }\circ \iota _{0}=f_{0}=\pi \circ {\tilde {f}}_{0}}
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There is a common generalization of the homotopy lifting property and the
710:{\displaystyle {\tilde {f}}_{0}=\left.{\tilde {f}}\right|_{Y\times \{0\}}}
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80:
1887:
1783:(Third Printing, 1965 ed.). New York: Academic Press Inc.
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is trivially the lift of a constant map to the image point of
633:{\displaystyle f_{\bullet }=\pi \circ {\tilde {f}}_{\bullet }}
722:
551:{\displaystyle {\tilde {f}}_{\bullet }\colon Y\times I\to E}
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1452:{\displaystyle \left.{\tilde {f}}\right|_{T}={\tilde {g}}}
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satisfies the homotopy lifting property with respect to
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Generalization: homotopy lifting extension property
412:{\displaystyle f_{0}=f_{\bullet }|_{Y\times \{0\}}}
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297:{\displaystyle f_{\bullet }\colon Y\times I\to B}
1307:{\displaystyle {\tilde {f}}\colon X\times \to E}
118:, where there need be no unique way of lifting.
720:The following diagram depicts this situation:
348:{\displaystyle {\tilde {f}}_{0}\colon Y\to E}
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1823:, Cambridge: Cambridge University Press,
1760:. Princeton: Princeton University Press.
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763:{\displaystyle {\tilde {f}}_{\bullet }}
1839:The Architecture of Modern Mathematics
1802:(Third ed.). New York: Springer.
1166:{\displaystyle f\colon X\times \to B}
27:Homotopy theory in algebraic topology
7:
849:, or one sometimes simply says that
1648:is irrelevant in that every map to
1576:The homotopy extension property of
1113:homotopy lifting extension property
1508:
1365:{\displaystyle \pi {\tilde {f}}=f}
46:(also known as an instance of the
25:
1463:The homotopy lifting property of
1065:{\displaystyle \pi \colon E\to B}
870:has the homotopy lifting property
151:{\displaystyle \pi \colon E\to B}
875:A weaker notion of fibration is
56:) is a technical condition on a
1628:to be a constant map, so that
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1756:The Topology of Fibre Bundles
1566:{\displaystyle X\times \{0\}}
1514:{\displaystyle Y=\emptyset }
1394:{\displaystyle {\tilde {g}}}
1040:. Given additionally a map
933:{\displaystyle X\supseteq Y}
578:{\displaystyle f_{\bullet }}
1873:Encyclopedia of Mathematics
940:, for simplicity we denote
912:homotopy extension property
772:homotopy extension property
1927:
1866:A.V. Chernavskii (2001) ,
1255:, there exists a homotopy
1104:{\displaystyle (X,Y,\pi )}
914:. Given a pair of spaces
87:to be moved "upstairs" to
1884:homotopy lifting property
1798:Husemoller, Dale (1994).
1752:Steenrod, Norman (1951).
44:homotopy lifting property
1488:{\displaystyle (X,\pi )}
1248:{\displaystyle g=f|_{T}}
504:there exists a homotopy
204:{\displaystyle (Y,\pi )}
1847:Oxford University Press
1817:Hatcher, Allen (2002),
640:) which also satisfies
54:covering homotopy axiom
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225:{\displaystyle \pi \,}
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49:right lifting property
1841:, J. Ferreiros &
1775:Hu, Sze-Tsen (1959).
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1601:{\displaystyle (X,Y)}
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1665:{\displaystyle \pi }
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734:are true. A lifting
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1868:"Covering homotopy"
172:{\displaystyle Y\,}
58:continuous function
34:, in particular in
1911:Algebraic topology
1820:Algebraic Topology
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1854:978-0-19-856793-6
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1534:{\displaystyle T}
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122:Formal definition
62:topological space
18:Covering homotopy
16:(Redirected from
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94:For example, a
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133:
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119:
117:
113:
109:
108:vector bundle
105:
101:
97:
92:
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63:
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1707:Hu, Sze-Tsen
1701:
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1073:
909:
881:CW complexes
874:
850:
845:is called a
802:
780:
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307:for any map
125:
112:fiber bundle
99:
96:covering map
93:
88:
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76:
72:
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64:
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29:
1845:, editors,
781:If the map
32:mathematics
1900:Categories
1746:References
1521:, so that
1878:EMS Press
1843:J.J. Gray
1687:Fibration
1660:π
1636:π
1616:π
1552:×
1509:∅
1480:π
1444:~
1419:~
1386:~
1351:~
1342:π
1299:→
1281:×
1275::
1269:~
1202:→
1196::
1190:~
1158:→
1140:×
1134::
1096:π
1057:→
1051::
1048:π
1013:×
1007:⊆
986:×
977:∪
962:×
925:⊇
858:π
847:fibration
833:π
789:π
756:∙
749:~
694:×
680:~
655:~
626:∙
619:~
609:∘
606:π
598:∙
571:∙
543:→
537:×
531::
526:∙
519:~
479:~
469:∘
466:π
441:ι
437:∘
432:∙
396:×
381:∙
340:→
334::
322:~
289:→
283:×
277::
272:∙
219:π
196:π
143:→
137::
134:π
116:fibration
1732:(1994).
1709:(1959).
1676:See also
1122:homotopy
1120:For any
1111:has the
558:lifting
355:lifting
258:homotopy
256:for any
81:homotopy
75:"above"
1886:at the
1719:page 24
825:, then
805:spaces
60:from a
52:or the
38:within
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252:, if:
100:unique
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1693:Notes
1173:, and
304:, and
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