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330:. We can lift such a path to the sphere by choosing one of the two sphere points mapping to the first point on the path, then maintain continuity. In this case, each of the two starting points forces a unique path on the sphere, the lift of the path in the projective plane. Thus in the
516:{\displaystyle {\begin{aligned}f\colon \,&\to \mathbb {RP} ^{2}&&\ {\text{ (projective plane path)}}\\g\colon \,&S^{2}\to \mathbb {RP} ^{2}&&\ {\text{ (covering map)}}\\h\colon \,&\to S^{2}&&\ {\text{ (sphere path)}}\end{aligned}}}
561:. They aim "to lift concepts to a relational level making them point free as well as quantifier free, thus liberating them from the style of first order predicate logic and approaching the clarity of algebraic reasoning."
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For maps of a circle, the definition of a lift to the real line is slightly different (a common application is the calculation of
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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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and
Michael Winter have illustrated the method of lifting traditional logical expressions of
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uses a lift of a homeomorphism of the circle to the real line.
326:. A path in the projective plane is a continuous map from the
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are formulated in terms of existence and (in the last case)
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allows idempotents to be lifted above the
Jacobson radical.
765:{\displaystyle F_{T}:\mathbb {R} \rightarrow \mathbb {R} }
805:{\displaystyle \pi :\mathbb {R} \rightarrow {\text{S}}}
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314:. For example, consider mapping opposite points on a
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Lifts are ubiquitous; for example, the definition of
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functional to lift simple operators to monadic form.
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545:are relegated to established domains and ranges of
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675:{\displaystyle T:{\text{S}}\rightarrow {\text{S}}}
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886:lifts p-adic varieties to characteristic zero.
630:denotes the identity relation on the range of
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1018:An Introduction to Chaotic Dynamical Systems
874:satisfies an infinitesimal lifting property.
278:; however, they might not always lift to an
850:{\displaystyle \pi \circ F_{T}=T\circ \pi }
772:, for which there exists a projection (or,
334:with continuous maps as morphisms, we have
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109:Learn how and when to remove this message
921:: Andrew Wiles (1995) modularity lifting
951:
557:to calculus of relations in their book
960:The Architecture of Modern Mathematics
282:. This leads to the definition of the
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47:adding citations to reliable sources
603:{\displaystyle M^{T};M\subseteq I}
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239:) and the valuative criteria of
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729:, is any map on the real line,
34:needs additional citations for
930:Monad (functional programming)
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571:corresponds to the inclusion
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332:category of topological spaces
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1020:, pp. 102-103, Addison-Wesley
646:). Given a map on a circle,
400: (projective plane path)
322:from the sphere covering the
994:Lecture Notes in Mathematics
884:Monsky–Washnitzer cohomology
1344:Constructions on categories
988:and Michael Winter (2018):
940:Tangent bundle § Lifts
539:first-order predicate logic
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1451:Higher-dimensional algebra
1016:Robert L. Devaney (1989):
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237:Homotopy lifting property
1261:Cokernels and quotients
1184:Universal constructions
968:Oxford University Press
903:of Siegel modular forms
58:"Lift" mathematics
1418:Higher category theory
1164:Natural transformation
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962:, J. Ferreiros &
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722:{\displaystyle F_{T}}
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533:Calculus of relations
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318:to the same point, a
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1287:Algebraic categories
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452: (covering map)
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43:improve this article
1456:Homotopy hypothesis
1134:Commutative diagram
990:Relational Topology
919:Arithmetic geometry
907:Saito–Kurokawa lift
872:Formally smooth map
559:Relational Topology
506: (sphere path)
298:A basic example in
264:homological algebra
135:commutative diagram
16:Term in mathematics
1169:Universal property
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260:algebraic topology
255:of certain lifts.
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1371:Quotient category
1366:Opposite category
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1006:978-3-319-74451-3
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867:Projective module
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695:{\displaystyle T}
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537:The notations of
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1441:Categorification
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1376:Product category
1361:Kleisli category
1356:Functor category
1201:Terminal objects
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1124:Adjoint functors
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531:Main article:
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320:continuous map
312:covering space
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294:Covering space
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280:exact sequence
268:tensor product
218:. We say that
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1519:Tetracategory
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1228:Inverse limit
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901:Miyawaki lift
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880:in categories
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328:unit interval
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302:is lifting a
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125:The morphism
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63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
1599:
1580:Categorified
1484:n-categories
1435:Key concepts
1273:Direct limit
1256:Coequalizers
1174:Yoneda lemma
1080:Key concepts
1070:Key concepts
1017:
1012:
989:
981:
959:
954:
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812:, such that
774:covering map
682:, a lift of
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99:January 2021
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
1548:-categories
1524:Kan complex
1514:Tricategory
1496:-categories
1386:Subcategory
1144:Exponential
1112:Preadditive
1107:Pre-abelian
996:vol. 2208,
966:, editors,
638:Circle maps
543:quantifiers
288:Ext functor
284:Tor functor
272:Hom functor
245:proper maps
147:mathematics
1566:3-category
1556:2-category
1529:∞-groupoid
1504:Bicategory
1251:Coproducts
1211:Equalizers
1117:Bicategory
946:References
896:Ikeda lift
253:uniqueness
233:fibrations
204:such that
149:, given a
69:newspapers
1615:Symmetric
1560:2-functor
1300:Relations
1223:Pullbacks
964:J.J. Gray
845:π
842:∘
823:∘
820:π
795:→
784:π
755:→
665:→
595:⊆
486:→
463::
427:→
411::
375:→
352::
241:separated
1690:Category
1675:Glossary
1655:Category
1629:n-monoid
1582:concepts
1238:Colimits
1206:Products
1159:Morphism
1102:Concrete
1097:Additive
1087:Category
890:SBI ring
861:See also
555:topology
300:topology
286:and the
270:and the
151:morphism
1665:Outline
1624:n-group
1589:2-group
1544:Strict
1534:∞-topos
1330:Modules
1268:Pushout
1216:Kernels
1149:Functor
1092:Abelian
306:in one
276:adjoint
249:schemes
182:lifting
83:scholar
1611:Traced
1594:2-ring
1324:Fields
1310:Groups
1305:Magmas
1193:Limits
1004:
973:
610:where
502:
448:
396:
316:sphere
85:
78:
71:
64:
56:
1605:-ring
1492:Weak
1476:Topos
1320:Rings
932:uses
235:(see
90:JSTOR
76:books
1295:Sets
1002:ISBN
971:ISBN
304:path
274:are
262:and
243:and
178:lift
176:, a
62:news
1139:End
1129:CCC
934:map
776:),
258:In
247:of
188:to
184:of
180:or
141:In
45:by
1692::
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1613:)(
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857:.
702:,
549:.
290:.
266:,
228:.
209:=
200:→
196::
172:→
168::
160:→
156::
1609:(
1602:n
1600:E
1562:)
1558:(
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1506:(
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1336:)
1332:(
1326:)
1322:(
1050:e
1043:t
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839:T
836:=
831:T
827:F
799:S
791:R
787::
759:R
751:R
747::
742:T
738:F
715:T
711:F
690:T
669:S
661:S
657::
654:T
632:M
618:I
598:I
592:M
589:;
584:T
580:M
569:M
494:2
490:S
483:]
480:1
477:,
474:0
471:[
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440:2
435:P
432:R
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408:g
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380:R
372:]
369:1
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360:[
349:f
226:h
220:f
215:h
213:∘
211:g
207:f
202:Z
198:X
194:h
190:Z
186:f
174:Y
170:Z
166:g
162:Y
158:X
154:f
137:)
133:(
131:f
127:h
112:)
106:(
101:)
97:(
87:·
80:·
73:·
66:·
39:.
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