221:
792:
734:
876:
175:
478:
667:
125:
395:
614:
352:
428:
279:
318:. The bottom two squares are pullback squares and they are contained in the top diagram as well: the first one as a trapezoid and the second one as a two-square rectangle.
544:
515:
308:
207:
909:
741:
988:
965:
674:
799:
130:
433:
621:
90:
361:
1011:
936:
584:
325:
1006:
24:
400:
251:
233:
80:
31:, used to construct a topos of sheaves. A Lawvere–Tierney topology is also sometimes also called a
928:
520:
491:
284:
180:
891:
are essentially the same as
Lawvere–Tierney topologies on the topos of presheaves of sets over
984:
961:
947:
978:
955:
551:
924:
920:
905:
48:
1000:
974:
951:
56:
932:
983:, Oxford Logic Guides, vol. 21, Oxford University Press, p. 196,
787:{\displaystyle s\subseteq w\Longrightarrow {\bar {s}}\subseteq {\bar {w}}}
220:
729:{\displaystyle {\overline {s\cap w}}\equiv {\bar {s}}\cap {\bar {w}}}
957:
Sheaves in geometry and logic. A first introduction to topos theory
219:
28:
871:{\displaystyle {\overline {f^{-1}(s)}}\equiv f^{-1}({\bar {s}})}
917:
919:, vol. 1, Paris: Gauthier-Villars, pp. 329–334,
170:{\displaystyle j\circ \wedge =\wedge \circ (j\times j)}
473:{\displaystyle {\bar {s}}:{\bar {S}}\rightarrowtail A}
111:
101:
802:
744:
677:
624:
587:
523:
494:
436:
403:
364:
328:
287:
254:
183:
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93:
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662:{\displaystyle {\bar {s}}\equiv {\bar {\bar {s}}}}
661:
608:
538:
509:
472:
422:
389:
346:
302:
273:
201:
169:
120:{\displaystyle j\circ {\mbox{true}}={\mbox{true}}}
119:
390:{\displaystyle \chi _{s}:A\rightarrow \Omega }
8:
887:Grothendieck topologies on a small category
980:Elementary Categories, Elementary Toposes
854:
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92:
52:
609:{\displaystyle s\subseteq {\bar {s}}}
7:
347:{\displaystyle s:S\rightarrowtail A}
565:-closure are (for some subobjects
384:
281:is the characteristic morphism of
240:is the characteristic morphism of
14:
224:Commutative diagrams showing how
423:{\displaystyle j\circ \chi _{s}}
274:{\displaystyle j\circ \chi _{s}}
671:preservation of intersections:
71:is a topos, then a topology on
16:Analog of Grothendieck topology
865:
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228:-closure operates. Ω and
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1:
829:
691:
127:), preserves intersections (
47:. They were introduced by
1028:
960:, Universitext, Springer,
796:stability under pullback:
539:{\displaystyle {\bar {s}}}
510:{\displaystyle {\bar {s}}}
430:defines another subobject
303:{\displaystyle {\bar {s}}}
202:{\displaystyle j\circ j=j}
910:"Quantifiers and sheaves"
561:Some theorems related to
21:Lawvere–Tierney topology
738:preservation of order:
581:inflationary property:
397:, then the composition
872:
788:
730:
663:
610:
540:
511:
474:
424:
391:
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177:), and is idempotent (
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25:Grothendieck topology
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234:subobject classifier
181:
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91:
81:subobject classifier
49:William Lawvere
948:Mac Lane, Saunders
868:
784:
726:
659:
606:
546:is said to be the
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507:
488:is a subobject of
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420:
387:
344:
322:Given a subobject
320:
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271:
244:as a subobject of
199:
167:
117:
115:
105:
45:geometric modality
23:is an analog of a
19:In mathematics, a
1012:Closure operators
990:978-0-19-158949-2
967:978-1-4612-0927-0
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87:preserves truth (
83:Ω to Ω such that
27:for an arbitrary
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935:, archived from
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358:with classifier
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17:
12:
11:
5:
1025:
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1009:
999:
998:
995:
994:
989:
975:McLarty, Colin
971:
966:
952:Moerdijk, Ieke
944:
906:Lawvere, F. W.
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109:
99:
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75:is a morphism
64:
61:
33:local operator
15:
13:
10:
9:
6:
4:
3:
2:
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986:
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949:
945:
942:on 2018-03-17
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696:
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670:
645:
637:
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618:idempotence:
617:
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559:
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527:
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467:
455:
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411:
407:
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378:
375:
370:
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357:
354:of an object
341:
335:
332:
329:
317:
313:
310:which is the
291:
266:
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137:
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107:
97:
94:
86:
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78:
74:
70:
62:
60:
58:
57:Myles Tierney
54:
50:
46:
42:
38:
34:
30:
26:
22:
1007:Topos theory
979:
956:
937:the original
916:
892:
888:
886:
574:
570:
566:
562:
560:
555:
547:
485:
481:
355:
321:
315:
314:-closure of
311:
245:
241:
229:
225:
213:
84:
76:
72:
68:
66:
44:
40:
36:
32:
20:
18:
1001:Categories
899:References
484:such that
63:Definition
977:(1995) ,
954:(2012) ,
860:¯
843:−
835:≡
830:¯
812:−
779:¯
770:⊆
764:¯
755:⟹
749:⊆
721:¯
712:∩
706:¯
697:≡
692:¯
685:∩
654:¯
649:¯
638:≡
632:¯
601:¯
592:⊆
531:¯
502:¯
465:↣
459:¯
444:¯
412:χ
408:∘
385:Ω
382:→
367:χ
339:↣
295:¯
263:χ
259:∘
188:∘
159:×
150:∘
147:∧
141:∧
138:∘
98:∘
79:from the
908:(1971),
883:Examples
236:. χ
232:are the
216:-closure
41:topology
37:coverage
933:2337874
925:0430021
552:closure
51: (
987:
964:
931:
923:
517:, and
55:) and
940:(PDF)
929:S2CID
913:(PDF)
29:topos
985:ISBN
962:ISBN
569:and
248:and
113:true
103:true
53:1971
577:):
573:of
554:of
480:of
209:).
67:If
43:or
39:or
35:or
1003::
950:;
927:,
921:MR
915:,
895:.
558:.
59:.
893:C
889:C
878:.
866:)
857:s
851:(
846:1
839:f
826:)
823:s
820:(
815:1
808:f
776:w
761:s
752:w
746:s
718:w
703:s
688:w
682:s
646:s
629:s
598:s
589:s
575:A
571:w
567:s
563:j
556:s
550:-
548:j
528:s
499:s
486:s
482:A
468:A
456:S
450::
441:s
416:s
405:j
379:A
376::
371:s
356:A
342:A
336:S
333::
330:s
316:s
312:j
292:s
267:s
256:j
246:A
242:s
238:s
230:t
226:j
214:j
197:j
194:=
191:j
185:j
165:)
162:j
156:j
153:(
144:=
135:j
108:=
95:j
85:j
77:j
73:E
69:E
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