236:
115:
546:
308:
834:
759:
699:
365:
68:
466:
618:
583:
663:
411:
727:
638:
857:
231:{\displaystyle \{S\in \operatorname {Ran} (U_{1}\cup \dots \cup U_{m})\mid S\cap U_{1}\neq \emptyset ,\dots ,S\cap U_{m}\neq \emptyset \}}
98:
936:
474:
1014:
250:
807:
732:
672:
338:
41:
1035:
987:
430:
372:
322:
378:
864:. Via this construction, one also obtains the topological chiral homology with coefficients in
368:
928:
592:
557:
920:
643:
549:
414:
314:
712:
623:
8:
912:
869:
785:
329:
972:
94:
932:
916:
90:
32:
1003:
968:
921:
881:
778:
762:
1029:
242:
83:
245:
20:
852:
is, roughly, a family of commutative algebras parametrized by points in
586:
71:
781:
777:
A theorem of
Beilinson and Drinfeld states that the Ran space of a
469:
977:
801:
109:
In general, the topology of the Ran space is generated by sets
75:
971:(2012). "Contractibility of the space of rational maps".
729:. A theorem of Beilinson and Drinfeld continues to hold:
995:
Tamagawa
Numbers via Nonabelian Poincare Duality (282y)
810:
735:
715:
675:
646:
626:
595:
560:
477:
433:
381:
341:
253:
118:
44:
836:, then its space of global sections is called the
828:
753:
721:
693:
657:
632:
612:
577:
540:
460:
405:
359:
302:
230:
62:
911:
1027:
988:"Homology and Cohomology of Stacks (Lecture 7)"
791:
225:
119:
868:. The construction is a generalization of
541:{\displaystyle (R,S,\mu )\to (R',S',\mu ')}
1019:Algebraic Topology: A Guide to Literature
976:
967:
927:. American Mathematical Society. p.
313:There is an analog of a Ran space for a
303:{\displaystyle U_{i}\subset X,i=1,...,m}
829:{\displaystyle \operatorname {Ran} (M)}
754:{\displaystyle \operatorname {Ran} (X)}
694:{\displaystyle \operatorname {Ran} (X)}
360:{\displaystyle \operatorname {Ran} (X)}
70:whose underlying set is the set of all
63:{\displaystyle \operatorname {Ran} (X)}
1028:
1001:
985:
951:
900:
413:consisting of a finitely generated
13:
1002:Lurie, Jacob (18 September 2017).
222:
191:
14:
1047:
986:Lurie, Jacob (19 February 2014).
1015:"Exponential space と Ran space"
945:
905:
894:
823:
817:
748:
742:
688:
682:
599:
564:
535:
502:
499:
496:
478:
461:{\displaystyle \mu :S\to X(R)}
455:
449:
443:
400:
382:
354:
348:
166:
134:
57:
51:
1:
961:
772:
104:
701:is a nonempty finite set of
97:. The notion is named after
7:
875:
838:topological chiral homology
792:Topological chiral homology
10:
1052:
406:{\displaystyle (R,S,\mu )}
887:
709:"with labels" given by
613:{\displaystyle S\to S'}
578:{\displaystyle R\to R'}
323:quasi-projective scheme
38:is a topological space
830:
755:
723:
695:
659:
634:
614:
579:
542:
462:
407:
361:
304:
232:
64:
844:with coefficients in
831:
756:
724:
696:
660:
658:{\displaystyle \mu '}
635:
615:
580:
553:-algebra homomorphism
543:
463:
408:
362:
305:
233:
65:
913:Beilinson, Alexander
808:
733:
722:{\displaystyle \mu }
713:
705:-rational points of
673:
644:
633:{\displaystyle \mu }
624:
593:
558:
475:
431:
379:
339:
251:
116:
42:
870:Hochschild homology
786:weakly contractible
620:that commutes with
1036:Topological spaces
954:, Theorem 5.5.3.11
917:Drinfeld, Vladimir
858:factorizable sheaf
856:, then there is a
826:
751:
719:
691:
655:
630:
610:
575:
538:
458:
427:and a map of sets
403:
357:
300:
228:
95:Hausdorff distance
93:is induced by the
60:
969:Gaitsgory, Dennis
804:on the Ran space
423:, a nonempty set
33:topological space
16:Topological space
1043:
1022:
1010:
1008:
1004:"Higher Algebra"
998:
992:
982:
980:
955:
949:
943:
942:
926:
909:
903:
898:
835:
833:
832:
827:
760:
758:
757:
752:
728:
726:
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720:
700:
698:
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692:
664:
662:
661:
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654:
639:
637:
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631:
619:
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611:
609:
584:
582:
581:
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574:
547:
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544:
539:
534:
523:
512:
467:
465:
464:
459:
412:
410:
409:
404:
366:
364:
363:
358:
309:
307:
306:
301:
263:
262:
237:
235:
234:
229:
218:
217:
187:
186:
165:
164:
146:
145:
69:
67:
66:
61:
1051:
1050:
1046:
1045:
1044:
1042:
1041:
1040:
1026:
1025:
1013:
1006:
990:
964:
959:
958:
950:
946:
939:
923:Chiral algebras
910:
906:
899:
895:
890:
882:Chiral homology
878:
809:
806:
805:
794:
775:
734:
731:
730:
714:
711:
710:
674:
671:
670:
647:
645:
642:
641:
625:
622:
621:
602:
594:
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567:
559:
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555:
527:
516:
505:
476:
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472:
432:
429:
428:
380:
377:
376:
340:
337:
336:
258:
254:
252:
249:
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213:
209:
182:
178:
160:
156:
141:
137:
117:
114:
113:
107:
43:
40:
39:
17:
12:
11:
5:
1049:
1039:
1038:
1024:
1023:
1011:
999:
983:
963:
960:
957:
956:
944:
937:
904:
892:
891:
889:
886:
885:
884:
877:
874:
860:associated to
825:
822:
819:
816:
813:
793:
790:
774:
771:
769:is connected.
750:
747:
744:
741:
738:
718:
690:
687:
684:
681:
678:
665:. Roughly, an
653:
650:
629:
608:
605:
601:
598:
573:
570:
566:
563:
537:
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530:
526:
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457:
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144:
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133:
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106:
103:
59:
56:
53:
50:
47:
15:
9:
6:
4:
3:
2:
1048:
1037:
1034:
1033:
1031:
1020:
1016:
1012:
1005:
1000:
996:
989:
984:
979:
974:
970:
966:
965:
953:
948:
940:
938:0-8218-3528-9
934:
930:
925:
924:
918:
914:
908:
902:
897:
893:
883:
880:
879:
873:
871:
867:
863:
859:
855:
851:
847:
843:
839:
820:
814:
811:
803:
799:
789:
787:
783:
780:
770:
768:
764:
745:
739:
736:
716:
708:
704:
685:
679:
676:
668:
651:
648:
627:
606:
603:
596:
588:
571:
568:
561:
554:
552:
548:consist of a
531:
528:
524:
520:
517:
513:
509:
506:
493:
490:
487:
484:
481:
471:
452:
446:
440:
437:
434:
426:
422:
419:
417:
397:
394:
391:
388:
385:
374:
370:
351:
345:
342:
335:, denoted by
334:
331:
327:
324:
320:
316:
311:
297:
294:
291:
288:
285:
282:
279:
276:
273:
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267:
264:
259:
255:
247:
244:
219:
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210:
206:
203:
200:
197:
194:
188:
183:
179:
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172:
169:
161:
157:
153:
150:
147:
142:
138:
131:
128:
125:
122:
112:
111:
110:
102:
100:
96:
92:
88:
85:
81:
77:
73:
54:
48:
45:
37:
34:
30:
26:
22:
1018:
994:
947:
922:
907:
896:
865:
861:
853:
849:
845:
841:
837:
797:
795:
776:
766:
706:
702:
666:
550:
468:, and whose
424:
420:
415:
375:are triples
332:
325:
319:Ran prestack
318:
312:
246:open subsets
240:
108:
86:
84:metric space
79:
35:
28:
24:
18:
29:Ran's space
21:mathematics
962:References
952:Lurie 2017
901:Lurie 2014
773:Properties
669:-point of
587:surjective
105:Definition
978:1108.1741
815:
779:connected
740:
717:μ
680:
649:μ
628:μ
600:→
565:→
529:μ
500:→
494:μ
470:morphisms
444:→
435:μ
398:μ
367:, is the
346:
265:⊂
223:∅
220:≠
207:∩
198:…
192:∅
189:≠
176:∩
170:∣
154:∪
151:⋯
148:∪
132:
126:∈
49:
25:Ran space
1030:Category
919:(2004).
876:See also
782:manifold
652:′
607:′
572:′
532:′
521:′
510:′
418:-algebra
369:category
243:disjoint
241:for any
91:topology
82:: for a
72:nonempty
1021:. 2018.
802:cosheaf
763:acyclic
373:objects
328:over a
99:Ziv Ran
76:subsets
74:finite
31:) of a
935:
585:and a
371:whose
317:: the
315:scheme
23:, the
1007:(PDF)
991:(PDF)
973:arXiv
888:Notes
848:. If
800:is a
330:field
321:of a
933:ISBN
640:and
589:map
89:the
27:(or
929:173
840:of
812:Ran
796:If
784:is
765:if
761:is
737:Ran
677:Ran
343:Ran
129:Ran
78:of
46:Ran
19:In
1032::
1017:.
993:.
931:.
915:;
872:.
788:.
310:.
101:.
1009:.
997:.
981:.
975::
941:.
866:A
862:A
854:M
850:A
846:F
842:M
824:)
821:M
818:(
798:F
767:X
749:)
746:X
743:(
707:X
703:R
689:)
686:X
683:(
667:R
604:S
597:S
569:R
562:R
551:k
536:)
525:,
518:S
514:,
507:R
503:(
497:)
491:,
488:S
485:,
482:R
479:(
456:)
453:R
450:(
447:X
441:S
438::
425:S
421:R
416:k
401:)
395:,
392:S
389:,
386:R
383:(
355:)
352:X
349:(
333:k
326:X
298:m
295:,
292:.
289:.
286:.
283:,
280:1
277:=
274:i
271:,
268:X
260:i
256:U
226:}
215:m
211:U
204:S
201:,
195:,
184:1
180:U
173:S
167:)
162:m
158:U
143:1
139:U
135:(
123:S
120:{
87:X
80:X
58:)
55:X
52:(
36:X
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