20:
623:
1048:
946:
799:
344:
394:
693:
500:
526:
447:
980:
865:
834:
728:
653:
474:
421:
302:
537:
991:
502:
to the "smallest possible" simplicial set so that the resulting simplicial set contains no non-degenerate simplices in degrees
1122:
1100:
1056:
The above constructions work for more general categories (instead of sets) as well, provided that the category has
273:
The above definition of the skeleton of a simplicial complex is a particular case of the notion of skeleton of a
1053:(The boundary and degeneracy morphisms are given by various projections and diagonal embeddings, respectively.)
877:
736:
1195:
346:, together with face and degeneracy maps between them satisfying a number of equations. The idea of the
696:
1200:
307:
353:
165:
983:
658:
169:
479:
505:
426:
958:
843:
812:
706:
631:
452:
399:
280:
116:
40:
8:
1065:
79:
1069:
157:
89:
71:
51:
1168:
1118:
1096:
161:
153:
64:
1150:
24:
1088:
274:
145:
1171:
1189:
1138:
1117:, Progress in Mathematics, vol. 174, Basel, Boston, Berlin: Birkhäuser,
1061:
1057:
39:
This article is about mathematics. For the concept in computer graphics, see
1092:
618:{\displaystyle i_{*}:\Delta ^{op}Sets\rightarrow \Delta _{\leq n}^{op}Sets}
1142:
47:
75:
1176:
1043:{\displaystyle \dots \rightarrow K_{0}\times K_{0}\rightarrow K_{0}.}
32:
214:
198:
93:
1095:, Abstract Regular Polytopes, Cambridge University Press, 2002.
19:
1149:, Lecture Notes in Mathematics, No. 100, Berlin, New York:
994:
961:
880:
846:
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739:
709:
661:
634:
540:
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356:
310:
283:
1060:. The coskeleton is needed to define the concept of
1042:
974:
940:
859:
828:
793:
722:
687:
647:
617:
520:
494:
468:
441:
415:
388:
338:
296:
1166:
1187:
176:has infinite dimension, in the sense that the
1112:
264:(cube) = 8 vertices, 12 edges, 6 square faces
449:and then to complete the collection of the
1137:
955:is the constant simplicial set defined by
941:{\displaystyle cosk_{n}(K):=i^{!}i_{*}K.}
304:can be described by a collection of sets
982:. The 0-coskeleton is given by the Cech
531:More precisely, the restriction functor
18:
794:{\displaystyle sk_{n}(K):=i^{*}i_{*}K.}
172:. They are particularly important when
156:. The skeletons of a space are used in
1188:
1113:Goerss, P. G.; Jardine, J. F. (1999),
268:
1167:
277:. Briefly speaking, a simplicial set
218:P (functionally represented as skel
13:
583:
555:
14:
1212:
1160:
703:-skeleton of some simplicial set
951:For example, the 0-skeleton of
695:are comparable with the one of
339:{\displaystyle K_{i},\ i\geq 0}
126:is obtained by stopping at the
1131:
1106:
1082:
1024:
998:
906:
900:
759:
753:
579:
383:
370:
192:
136:These subspaces increase with
1:
1075:
804:
396:is to first discard the sets
389:{\displaystyle sk_{n}(K_{*})}
257:(cube) = 8 vertices, 12 edges
16:Concept in algebraic topology
628:has a left adjoint, denoted
235:elements of dimension up to
7:
688:{\displaystyle i^{*},i_{*}}
10:
1217:
1115:Simplicial Homotopy Theory
871:-coskeleton is defined as
697:image functors for sheaves
183:do not become constant as
38:
115:In other words, given an
168:, and generally to make
495:{\displaystyle i\leq n}
1044:
976:
942:
861:
830:
795:
724:
689:
649:
619:
522:
521:{\displaystyle i>n}
496:
470:
443:
442:{\displaystyle i>n}
417:
390:
340:
298:
36:
1045:
977:
975:{\displaystyle K_{0}}
943:
862:
860:{\displaystyle i^{!}}
831:
829:{\displaystyle i_{*}}
796:
725:
723:{\displaystyle K_{*}}
690:
650:
648:{\displaystyle i^{*}}
620:
523:
497:
471:
469:{\displaystyle K_{i}}
444:
418:
416:{\displaystyle K_{i}}
391:
341:
299:
297:{\displaystyle K_{*}}
22:
992:
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878:
844:
813:
737:
707:
659:
632:
538:
506:
480:
453:
427:
400:
354:
308:
281:
117:inductive definition
41:topological skeleton
1066:homotopical algebra
602:
269:For simplicial sets
250:(cube) = 8 vertices
228:)) consists of all
170:inductive arguments
1196:Algebraic topology
1169:Weisstein, Eric W.
1070:algebraic geometry
1040:
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466:
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386:
336:
294:
162:spectral sequences
158:obstruction theory
119:of a complex, the
72:simplicial complex
52:algebraic topology
50:, particularly in
37:
1124:978-3-7643-6064-1
655:. (The notations
326:
154:topological graph
65:topological space
1208:
1201:General topology
1182:
1181:
1154:
1153:
1135:
1129:
1128:, section IV.3.2
1127:
1110:
1104:
1086:
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104:) of dimensions
103:
100:(resp. cells of
99:
87:
78:) refers to the
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62:
59:
30:
1216:
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1151:Springer-Verlag
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184:
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177:
173:
160:, to construct
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137:
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105:
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70:presented as a
67:
57:
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44:
28:
25:hypercube graph
17:
12:
11:
5:
1214:
1204:
1203:
1198:
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1183:
1162:
1161:External links
1159:
1156:
1155:
1147:Etale homotopy
1139:Artin, Michael
1130:
1123:
1105:
1089:Peter McMullen
1080:
1079:
1077:
1074:
1058:fiber products
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275:simplicial set
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146:discrete space
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1101:0-521-81496-0
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1063:
1062:hypercovering
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242:For example:
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1143:Mazur, Barry
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1093:Egon Schulte
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185:
164:by means of
135:
110:
106:
88:that is the
56:
45:
193:In geometry
166:filtrations
48:mathematics
1190:Categories
1172:"Skeleton"
1076:References
809:Moreover,
805:Coskeleton
350:-skeleton
150:1-skeleton
148:, and the
142:0-skeleton
76:CW complex
29:1-skeleton
1177:MathWorld
1103:(Page 29)
1025:→
1012:×
999:→
996:⋯
928:∗
822:∗
781:∗
771:∗
716:∗
681:∗
668:∗
641:∗
588:≤
584:Δ
580:→
556:Δ
547:∗
487:≤
379:∗
331:≥
290:∗
233:-polytope
206:-skeleton
124:-skeleton
94:simplices
60:-skeleton
33:tesseract
1145:(1969),
840:adjoint
215:polytope
199:geometry
131:-th step
80:subspace
699:.) The
92:of the
74:(resp.
31:of the
27:is the
1121:
1099:
867:. The
836:has a
325:
140:. The
54:, the
984:nerve
838:right
476:with
423:with
144:is a
90:union
63:of a
23:This
1119:ISBN
1097:ISBN
1068:and
513:>
434:>
260:skel
253:skel
246:skel
201:, a
188:→ ∞.
1064:in
208:of
197:In
96:of
46:In
1192::
1174:.
1141:;
1091:,
1072:.
910::=
763::=
528:.
239:.
152:a
133:.
109:≤
1180:.
1038:.
1033:0
1029:K
1020:0
1016:K
1007:0
1003:K
968:0
964:K
953:K
936:.
933:K
924:i
918:!
914:i
907:)
904:K
901:(
896:n
892:k
888:s
885:o
882:c
869:n
853:!
849:i
818:i
789:.
786:K
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767:i
760:)
757:K
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637:i
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604:S
599:p
596:o
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574:t
571:e
568:S
563:p
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543:i
516:n
510:i
490:n
484:i
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458:K
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409:i
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375:K
371:(
366:n
362:k
358:s
348:n
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328:i
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211:n
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111:n
107:m
102:X
98:X
85:n
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68:X
58:n
43:.
35:.
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