31:
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355:
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991:
876:
845:
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664:
485:
432:
313:
548:
1133:
1002:
1111:
513:
to the "smallest possible" simplicial set so that the resulting simplicial set contains no non-degenerate simplices in degrees
1206:
1067:
The above constructions work for more general categories (instead of sets) as well, provided that the category has
707:
1211:
284:
The above definition of the skeleton of a simplicial complex is a particular case of the notion of skeleton of a
1064:(The boundary and degeneracy morphisms are given by various projections and diagonal embeddings, respectively.)
888:
176:
747:
994:
180:
357:, together with face and degeneracy maps between them satisfying a number of equations. The idea of the
318:
127:
51:
1076:
364:
90:
1080:
669:
168:
100:
82:
62:
1179:
1129:
1107:
172:
164:
75:
490:
516:
437:
969:
854:
823:
717:
642:
463:
410:
291:
1161:
35:
1099:
285:
156:
1200:
1149:
1072:
1068:
1103:
1153:
58:
1182:
86:
1128:, Progress in Mathematics, vol. 174, Basel, Boston, Berlin: Birkhäuser,
50:
This article is about mathematics. For the concept in computer graphics, see
1187:
43:
629:{\displaystyle i_{*}:\Delta ^{op}Sets\rightarrow \Delta _{\leq n}^{op}Sets}
30:
17:
225:
209:
104:
1054:{\displaystyle \dots \rightarrow K_{0}\times K_{0}\rightarrow K_{0}.}
29:
1106:, Abstract Regular Polytopes, Cambridge University Press, 2002.
1160:, Lecture Notes in Mathematics, No. 100, Berlin, New York:
1005:
972:
891:
857:
826:
750:
720:
672:
645:
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519:
493:
466:
440:
413:
367:
321:
294:
1071:. The coskeleton is needed to define the concept of
1053:
985:
951:
870:
839:
804:
733:
698:
658:
628:
531:
505:
479:
452:
426:
399:
349:
307:
187:has infinite dimension, in the sense that the
275:(cube) = 8 vertices, 12 edges, 6 square faces
8:
460:and then to complete the collection of the
966:is the constant simplicial set defined by
1042:
1029:
1016:
1004:
977:
971:
952:{\displaystyle cosk_{n}(K):=i^{!}i_{*}K.}
937:
927:
905:
890:
862:
856:
831:
825:
790:
780:
758:
749:
725:
719:
690:
677:
671:
650:
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605:
597:
569:
556:
550:
518:
492:
471:
465:
439:
418:
412:
388:
375:
366:
326:
320:
315:can be described by a collection of sets
299:
293:
993:. The 0-coskeleton is given by the Cech
542:More precisely, the restriction functor
1092:
805:{\displaystyle sk_{n}(K):=i^{*}i_{*}K.}
183:. They are particularly important when
167:. The skeletons of a space are used in
1124:Goerss, P. G.; Jardine, J. F. (1999),
288:. Briefly speaking, a simplicial set
7:
229:P (functionally represented as skel
594:
566:
25:
714:-skeleton of some simplicial set
962:For example, the 0-skeleton of
706:are comparable with the one of
350:{\displaystyle K_{i},\ i\geq 0}
137:is obtained by stopping at the
1035:
1009:
917:
911:
770:
764:
590:
394:
381:
147:These subspaces increase with
1:
407:is to first discard the sets
400:{\displaystyle sk_{n}(K_{*})}
268:(cube) = 8 vertices, 12 edges
27:Concept in algebraic topology
639:has a left adjoint, denoted
246:elements of dimension up to
699:{\displaystyle i^{*},i_{*}}
1228:
1126:Simplicial Homotopy Theory
882:-coskeleton is defined as
708:image functors for sheaves
194:do not become constant as
49:
126:In other words, given an
179:, and generally to make
506:{\displaystyle i\leq n}
1055:
987:
953:
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532:{\displaystyle i>n}
507:
481:
454:
453:{\displaystyle i>n}
428:
401:
351:
309:
47:
1056:
988:
986:{\displaystyle K_{0}}
954:
873:
871:{\displaystyle i^{!}}
842:
840:{\displaystyle i_{*}}
807:
736:
734:{\displaystyle K_{*}}
701:
661:
659:{\displaystyle i^{*}}
631:
534:
508:
482:
480:{\displaystyle K_{i}}
455:
429:
427:{\displaystyle K_{i}}
402:
352:
310:
308:{\displaystyle K_{*}}
33:
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491:
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411:
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319:
292:
128:inductive definition
52:topological skeleton
1077:homotopical algebra
613:
280:For simplicial sets
261:(cube) = 8 vertices
239:)) consists of all
181:inductive arguments
1207:Algebraic topology
1180:Weisstein, Eric W.
1081:algebraic geometry
1051:
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477:
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424:
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305:
173:spectral sequences
169:obstruction theory
130:of a complex, the
83:simplicial complex
63:algebraic topology
61:, particularly in
48:
1135:978-3-7643-6064-1
666:. (The notations
337:
165:topological graph
76:topological space
16:(Redirected from
1219:
1212:General topology
1193:
1192:
1165:
1164:
1146:
1140:
1139:, section IV.3.2
1138:
1121:
1115:
1097:
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154:
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143:
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136:
134:
125:
115:) of dimensions
114:
111:(resp. cells of
110:
98:
89:) refers to the
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41:
21:
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1222:
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1162:Springer-Verlag
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171:, to construct
160:
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139:
138:
132:
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116:
112:
108:
97:
93:
81:presented as a
78:
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66:
55:
39:
36:hypercube graph
28:
23:
22:
15:
12:
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1225:
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1199:
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1173:
1172:External links
1170:
1167:
1166:
1158:Etale homotopy
1150:Artin, Michael
1141:
1134:
1116:
1100:Peter McMullen
1091:
1090:
1088:
1085:
1069:fiber products
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157:discrete space
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1073:hypercovering
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1154:Mazur, Barry
1144:
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1119:
1104:Egon Schulte
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175:by means of
146:
121:
117:
99:that is the
67:
56:
204:In geometry
177:filtrations
59:mathematics
1201:Categories
1183:"Skeleton"
1087:References
820:Moreover,
816:Coskeleton
361:-skeleton
161:1-skeleton
159:, and the
153:0-skeleton
87:CW complex
40:1-skeleton
18:0-skeleton
1188:MathWorld
1114:(Page 29)
1036:→
1023:×
1010:→
1007:⋯
939:∗
833:∗
792:∗
782:∗
727:∗
692:∗
679:∗
652:∗
599:≤
595:Δ
591:→
567:Δ
558:∗
498:≤
390:∗
342:≥
301:∗
244:-polytope
217:-skeleton
135:-skeleton
105:simplices
71:-skeleton
44:tesseract
1156:(1969),
851:adjoint
226:polytope
210:geometry
142:-th step
91:subspace
710:.) The
103:of the
85:(resp.
42:of the
38:is the
1132:
1110:
878:. The
847:has a
336:
151:. The
65:, the
995:nerve
849:right
487:with
434:with
155:is a
101:union
74:of a
34:This
1130:ISBN
1108:ISBN
1079:and
524:>
445:>
271:skel
264:skel
257:skel
212:, a
199:→ ∞.
1075:in
219:of
208:In
107:of
57:In
1203::
1185:.
1152:;
1102:,
1083:.
921::=
774::=
539:.
250:.
163:a
144:.
120:≤
1191:.
1049:.
1044:0
1040:K
1031:0
1027:K
1018:0
1014:K
979:0
975:K
964:K
947:.
944:K
935:i
929:!
925:i
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915:K
912:(
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903:k
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864:!
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778:i
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768:K
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615:S
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189:X
185:X
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122:n
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113:X
109:X
96:n
94:X
79:X
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46:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
