1086:
1625:
2992:
2181:
of simplicial sets there is a model category structure where the fibrations are precisely the Kan fibrations, cofibrations are all injective maps, and weak equivalences are simplicial maps which become homotopy equivalences after applying the geometric realization functor.
1394:
which has the same long exact sequence in singular homology associated to a cofibration, but it is not equivalent. The notion of cofibration is distinguished from these because its homotopy-theoretic definition is more amenable to formal analysis and generalization.
1075:
1163:
Topologists have long studied notions of "good subspace embedding", many of which imply that the map is a cofibration, or the converse, or have similar formal properties with regards to homology. In 1937, Borsuk proved that if
2912:
1344:
972:
2570:
2078:
2119:
2845:
2391:
2445:
959:
524:
409:
1979:
2307:
2179:
1789:
916:
878:
284:
179:
3034:
2615:
2346:
2268:
2236:
2147:
1932:
841:
748:
366:
2025:
1577:
243:
2514:
1545:
1510:
1232:
2900:
2772:
2647:
1740:
1661:
1428:
784:
211:
64:
454:
329:
2708:
1874:
576:
550:
1844:
1117:
2740:
1900:
2670:
1451:
713:
2868:
2469:
1999:
1809:
1708:
1688:
1617:
1597:
1475:
1384:
1364:
1292:
1272:
1252:
1206:
1182:
1141:
804:
596:
429:
304:
136:
116:
96:
2676:. In fact, if we work in just the category of topological spaces, the cofibrant replacement for any map from a point to a space forms a cofibrant replacement.
628:
652:
as a formal framework for doing homotopy theory in more general categories; a model category is endowed with three distinguished classes of morphisms called
1876:
is a cofibration). This follows from the previous fact and the fact that cofibrations are stable under pushout, because pushouts are the gluing maps to the
967:
2987:{\displaystyle {\underset {\to }{\text{hocolim}}}\left({\begin{matrix}X&\xrightarrow {f} &Y\\\downarrow &&\\\end{matrix}}\right)=C_{f}}
4067:
3258:
1619:
is a homotopy equivalence. This result can be summarized by saying that "every map is equivalent in the homotopy category to a cofibration."
4062:
3349:
3373:
3568:
1297:
3438:
3213:
3170:
3664:
3717:
3245:
4001:
3097:
2519:
2037:
3766:
2086:
3749:
3358:
2784:
2579:
Similar equivalences can be stated for deformation-retract pairs, and for neighborhood deformation-retract pairs.
2351:
2396:
3961:
3368:
4110:
3946:
3669:
3443:
3199:
1944:
3991:
75:
2195:, every cofibration is a closed inclusion (injective with closed image); the result also generalizes to
3996:
3966:
3674:
3630:
3611:
3378:
3322:
921:
642:
459:
371:
2273:
2160:
1748:
883:
251:
3533:
3398:
3000:
2573:
2203:
638:
2596:
2319:
2241:
2209:
2128:
1913:
1254:
has the homotopy extension property with respect to any absolute neighborhood retract. Likewise, if
721:
4115:
3918:
3783:
3475:
3317:
3615:
3585:
3509:
3499:
3455:
3285:
3238:
846:
141:
2004:
1550:
216:
3956:
3575:
3470:
3383:
3290:
3192:
3041:
2493:
1515:
1480:
1211:
809:
334:
2873:
2745:
2620:
1717:
1634:
1401:
757:
648:
Cofibrations are a fundamental concept of homotopy theory. Quillen has proposed the notion of
184:
37:
3605:
3600:
2687:
2149:. It follows that the cofibrant objects are the complexes whose objects are all projective.
1853:
555:
529:
3936:
3874:
3722:
3426:
3416:
3388:
3363:
3273:
2196:
1817:
1814:
A frequently used fact is that a cellular inclusion is a cofibration (so, for instance, if
1120:
1095:
8:
4074:
3756:
3634:
3619:
3548:
3307:
2717:
2487:
1879:
1430:
is a continuous map between topological spaces, there is an associated topological space
4047:
2652:
1433:
680:
434:
309:
4016:
3971:
3868:
3739:
3543:
3231:
2853:
2454:
2448:
1984:
1794:
1693:
1673:
1663:
factors as the composition of a cofibration and a homotopy equivalence which is also a
1602:
1582:
1460:
1369:
1349:
1277:
1257:
1237:
1191:
1167:
1126:
789:
581:
414:
289:
121:
101:
81:
25:
3553:
2471:
is a cofibration precisely when a mapping cylinder can be constructed for every space
601:
3951:
3931:
3926:
3833:
3744:
3558:
3538:
3393:
3332:
3209:
3176:
3166:
3103:
3093:
2122:
2028:
29:
4089:
3883:
3838:
3761:
3732:
3590:
3523:
3518:
3513:
3503:
3295:
3278:
3058:
3037:
2903:
2313:
1935:
1454:
3195: : chapter 6 defines and discusses cofibrations, and they are used throughout
1070:{\displaystyle {\begin{aligned}H(a,0)&=f(a)\\H'(x,0)&=f'(x)\end{aligned}}}
4032:
3941:
3771:
3727:
3493:
2192:
21:
3898:
3823:
3793:
3691:
3684:
3624:
3595:
3465:
3460:
3421:
3063:
2027:, then there is a model category structure where the weak equivalences are the
1624:
1147:
649:
4104:
4084:
3908:
3903:
3888:
3878:
3828:
3805:
3679:
3639:
3580:
3528:
3327:
3180:
3107:
4011:
4006:
3848:
3815:
3788:
3696:
3337:
3160:
3843:
3800:
3701:
3302:
3087:
17:
4079:
4037:
3863:
3776:
3408:
3312:
3223:
3203:
1085:
3893:
3858:
3563:
3450:
3053:
1664:
1631:
Arne Strøm has proved a strengthening of this result, that every map
634:
2943:
2870:. Homotopically, the cofiber acts as a homotopy cokernel of the map
2031:, the fibrations are the epimorphisms, and the cofibrations are maps
4057:
4052:
4042:
3433:
3254:
246:
2238:
is any (continuous) map (between compactly generated spaces), and
1791:
of the boundary sphere of a solid disk is a cofibration for every
1847:
1080:
We can encode this condition in the following commutative diagram
641:
with respect to all spaces; this is one instance of the broader
3649:
2576:
and thus induces maps to every space sensible in the diagram.
1346:
is an absolute neighborhood retract, then the inclusion of
1339:{\displaystyle A\times I\cup X\times {1}\subset X\times I}
1146:
For the notion of a cofibration in a model category, see
3146:
Arne Strøm, The homotopy category is a homotopy category
1579:
as pictured in the commutative diagram below. Moreover,
2931:
3003:
2915:
2876:
2856:
2787:
2748:
2720:
2690:
2655:
2623:
2599:
2522:
2496:
2457:
2399:
2354:
2322:
2276:
2244:
2212:
2163:
2131:
2089:
2040:
2007:
1987:
1947:
1916:
1882:
1856:
1820:
1797:
1751:
1720:
1696:
1676:
1637:
1605:
1585:
1553:
1518:
1483:
1463:
1436:
1404:
1372:
1352:
1300:
1280:
1260:
1240:
1214:
1194:
1170:
1129:
1098:
970:
924:
886:
849:
812:
792:
760:
724:
683:
664:
satisfying certain lifting and factorization axioms.
604:
584:
558:
532:
462:
437:
417:
374:
337:
312:
292:
254:
219:
187:
144:
124:
104:
84:
40:
2348:
and the embedding (at one end of the unit interval)
2083:
which are degreewise monic and the cokernel complex
3193:
Peter May, "A Concise Course in
Algebraic Topology"
2393:. That is, the mapping cylinder can be defined as
3158:
3028:
2986:
2894:
2862:
2839:
2766:
2734:
2702:
2664:
2641:
2609:
2583:
2564:
2508:
2463:
2439:
2385:
2340:
2301:
2262:
2230:
2173:
2141:
2113:
2072:
2019:
1993:
1973:
1926:
1894:
1868:
1838:
1803:
1783:
1734:
1702:
1682:
1655:
1611:
1591:
1571:
1539:
1504:
1469:
1445:
1422:
1386:is a cofibration. Hatcher's introductory textbook
1378:
1358:
1338:
1286:
1266:
1246:
1226:
1208:is normal, and its product with the unit interval
1200:
1176:
1135:
1111:
1069:
953:
910:
872:
835:
798:
778:
742:
707:
622:
590:
570:
544:
518:
448:
423:
403:
360:
323:
298:
278:
237:
205:
173:
130:
110:
90:
58:
4102:
2649:is not a cofibration, then the mapping cylinder
3220:Chapter 7 has many results not found elsewhere.
2902:. In fact, for pointed topological spaces, the
2565:{\displaystyle (A\times I)\cup (X\times \{0\})}
2206:of a cofibration is a cofibration. That is, if
118:is a cofibration if for each topological space
3085:
2073:{\displaystyle i:C_{\bullet }\to D_{\bullet }}
633:This definition is formally dual to that of a
3239:
1981:be the category of chain complexes which are
2831:
2825:
2556:
2550:
2114:{\displaystyle {\text{Coker}}(i)_{\bullet }}
2840:{\displaystyle C_{f}=M_{f}/(X\times \{0\})}
3246:
3232:
2386:{\displaystyle i_{0}\colon A\to A\times I}
1623:
1477:. There is a canonical subspace embedding
1084:
1234:is normal) then every closed subspace of
1143:equipped with the compact-open topology.
3253:
3092:. Chicago: University of Chicago Press.
2588:
2440:{\displaystyle Mi=X\cup _{i}(A\times I)}
2270:is a cofibration, then the induced map
78:with respect to all topological spaces
4103:
3135:Topics in Topology and Homotopy Theory
3089:A concise course in algebraic topology
3227:
1974:{\displaystyle C_{+}({\mathcal {A}})}
1905:
3154:
3152:
3081:
3079:
2316:can be understood as the pushout of
2166:
880:, we can extend a homotopy of maps
786:such that there is an extension to
637:, which is required to satisfy the
13:
2602:
2152:
2134:
1963:
1919:
750:of topological spaces is called a
672:
14:
4127:
3198:
3149:
3076:
2714:to be the induced quotient space
954:{\displaystyle H':X\times I\to S}
519:{\displaystyle h'(i(a),t)=h(a,t)}
404:{\displaystyle h':X\times I\to S}
2778:is defined as the quotient space
2302:{\displaystyle B\to B\cup _{g}X}
2174:{\displaystyle {\textbf {SSet}}}
1784:{\displaystyle S^{n-1}\to D^{n}}
911:{\displaystyle H:A\times I\to S}
279:{\displaystyle h:A\times I\to S}
3029:{\displaystyle X\to Y\to C_{f}}
2584:Constructions with cofibrations
3286:Differentiable/Smooth manifold
3140:
3127:
3114:
3013:
3007:
2997:In fact, the sequence of maps
2958:
2921:
2886:
2834:
2816:
2758:
2694:
2633:
2610:{\displaystyle {\mathcal {M}}}
2593:Note that in a model category
2559:
2541:
2535:
2523:
2434:
2422:
2371:
2341:{\displaystyle i\colon A\to X}
2332:
2280:
2263:{\displaystyle i\colon A\to X}
2254:
2231:{\displaystyle g\colon A\to B}
2222:
2142:{\displaystyle {\mathcal {A}}}
2102:
2095:
2057:
1968:
1958:
1927:{\displaystyle {\mathcal {A}}}
1860:
1833:
1821:
1768:
1726:
1647:
1531:
1493:
1414:
1158:
1060:
1054:
1036:
1024:
1009:
1003:
990:
978:
945:
902:
827:
770:
743:{\displaystyle i\colon A\to X}
734:
702:
690:
617:
605:
513:
501:
492:
483:
477:
471:
395:
352:
270:
197:
165:
138:, and for any continuous maps
50:
1:
3069:
2850:which is the mapping cone of
2486:), if and only if there is a
2185:
1690:with distinguished basepoint
667:
3044:in triangulated categories.
331:, there is a continuous map
7:
3992:Classification of manifolds
3165:. Berlin: Springer-Verlag.
3159:Quillen, Daniel G. (1967).
3047:
1390:uses a technical notion of
1294:and the subspace inclusion
1153:
873:{\displaystyle f'\circ i=f}
174:{\displaystyle f,f':A\to S}
76:homotopy extension property
10:
4132:
2679:
2020:{\displaystyle q<<0}
1572:{\displaystyle r\circ i=f}
715:denote the unit interval.
598:denotes the unit interval
238:{\displaystyle g\circ i=f}
4068:over commutative algebras
4025:
3984:
3917:
3814:
3710:
3657:
3648:
3484:
3407:
3346:
3266:
2509:{\displaystyle X\times I}
1938:with enough projectives.
1540:{\displaystyle r:Mf\to Y}
1505:{\displaystyle i:X\to Mf}
1227:{\displaystyle X\times I}
836:{\displaystyle f':X\to S}
806:, meaning there is a map
639:homotopy lifting property
361:{\displaystyle g':X\to S}
3784:Riemann curvature tensor
3036:comes equipped with the
2895:{\displaystyle f:X\to Y}
2767:{\displaystyle f:X\to Y}
2642:{\displaystyle i:*\to X}
2478:There is a cofibration (
1735:{\displaystyle {x}\to X}
1656:{\displaystyle f:X\to Y}
1423:{\displaystyle f:X\to Y}
1274:is a closed subspace of
779:{\displaystyle f:A\to S}
206:{\displaystyle g:X\to S}
59:{\displaystyle i:A\to X}
3086:May, J. Peter. (1999).
3576:Manifold with boundary
3291:Differential structure
3205:Topology and Groupoids
3042:distinguished triangle
3030:
2995:
2988:
2896:
2864:
2848:
2841:
2768:
2736:
2704:
2703:{\displaystyle A\to X}
2666:
2643:
2611:
2566:
2510:
2465:
2441:
2387:
2342:
2303:
2264:
2232:
2175:
2143:
2115:
2081:
2074:
2021:
1995:
1975:
1928:
1896:
1870:
1869:{\displaystyle A\to X}
1840:
1805:
1785:
1736:
1704:
1684:
1657:
1613:
1593:
1573:
1541:
1506:
1471:
1447:
1424:
1380:
1360:
1340:
1288:
1268:
1248:
1228:
1202:
1178:
1137:
1113:
1090:
1078:
1071:
955:
918:to a homotopy of maps
912:
874:
837:
800:
780:
744:
709:
643:EckmannâHilton duality
624:
592:
572:
571:{\displaystyle t\in I}
546:
545:{\displaystyle a\in A}
520:
450:
425:
405:
362:
325:
300:
280:
239:
207:
175:
132:
112:
92:
60:
3202:. "7. Cofibrations".
3031:
2989:
2908:
2897:
2865:
2842:
2780:
2769:
2737:
2705:
2674:cofibrant replacement
2667:
2644:
2612:
2589:Cofibrant replacement
2567:
2511:
2466:
2442:
2388:
2343:
2304:
2265:
2233:
2197:weak Hausdorff spaces
2176:
2144:
2116:
2075:
2033:
2022:
1996:
1976:
1929:
1897:
1871:
1841:
1839:{\displaystyle (X,A)}
1806:
1786:
1737:
1714:if the inclusion map
1705:
1685:
1658:
1614:
1599:is a cofibration and
1594:
1574:
1542:
1512:and a projection map
1507:
1472:
1448:
1425:
1381:
1361:
1341:
1289:
1269:
1249:
1229:
1203:
1179:
1138:
1114:
1112:{\displaystyle S^{I}}
1082:
1072:
963:
956:
913:
875:
838:
801:
781:
745:
710:
677:In what follows, let
625:
593:
573:
547:
521:
451:
426:
406:
363:
326:
301:
281:
240:
208:
176:
133:
113:
93:
61:
3723:Covariant derivative
3274:Topological manifold
3001:
2913:
2874:
2854:
2785:
2746:
2718:
2688:
2653:
2621:
2597:
2572:, since this is the
2520:
2494:
2455:
2397:
2352:
2320:
2274:
2242:
2210:
2161:
2129:
2087:
2038:
2005:
1985:
1945:
1914:
1880:
1854:
1818:
1795:
1749:
1718:
1694:
1674:
1670:A topological space
1635:
1603:
1583:
1551:
1516:
1481:
1461:
1434:
1402:
1370:
1350:
1298:
1278:
1258:
1238:
1212:
1192:
1168:
1127:
1096:
968:
922:
884:
847:
810:
790:
758:
722:
681:
602:
582:
556:
530:
460:
435:
415:
372:
335:
310:
290:
252:
217:
185:
142:
122:
102:
82:
38:
4111:Homotopical algebra
3757:Exterior derivative
3359:AtiyahâSinger index
3308:Riemannian manifold
3162:Homotopical algebra
2947:
2735:{\displaystyle X/A}
1895:{\displaystyle n-1}
4063:Secondary calculus
4017:Singularity theory
3972:Parallel transport
3740:De Rham cohomology
3379:Generalized Stokes
3122:Algebraic Topology
3040:which acts like a
3026:
2984:
2965:
2924:
2892:
2860:
2837:
2764:
2742:. In general, for
2732:
2700:
2684:For a cofibration
2665:{\displaystyle Mi}
2662:
2639:
2607:
2562:
2506:
2461:
2449:universal property
2437:
2383:
2338:
2299:
2260:
2228:
2171:
2139:
2123:projective objects
2111:
2070:
2029:quasi-isomorphisms
2017:
1991:
1971:
1924:
1906:In chain complexes
1892:
1866:
1836:
1801:
1781:
1745:The inclusion map
1742:is a cofibration.
1732:
1700:
1680:
1653:
1609:
1589:
1569:
1537:
1502:
1467:
1446:{\displaystyle Mf}
1443:
1420:
1388:Algebraic Topology
1376:
1356:
1336:
1284:
1264:
1244:
1224:
1198:
1174:
1133:
1109:
1067:
1065:
951:
908:
870:
833:
796:
776:
740:
708:{\displaystyle I=}
705:
620:
588:
568:
542:
516:
449:{\displaystyle g'}
446:
421:
401:
358:
324:{\displaystyle f'}
321:
296:
276:
235:
203:
171:
128:
108:
88:
56:
30:topological spaces
26:continuous mapping
4098:
4097:
3980:
3979:
3745:Differential form
3399:Whitney embedding
3333:Differential form
3215:978-1-4196-2722-4
3172:978-3-540-03914-3
2948:
2920:
2917:
2863:{\displaystyle f}
2464:{\displaystyle i}
2309:is a cofibration.
2168:
2093:
1994:{\displaystyle 0}
1804:{\displaystyle n}
1703:{\displaystyle x}
1683:{\displaystyle X}
1612:{\displaystyle r}
1592:{\displaystyle i}
1470:{\displaystyle f}
1379:{\displaystyle X}
1359:{\displaystyle A}
1287:{\displaystyle X}
1267:{\displaystyle A}
1247:{\displaystyle X}
1201:{\displaystyle X}
1177:{\displaystyle X}
1136:{\displaystyle S}
799:{\displaystyle X}
662:weak equivalences
591:{\displaystyle I}
424:{\displaystyle g}
299:{\displaystyle f}
131:{\displaystyle S}
111:{\displaystyle i}
91:{\displaystyle S}
4123:
4090:Stratified space
4048:FrĂŠchet manifold
3762:Interior product
3655:
3654:
3352:
3248:
3241:
3234:
3225:
3224:
3219:
3185:
3184:
3156:
3147:
3144:
3138:
3131:
3125:
3118:
3112:
3111:
3083:
3059:Homotopy colimit
3038:cofiber sequence
3035:
3033:
3032:
3027:
3025:
3024:
2993:
2991:
2990:
2985:
2983:
2982:
2970:
2966:
2963:
2962:
2939:
2925:
2918:
2904:homotopy colimit
2901:
2899:
2898:
2893:
2869:
2867:
2866:
2861:
2846:
2844:
2843:
2838:
2815:
2810:
2809:
2797:
2796:
2773:
2771:
2770:
2765:
2741:
2739:
2738:
2733:
2728:
2709:
2707:
2706:
2701:
2671:
2669:
2668:
2663:
2648:
2646:
2645:
2640:
2616:
2614:
2613:
2608:
2606:
2605:
2571:
2569:
2568:
2563:
2515:
2513:
2512:
2507:
2470:
2468:
2467:
2462:
2451:of the pushout,
2446:
2444:
2443:
2438:
2421:
2420:
2392:
2390:
2389:
2384:
2364:
2363:
2347:
2345:
2344:
2339:
2314:mapping cylinder
2308:
2306:
2305:
2300:
2295:
2294:
2269:
2267:
2266:
2261:
2237:
2235:
2234:
2229:
2193:Hausdorff spaces
2180:
2178:
2177:
2172:
2170:
2169:
2148:
2146:
2145:
2140:
2138:
2137:
2121:is a complex of
2120:
2118:
2117:
2112:
2110:
2109:
2094:
2091:
2079:
2077:
2076:
2071:
2069:
2068:
2056:
2055:
2026:
2024:
2023:
2018:
2000:
1998:
1997:
1992:
1980:
1978:
1977:
1972:
1967:
1966:
1957:
1956:
1936:Abelian category
1933:
1931:
1930:
1925:
1923:
1922:
1901:
1899:
1898:
1893:
1875:
1873:
1872:
1867:
1845:
1843:
1842:
1837:
1810:
1808:
1807:
1802:
1790:
1788:
1787:
1782:
1780:
1779:
1767:
1766:
1741:
1739:
1738:
1733:
1725:
1709:
1707:
1706:
1701:
1689:
1687:
1686:
1681:
1662:
1660:
1659:
1654:
1627:
1618:
1616:
1615:
1610:
1598:
1596:
1595:
1590:
1578:
1576:
1575:
1570:
1546:
1544:
1543:
1538:
1511:
1509:
1508:
1503:
1476:
1474:
1473:
1468:
1455:mapping cylinder
1452:
1450:
1449:
1444:
1429:
1427:
1426:
1421:
1385:
1383:
1382:
1377:
1365:
1363:
1362:
1357:
1345:
1343:
1342:
1337:
1323:
1293:
1291:
1290:
1285:
1273:
1271:
1270:
1265:
1253:
1251:
1250:
1245:
1233:
1231:
1230:
1225:
1207:
1205:
1204:
1199:
1183:
1181:
1180:
1175:
1142:
1140:
1139:
1134:
1118:
1116:
1115:
1110:
1108:
1107:
1088:
1076:
1074:
1073:
1068:
1066:
1053:
1023:
960:
958:
957:
952:
932:
917:
915:
914:
909:
879:
877:
876:
871:
857:
842:
840:
839:
834:
820:
805:
803:
802:
797:
785:
783:
782:
777:
749:
747:
746:
741:
714:
712:
711:
706:
629:
627:
626:
623:{\displaystyle }
621:
597:
595:
594:
589:
577:
575:
574:
569:
551:
549:
548:
543:
525:
523:
522:
517:
470:
455:
453:
452:
447:
445:
430:
428:
427:
422:
410:
408:
407:
402:
382:
367:
365:
364:
359:
345:
330:
328:
327:
322:
320:
305:
303:
302:
297:
285:
283:
282:
277:
244:
242:
241:
236:
212:
210:
209:
204:
180:
178:
177:
172:
158:
137:
135:
134:
129:
117:
115:
114:
109:
97:
95:
94:
89:
65:
63:
62:
57:
20:, in particular
4131:
4130:
4126:
4125:
4124:
4122:
4121:
4120:
4116:Homotopy theory
4101:
4100:
4099:
4094:
4033:Banach manifold
4026:Generalizations
4021:
3976:
3913:
3810:
3772:Ricci curvature
3728:Cotangent space
3706:
3644:
3486:
3480:
3439:Exponential map
3403:
3348:
3342:
3262:
3252:
3216:
3189:
3188:
3173:
3157:
3150:
3145:
3141:
3132:
3128:
3120:Edwin Spanier,
3119:
3115:
3100:
3084:
3077:
3072:
3050:
3020:
3016:
3002:
2999:
2998:
2978:
2974:
2964:
2961:
2955:
2954:
2949:
2937:
2930:
2926:
2916:
2914:
2911:
2910:
2875:
2872:
2871:
2855:
2852:
2851:
2811:
2805:
2801:
2792:
2788:
2786:
2783:
2782:
2747:
2744:
2743:
2724:
2719:
2716:
2715:
2689:
2686:
2685:
2682:
2654:
2651:
2650:
2622:
2619:
2618:
2601:
2600:
2598:
2595:
2594:
2591:
2586:
2521:
2518:
2517:
2495:
2492:
2491:
2456:
2453:
2452:
2416:
2412:
2398:
2395:
2394:
2359:
2355:
2353:
2350:
2349:
2321:
2318:
2317:
2290:
2286:
2275:
2272:
2271:
2243:
2240:
2239:
2211:
2208:
2207:
2188:
2165:
2164:
2162:
2159:
2158:
2155:
2153:Simplicial sets
2133:
2132:
2130:
2127:
2126:
2105:
2101:
2090:
2088:
2085:
2084:
2064:
2060:
2051:
2047:
2039:
2036:
2035:
2006:
2003:
2002:
1986:
1983:
1982:
1962:
1961:
1952:
1948:
1946:
1943:
1942:
1918:
1917:
1915:
1912:
1911:
1908:
1881:
1878:
1877:
1855:
1852:
1851:
1819:
1816:
1815:
1796:
1793:
1792:
1775:
1771:
1756:
1752:
1750:
1747:
1746:
1721:
1719:
1716:
1715:
1695:
1692:
1691:
1675:
1672:
1671:
1636:
1633:
1632:
1604:
1601:
1600:
1584:
1581:
1580:
1552:
1549:
1548:
1517:
1514:
1513:
1482:
1479:
1478:
1462:
1459:
1458:
1435:
1432:
1431:
1403:
1400:
1399:
1371:
1368:
1367:
1351:
1348:
1347:
1319:
1299:
1296:
1295:
1279:
1276:
1275:
1259:
1256:
1255:
1239:
1236:
1235:
1213:
1210:
1209:
1193:
1190:
1189:
1169:
1166:
1165:
1161:
1156:
1128:
1125:
1124:
1103:
1099:
1097:
1094:
1093:
1064:
1063:
1046:
1039:
1016:
1013:
1012:
993:
971:
969:
966:
965:
925:
923:
920:
919:
885:
882:
881:
850:
848:
845:
844:
813:
811:
808:
807:
791:
788:
787:
759:
756:
755:
754:if for any map
723:
720:
719:
682:
679:
678:
675:
673:Homotopy theory
670:
603:
600:
599:
583:
580:
579:
557:
554:
553:
531:
528:
527:
463:
461:
458:
457:
438:
436:
433:
432:
416:
413:
412:
375:
373:
370:
369:
368:and a homotopy
338:
336:
333:
332:
313:
311:
308:
307:
291:
288:
287:
253:
250:
249:
218:
215:
214:
186:
183:
182:
151:
143:
140:
139:
123:
120:
119:
103:
100:
99:
83:
80:
79:
39:
36:
35:
22:homotopy theory
12:
11:
5:
4129:
4119:
4118:
4113:
4096:
4095:
4093:
4092:
4087:
4082:
4077:
4072:
4071:
4070:
4060:
4055:
4050:
4045:
4040:
4035:
4029:
4027:
4023:
4022:
4020:
4019:
4014:
4009:
4004:
3999:
3994:
3988:
3986:
3982:
3981:
3978:
3977:
3975:
3974:
3969:
3964:
3959:
3954:
3949:
3944:
3939:
3934:
3929:
3923:
3921:
3915:
3914:
3912:
3911:
3906:
3901:
3896:
3891:
3886:
3881:
3871:
3866:
3861:
3851:
3846:
3841:
3836:
3831:
3826:
3820:
3818:
3812:
3811:
3809:
3808:
3803:
3798:
3797:
3796:
3786:
3781:
3780:
3779:
3769:
3764:
3759:
3754:
3753:
3752:
3742:
3737:
3736:
3735:
3725:
3720:
3714:
3712:
3708:
3707:
3705:
3704:
3699:
3694:
3689:
3688:
3687:
3677:
3672:
3667:
3661:
3659:
3652:
3646:
3645:
3643:
3642:
3637:
3627:
3622:
3608:
3603:
3598:
3593:
3588:
3586:Parallelizable
3583:
3578:
3573:
3572:
3571:
3561:
3556:
3551:
3546:
3541:
3536:
3531:
3526:
3521:
3516:
3506:
3496:
3490:
3488:
3482:
3481:
3479:
3478:
3473:
3468:
3466:Lie derivative
3463:
3461:Integral curve
3458:
3453:
3448:
3447:
3446:
3436:
3431:
3430:
3429:
3422:Diffeomorphism
3419:
3413:
3411:
3405:
3404:
3402:
3401:
3396:
3391:
3386:
3381:
3376:
3371:
3366:
3361:
3355:
3353:
3344:
3343:
3341:
3340:
3335:
3330:
3325:
3320:
3315:
3310:
3305:
3300:
3299:
3298:
3293:
3283:
3282:
3281:
3270:
3268:
3267:Basic concepts
3264:
3263:
3251:
3250:
3243:
3236:
3228:
3222:
3221:
3214:
3196:
3187:
3186:
3171:
3148:
3139:
3133:Garth Warner,
3126:
3124:, 1966, p. 57.
3113:
3098:
3074:
3073:
3071:
3068:
3067:
3066:
3064:Homotopy fiber
3061:
3056:
3049:
3046:
3023:
3019:
3015:
3012:
3009:
3006:
2981:
2977:
2973:
2969:
2960:
2957:
2956:
2953:
2950:
2946:
2942:
2938:
2936:
2933:
2932:
2929:
2923:
2891:
2888:
2885:
2882:
2879:
2859:
2836:
2833:
2830:
2827:
2824:
2821:
2818:
2814:
2808:
2804:
2800:
2795:
2791:
2763:
2760:
2757:
2754:
2751:
2731:
2727:
2723:
2710:we define the
2699:
2696:
2693:
2681:
2678:
2661:
2658:
2638:
2635:
2632:
2629:
2626:
2604:
2590:
2587:
2585:
2582:
2581:
2580:
2577:
2561:
2558:
2555:
2552:
2549:
2546:
2543:
2540:
2537:
2534:
2531:
2528:
2525:
2505:
2502:
2499:
2476:
2460:
2436:
2433:
2430:
2427:
2424:
2419:
2415:
2411:
2408:
2405:
2402:
2382:
2379:
2376:
2373:
2370:
2367:
2362:
2358:
2337:
2334:
2331:
2328:
2325:
2310:
2298:
2293:
2289:
2285:
2282:
2279:
2259:
2256:
2253:
2250:
2247:
2227:
2224:
2221:
2218:
2215:
2200:
2187:
2184:
2154:
2151:
2136:
2108:
2104:
2100:
2097:
2067:
2063:
2059:
2054:
2050:
2046:
2043:
2016:
2013:
2010:
1990:
1970:
1965:
1960:
1955:
1951:
1921:
1907:
1904:
1891:
1888:
1885:
1865:
1862:
1859:
1835:
1832:
1829:
1826:
1823:
1800:
1778:
1774:
1770:
1765:
1762:
1759:
1755:
1731:
1728:
1724:
1710:is said to be
1699:
1679:
1652:
1649:
1646:
1643:
1640:
1629:
1628:
1608:
1588:
1568:
1565:
1562:
1559:
1556:
1536:
1533:
1530:
1527:
1524:
1521:
1501:
1498:
1495:
1492:
1489:
1486:
1466:
1442:
1439:
1419:
1416:
1413:
1410:
1407:
1375:
1355:
1335:
1332:
1329:
1326:
1322:
1318:
1315:
1312:
1309:
1306:
1303:
1283:
1263:
1243:
1223:
1220:
1217:
1197:
1186:binormal space
1173:
1160:
1157:
1155:
1152:
1148:model category
1132:
1106:
1102:
1062:
1059:
1056:
1052:
1049:
1045:
1042:
1040:
1038:
1035:
1032:
1029:
1026:
1022:
1019:
1015:
1014:
1011:
1008:
1005:
1002:
999:
996:
994:
992:
989:
986:
983:
980:
977:
974:
973:
950:
947:
944:
941:
938:
935:
931:
928:
907:
904:
901:
898:
895:
892:
889:
869:
866:
863:
860:
856:
853:
832:
829:
826:
823:
819:
816:
795:
775:
772:
769:
766:
763:
739:
736:
733:
730:
727:
704:
701:
698:
695:
692:
689:
686:
674:
671:
669:
666:
650:model category
619:
616:
613:
610:
607:
587:
567:
564:
561:
541:
538:
535:
515:
512:
509:
506:
503:
500:
497:
494:
491:
488:
485:
482:
479:
476:
473:
469:
466:
444:
441:
420:
400:
397:
394:
391:
388:
385:
381:
378:
357:
354:
351:
348:
344:
341:
319:
316:
295:
275:
272:
269:
266:
263:
260:
257:
234:
231:
228:
225:
222:
202:
199:
196:
193:
190:
170:
167:
164:
161:
157:
154:
150:
147:
127:
107:
87:
74:if it has the
68:
67:
55:
52:
49:
46:
43:
9:
6:
4:
3:
2:
4128:
4117:
4114:
4112:
4109:
4108:
4106:
4091:
4088:
4086:
4085:Supermanifold
4083:
4081:
4078:
4076:
4073:
4069:
4066:
4065:
4064:
4061:
4059:
4056:
4054:
4051:
4049:
4046:
4044:
4041:
4039:
4036:
4034:
4031:
4030:
4028:
4024:
4018:
4015:
4013:
4010:
4008:
4005:
4003:
4000:
3998:
3995:
3993:
3990:
3989:
3987:
3983:
3973:
3970:
3968:
3965:
3963:
3960:
3958:
3955:
3953:
3950:
3948:
3945:
3943:
3940:
3938:
3935:
3933:
3930:
3928:
3925:
3924:
3922:
3920:
3916:
3910:
3907:
3905:
3902:
3900:
3897:
3895:
3892:
3890:
3887:
3885:
3882:
3880:
3876:
3872:
3870:
3867:
3865:
3862:
3860:
3856:
3852:
3850:
3847:
3845:
3842:
3840:
3837:
3835:
3832:
3830:
3827:
3825:
3822:
3821:
3819:
3817:
3813:
3807:
3806:Wedge product
3804:
3802:
3799:
3795:
3792:
3791:
3790:
3787:
3785:
3782:
3778:
3775:
3774:
3773:
3770:
3768:
3765:
3763:
3760:
3758:
3755:
3751:
3750:Vector-valued
3748:
3747:
3746:
3743:
3741:
3738:
3734:
3731:
3730:
3729:
3726:
3724:
3721:
3719:
3716:
3715:
3713:
3709:
3703:
3700:
3698:
3695:
3693:
3690:
3686:
3683:
3682:
3681:
3680:Tangent space
3678:
3676:
3673:
3671:
3668:
3666:
3663:
3662:
3660:
3656:
3653:
3651:
3647:
3641:
3638:
3636:
3632:
3628:
3626:
3623:
3621:
3617:
3613:
3609:
3607:
3604:
3602:
3599:
3597:
3594:
3592:
3589:
3587:
3584:
3582:
3579:
3577:
3574:
3570:
3567:
3566:
3565:
3562:
3560:
3557:
3555:
3552:
3550:
3547:
3545:
3542:
3540:
3537:
3535:
3532:
3530:
3527:
3525:
3522:
3520:
3517:
3515:
3511:
3507:
3505:
3501:
3497:
3495:
3492:
3491:
3489:
3483:
3477:
3474:
3472:
3469:
3467:
3464:
3462:
3459:
3457:
3454:
3452:
3449:
3445:
3444:in Lie theory
3442:
3441:
3440:
3437:
3435:
3432:
3428:
3425:
3424:
3423:
3420:
3418:
3415:
3414:
3412:
3410:
3406:
3400:
3397:
3395:
3392:
3390:
3387:
3385:
3382:
3380:
3377:
3375:
3372:
3370:
3367:
3365:
3362:
3360:
3357:
3356:
3354:
3351:
3347:Main results
3345:
3339:
3336:
3334:
3331:
3329:
3328:Tangent space
3326:
3324:
3321:
3319:
3316:
3314:
3311:
3309:
3306:
3304:
3301:
3297:
3294:
3292:
3289:
3288:
3287:
3284:
3280:
3277:
3276:
3275:
3272:
3271:
3269:
3265:
3260:
3256:
3249:
3244:
3242:
3237:
3235:
3230:
3229:
3226:
3217:
3211:
3207:
3206:
3201:
3200:Brown, Ronald
3197:
3194:
3191:
3190:
3182:
3178:
3174:
3168:
3164:
3163:
3155:
3153:
3143:
3136:
3130:
3123:
3117:
3109:
3105:
3101:
3099:0-226-51182-0
3095:
3091:
3090:
3082:
3080:
3075:
3065:
3062:
3060:
3057:
3055:
3052:
3051:
3045:
3043:
3039:
3021:
3017:
3010:
3004:
2994:
2979:
2975:
2971:
2967:
2951:
2944:
2940:
2934:
2927:
2907:
2905:
2889:
2883:
2880:
2877:
2857:
2847:
2828:
2822:
2819:
2812:
2806:
2802:
2798:
2793:
2789:
2779:
2777:
2761:
2755:
2752:
2749:
2729:
2725:
2721:
2713:
2697:
2691:
2677:
2675:
2659:
2656:
2636:
2630:
2627:
2624:
2578:
2575:
2553:
2547:
2544:
2538:
2532:
2529:
2526:
2503:
2500:
2497:
2489:
2485:
2481:
2477:
2474:
2458:
2450:
2431:
2428:
2425:
2417:
2413:
2409:
2406:
2403:
2400:
2380:
2377:
2374:
2368:
2365:
2360:
2356:
2335:
2329:
2326:
2323:
2315:
2311:
2296:
2291:
2287:
2283:
2277:
2257:
2251:
2248:
2245:
2225:
2219:
2216:
2213:
2205:
2201:
2198:
2194:
2190:
2189:
2183:
2157:The category
2150:
2124:
2106:
2098:
2080:
2065:
2061:
2052:
2048:
2044:
2041:
2032:
2030:
2014:
2011:
2008:
1988:
1953:
1949:
1939:
1937:
1903:
1889:
1886:
1883:
1863:
1857:
1849:
1830:
1827:
1824:
1812:
1798:
1776:
1772:
1763:
1760:
1757:
1753:
1743:
1729:
1722:
1713:
1697:
1677:
1668:
1666:
1650:
1644:
1641:
1638:
1626:
1622:
1621:
1620:
1606:
1586:
1566:
1563:
1560:
1557:
1554:
1534:
1528:
1525:
1522:
1519:
1499:
1496:
1490:
1487:
1484:
1464:
1456:
1440:
1437:
1417:
1411:
1408:
1405:
1396:
1393:
1389:
1373:
1353:
1333:
1330:
1327:
1324:
1320:
1316:
1313:
1310:
1307:
1304:
1301:
1281:
1261:
1241:
1221:
1218:
1215:
1195:
1187:
1171:
1151:
1149:
1144:
1130:
1122:
1104:
1100:
1089:
1087:
1081:
1077:
1057:
1050:
1047:
1043:
1041:
1033:
1030:
1027:
1020:
1017:
1006:
1000:
997:
995:
987:
984:
981:
975:
962:
948:
942:
939:
936:
933:
929:
926:
905:
899:
896:
893:
890:
887:
867:
864:
861:
858:
854:
851:
830:
824:
821:
817:
814:
793:
773:
767:
764:
761:
753:
737:
731:
728:
725:
716:
699:
696:
693:
687:
684:
665:
663:
659:
655:
651:
646:
645:in topology.
644:
640:
636:
631:
614:
611:
608:
585:
565:
562:
559:
539:
536:
533:
510:
507:
504:
498:
495:
489:
486:
480:
474:
467:
464:
442:
439:
418:
398:
392:
389:
386:
383:
379:
376:
355:
349:
346:
342:
339:
317:
314:
293:
273:
267:
264:
261:
258:
255:
248:
232:
229:
226:
223:
220:
200:
194:
191:
188:
168:
162:
159:
155:
152:
148:
145:
125:
105:
85:
77:
73:
53:
47:
44:
41:
34:
33:
32:
31:
27:
23:
19:
4012:Moving frame
4007:Morse theory
3997:Gauge theory
3854:
3789:Tensor field
3718:Closed/Exact
3697:Vector field
3665:Distribution
3606:Hypercomplex
3601:Quaternionic
3338:Vector field
3296:Smooth atlas
3204:
3161:
3142:
3137:, section 6.
3134:
3129:
3121:
3116:
3088:
2996:
2909:
2849:
2781:
2775:
2711:
2683:
2673:
2592:
2483:
2479:
2472:
2156:
2082:
2034:
1940:
1909:
1813:
1744:
1712:well-pointed
1711:
1669:
1630:
1397:
1391:
1387:
1185:
1162:
1145:
1091:
1083:
1079:
964:
751:
717:
676:
661:
658:cofibrations
657:
653:
647:
632:
98:. That is,
71:
69:
15:
3957:Levi-Civita
3947:Generalized
3919:Connections
3869:Lie algebra
3801:Volume form
3702:Vector flow
3675:Pushforward
3670:Lie bracket
3569:Lie algebra
3534:G-structure
3323:Pushforward
3303:Submanifold
2001:in degrees
1453:called the
1159:In topology
752:cofibration
72:cofibration
18:mathematics
4105:Categories
4080:Stratifold
4038:Diffeology
3834:Associated
3635:Symplectic
3620:Riemannian
3549:Hyperbolic
3476:Submersion
3384:HopfâRinow
3318:Submersion
3313:Smooth map
3070:References
2488:retraction
2447:. By the
2186:Properties
1941:If we let
1902:skeleton.
1547:such that
1121:path space
843:such that
668:Definition
654:fibrations
456:such that
245:, for any
3962:Principal
3937:Ehresmann
3894:Subbundle
3884:Principal
3859:Fibration
3839:Cotangent
3711:Covectors
3564:Lie group
3544:Hermitian
3487:manifolds
3456:Immersion
3451:Foliation
3389:Noether's
3374:Frobenius
3369:De Rham's
3364:Darboux's
3255:Manifolds
3181:294862881
3054:Fibration
3014:→
3008:→
2959:↓
2922:→
2887:→
2823:×
2759:→
2695:→
2634:→
2631:∗
2548:×
2539:∪
2530:×
2501:×
2429:×
2414:∪
2378:×
2372:→
2366::
2333:→
2327::
2288:∪
2281:→
2255:→
2249::
2223:→
2217::
2107:∙
2066:∙
2058:→
2053:∙
1887:−
1861:→
1769:→
1761:−
1727:→
1665:fibration
1648:→
1558:∘
1532:→
1494:→
1415:→
1392:good pair
1331:×
1325:⊂
1317:×
1311:∪
1305:×
1219:×
946:→
940:×
903:→
897:×
859:∘
828:→
771:→
735:→
729::
635:fibration
578:. (Here,
563:∈
537:∈
396:→
390:×
353:→
271:→
265:×
224:∘
198:→
166:→
51:→
4058:Orbifold
4053:K-theory
4043:Diffiety
3767:Pullback
3581:Oriented
3559:Kenmotsu
3539:Hadamard
3485:Types of
3434:Geodesic
3259:Glossary
3108:41266205
3048:See also
2941:→
2672:forms a
2012:<<
1154:Examples
1051:′
1021:′
930:′
855:′
818:′
526:for all
468:′
443:′
380:′
343:′
318:′
247:homotopy
156:′
28:between
4002:History
3985:Related
3899:Tangent
3877:)
3857:)
3824:Adjoint
3816:Bundles
3794:density
3692:Torsion
3658:Vectors
3650:Tensors
3633:)
3618:)
3614:,
3612:Pseudoâ
3591:Poisson
3524:Finsler
3519:Fibered
3514:Contact
3512:)
3504:Complex
3502:)
3471:Section
2919:hocolim
2776:cofiber
2712:cofiber
2680:Cofiber
2574:pushout
2204:pushout
1850:, then
1848:CW pair
1119:is the
961:, where
3967:Vector
3952:Koszul
3932:Cartan
3927:Affine
3909:Vector
3904:Tensor
3889:Spinor
3879:Normal
3875:Stable
3829:Affine
3733:bundle
3685:bundle
3631:Almost
3554:Kähler
3510:Almost
3500:Almost
3494:Closed
3394:Sard's
3350:(list)
3212:
3179:
3169:
3106:
3096:
2774:, the
1934:be an
1092:where
718:A map
4075:Sheaf
3849:Fiber
3625:Rizza
3596:Prime
3427:Local
3417:Curve
3279:Atlas
2490:from
2092:Coker
1846:is a
1366:into
1184:is a
411:from
286:from
213:with
70:is a
3942:Form
3844:Dual
3777:flow
3640:Tame
3616:Subâ
3529:Flat
3409:Maps
3210:ISBN
3177:OCLC
3167:ISBN
3104:OCLC
3094:ISBN
2312:The
2202:The
2191:For
2167:SSet
1910:Let
660:and
552:and
181:and
24:, a
3864:Jet
2617:if
2516:to
2125:in
1457:of
1398:If
1123:of
630:.)
431:to
306:to
16:In
4107::
3855:Co
3208:.
3175:.
3151:^
3102:.
3078:^
2906:of
2482:,
1811:.
1667:.
1150:.
656:,
3873:(
3853:(
3629:(
3610:(
3508:(
3498:(
3261:)
3257:(
3247:e
3240:t
3233:v
3218:.
3183:.
3110:.
3022:f
3018:C
3011:Y
3005:X
2980:f
2976:C
2972:=
2968:)
2952:Y
2945:f
2935:X
2928:(
2890:Y
2884:X
2881::
2878:f
2858:f
2835:)
2832:}
2829:0
2826:{
2820:X
2817:(
2813:/
2807:f
2803:M
2799:=
2794:f
2790:C
2762:Y
2756:X
2753::
2750:f
2730:A
2726:/
2722:X
2698:X
2692:A
2660:i
2657:M
2637:X
2628::
2625:i
2603:M
2560:)
2557:}
2554:0
2551:{
2545:X
2542:(
2536:)
2533:I
2527:A
2524:(
2504:I
2498:X
2484:X
2480:A
2475:.
2473:X
2459:i
2435:)
2432:I
2426:A
2423:(
2418:i
2410:X
2407:=
2404:i
2401:M
2381:I
2375:A
2369:A
2361:0
2357:i
2336:X
2330:A
2324:i
2297:X
2292:g
2284:B
2278:B
2258:X
2252:A
2246:i
2226:B
2220:A
2214:g
2199:.
2135:A
2103:)
2099:i
2096:(
2062:D
2049:C
2045::
2042:i
2015:0
2009:q
1989:0
1969:)
1964:A
1959:(
1954:+
1950:C
1920:A
1890:1
1884:n
1864:X
1858:A
1834:)
1831:A
1828:,
1825:X
1822:(
1799:n
1777:n
1773:D
1764:1
1758:n
1754:S
1730:X
1723:x
1698:x
1678:X
1651:Y
1645:X
1642::
1639:f
1607:r
1587:i
1567:f
1564:=
1561:i
1555:r
1535:Y
1529:f
1526:M
1523::
1520:r
1500:f
1497:M
1491:X
1488::
1485:i
1465:f
1441:f
1438:M
1418:Y
1412:X
1409::
1406:f
1374:X
1354:A
1334:I
1328:X
1321:1
1314:X
1308:I
1302:A
1282:X
1262:A
1242:X
1222:I
1216:X
1196:X
1188:(
1172:X
1131:S
1105:I
1101:S
1061:)
1058:x
1055:(
1048:f
1044:=
1037:)
1034:0
1031:,
1028:x
1025:(
1018:H
1010:)
1007:a
1004:(
1001:f
998:=
991:)
988:0
985:,
982:a
979:(
976:H
949:S
943:I
937:X
934::
927:H
906:S
900:I
894:A
891::
888:H
868:f
865:=
862:i
852:f
831:S
825:X
822::
815:f
794:X
774:S
768:A
765::
762:f
738:X
732:A
726:i
703:]
700:1
697:,
694:0
691:[
688:=
685:I
618:]
615:1
612:,
609:0
606:[
586:I
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560:t
540:A
534:a
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511:t
508:,
505:a
502:(
499:h
496:=
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490:t
487:,
484:)
481:a
478:(
475:i
472:(
465:h
440:g
419:g
399:S
393:I
387:X
384::
377:h
356:S
350:X
347::
340:g
315:f
294:f
274:S
268:I
262:A
259::
256:h
233:f
230:=
227:i
221:g
201:S
195:X
192::
189:g
169:S
163:A
160::
153:f
149:,
146:f
126:S
106:i
86:S
66:,
54:X
48:A
45::
42:i
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