1758:
There are three natural examples of acyclic classes, although doubtless others exist. The first is that of homotopy contractible complexes. The second is that of acyclic complexes. In functor categories (e.g. the category of all functors from topological spaces to abelian groups), there is a class of
1203:
as the abelian category. Free functors are projective objects in that category. The morphisms in the functor category are natural transformations, so the constructed chain maps and homotopies are all natural. The difference is that in the above version,
51:. They discovered that, when topologists were writing proofs to establish equivalence of various homology theories, there were numerous similarities in the processes. Eilenberg and MacLane then discovered the theorem to generalize this process.
1759:
complexes that are contractible on each object, but where the contractions might not be given by natural transformations. Another example is again in functor categories but this time the complexes are acyclic only at certain objects.
2286:
3090:
2130:
670:
351:
228:
850:
1416:
1201:
745:
1826:
does not necessarily have a calculus of either right or left fractions, it has weaker properties of having homotopy classes of both left and right fractions that permit forming the class
2599:
3499:
965:
928:
2382:
435:
1955:
1374:
129:
3214:
2709:
2638:
1460:
1341:
1249:
1130:
92:
3423:
is presentable and acyclic is not entirely straightforward and uses a detour through simplicial subdivision, which can also be handled using the above theorem). The class
1857:
584:
281:
1646:
1620:
3121:
888:
2988:
2670:
777:
3441:
3301:
3272:
2448:
2428:
2402:
2153:
1877:
1824:
1804:
1780:
1752:
1712:
1669:
1531:
1488:
1436:
377:
1156:
610:
307:
2816:
533:
404:
3519:
3421:
3401:
3381:
3361:
3341:
3321:
3234:
3181:
3161:
3141:
3008:
2952:
2928:
2908:
2888:
2864:
2836:
2789:
2769:
2749:
2729:
2528:
2508:
2488:
2468:
2330:
2310:
2039:
2015:
1995:
1975:
1920:
1900:
1732:
1692:
1594:
1574:
1551:
1511:
1309:
1289:
1269:
1222:
1090:
1066:
1028:
1005:
985:
501:
481:
455:
173:
149:
2161:
3016:
2047:
615:
3683:
315:
178:
785:
3648:
1783:
1379:
1164:
3600:
3567:
686:
2533:
3446:
933:
1048:
What is above is one of the earliest versions of the theorem. Another version is the one that says that if
893:
3678:
55:
2339:
409:
1925:
1346:
101:
3186:
2675:
2604:
3591:, A Series of Comprehensive Studies in Mathematics, vol. 200 (2nd ed.), Berlin, New York:
1441:
1322:
1230:
1095:
73:
1829:
538:
240:
1625:
1599:
3099:
855:
2931:
681:
95:
2961:
2643:
750:
3426:
3277:
3239:
2433:
2413:
2387:
2138:
1862:
1809:
1789:
1765:
1737:
1697:
1654:
1516:
1473:
1421:
356:
3524:
There are many other examples in both algebra and topology, some of which are described in
1135:
589:
286:
2794:
506:
382:
152:
8:
2955:
1291:
being acyclic at the models (there is only one) means nothing else than that the complex
3504:
3406:
3386:
3366:
3346:
3326:
3306:
3219:
3166:
3146:
3126:
2993:
2937:
2913:
2893:
2873:
2867:
2849:
2821:
2774:
2754:
2734:
2714:
2513:
2493:
2473:
2453:
2315:
2295:
2024:
2000:
1980:
1960:
1905:
1885:
1717:
1677:
1579:
1559:
1536:
1496:
1294:
1274:
1254:
1207:
1075:
1051:
1013:
990:
970:
486:
466:
440:
158:
134:
20:
3644:
3596:
3563:
231:
32:
3555:
2450:
the corresponding class of arrows in the category of chain complexes. Suppose that
1224:
being acyclic is a stronger assumption than being acyclic only at certain objects.
1069:
48:
44:
3620:
2281:{\displaystyle \sum (-1)^{i}K_{n}G^{i}\epsilon G^{m-i}:K_{n}G^{m+1}\to K_{n}G^{m}}
3592:
3085:{\displaystyle \sum _{n\geq 0}\sum _{{\textrm {Hom}}(\Delta _{n},X)}\Delta _{n}}
1031:
59:
1161:
This specializes almost to the above theorem if one uses the functor category
3672:
3584:
1227:
On the other hand, the above version almost implies this version by letting
3501:
and so we conclude that singular and simplicial homology are isomorphic on
458:
24:
36:
2125:{\displaystyle \cdots K_{n}G^{m+1}\to K_{n}G^{m}\to \cdots \to K_{n}}
3443:
is the class of homology equivalences. It is rather obvious that
40:
665:{\displaystyle M\in {\mathcal {M}}_{k}\cup {\mathcal {M}}_{k+1}}
1319:
There is a grand theorem that unifies both of the above. Let
16:
Generalizes showing that two homology theories are isomorphic
346:{\displaystyle {\mathcal {M}}_{k}\subseteq {\mathcal {K}}}
2711:
up to chain homotopies. If we suppose, in addition, that
2846:
Here is an example of this last theorem in action. Let
1271:
is basically just a free (and hence projective) module.
3539:
S. Eilenberg and S. Mac Lane (1953), "Acyclic Models."
1782:
denote the class of chain maps between complexes whose
1251:
a category with only one object. Then the free functor
223:{\displaystyle F,V:{\mathcal {K}}\to {\mathcal {C}}(R)}
3507:
3449:
3429:
3409:
3389:
3369:
3349:
3329:
3309:
3280:
3242:
3222:
3189:
3169:
3149:
3129:
3102:
3019:
2996:
2964:
2940:
2916:
2896:
2876:
2852:
2824:
2797:
2777:
2757:
2737:
2717:
2678:
2646:
2607:
2536:
2516:
2496:
2476:
2456:
2436:
2416:
2390:
2342:
2318:
2298:
2164:
2141:
2050:
2027:
2003:
1983:
1963:
1928:
1908:
1888:
1865:
1832:
1812:
1792:
1768:
1740:
1720:
1700:
1680:
1657:
1628:
1602:
1582:
1562:
1539:
1519:
1499:
1476:
1444:
1424:
1382:
1349:
1325:
1297:
1277:
1257:
1233:
1210:
1167:
1138:
1098:
1092:
is an acyclic complex in that category, then any map
1078:
1054:
1016:
993:
973:
936:
896:
858:
788:
753:
689:
618:
592:
541:
509:
489:
469:
443:
412:
385:
359:
318:
289:
243:
181:
161:
137:
104:
76:
2890:
be the category of abelian group valued functors on
1043:
845:{\displaystyle \varphi ,\psi :H_{0}(F)\to H_{0}(V)}
3513:
3493:
3435:
3415:
3395:
3375:
3355:
3335:
3315:
3295:
3266:
3228:
3208:
3175:
3155:
3135:
3115:
3084:
3002:
2982:
2946:
2922:
2902:
2882:
2858:
2830:
2810:
2783:
2763:
2743:
2723:
2703:
2664:
2632:
2593:
2522:
2502:
2482:
2462:
2442:
2422:
2396:
2376:
2324:
2304:
2280:
2147:
2124:
2033:
2009:
1989:
1969:
1949:
1922:, meaning there is given a natural transformation
1914:
1894:
1871:
1851:
1818:
1798:
1774:
1746:
1726:
1706:
1686:
1663:
1640:
1614:
1588:
1568:
1545:
1525:
1505:
1482:
1454:
1430:
1410:
1368:
1335:
1303:
1283:
1263:
1243:
1216:
1195:
1150:
1124:
1084:
1060:
1022:
999:
979:
959:
922:
882:
844:
771:
739:
664:
604:
578:
527:
495:
475:
449:
429:
398:
371:
345:
301:
275:
222:
167:
143:
123:
86:
967:, then there is a natural chain homotopy between
3670:
1411:{\displaystyle {\mathcal {C}}(R)^{\mathcal {K}}}
1196:{\displaystyle {\mathcal {C}}(R)^{\mathcal {K}}}
2640:to a natural transformation of chain functors
1694:is a double complex, all of whose rows are in
740:{\displaystyle \varphi :H_{0}(F)\to H_{0}(V)}
3615:
3613:
3611:
3631:
3629:
535:-acyclic at these models, which means that
65:
2990:be the functor that assigns to each space
2594:{\displaystyle f_{0}:H_{0}(K)\to H_{0}(L)}
2530:-acyclic. Then any natural transformation
3658:Schon, R. "Acyclic models and excision."
3608:
3626:
3494:{\displaystyle H_{0}(K)\simeq H_{0}(L)}
960:{\displaystyle M\in {\mathcal {M}}_{0}}
3671:
2292:We say that the chain complex functor
923:{\displaystyle \varphi ^{M}=\psi ^{M}}
131:be the category of chain complexes of
3560:An Introduction to Algebraic Topology
3143:-simplex and this functor assigns to
1343:be an abelian category (for example,
890:are natural chain maps as before and
3583:
2155:. The boundary operator is given by
676:Then the following assertions hold:
3274:. There is an obvious augmentation
13:
3430:
3191:
3163:the sum of as many copies of each
3104:
3073:
3052:
2680:
2609:
2437:
2417:
2391:
2377:{\displaystyle L\to H_{0}(L)\to 0}
2142:
1866:
1859:gotten by inverting the arrows in
1834:
1813:
1793:
1769:
1741:
1701:
1658:
1635:
1609:
1520:
1477:
1447:
1425:
1402:
1385:
1352:
1328:
1314:
1236:
1187:
1170:
1068:is a complex of projectives in an
1038:
946:
645:
628:
430:{\displaystyle {\mathcal {M}}_{k}}
416:
338:
322:
206:
196:
107:
79:
14:
3695:
1977:). We say that the chain complex
1950:{\displaystyle \epsilon :G\to Id}
1533:if and only if the suspension of
1369:{\displaystyle {\mathcal {C}}(R)}
124:{\displaystyle {\mathcal {C}}(R)}
3643:, American Mathematical Society
3209:{\displaystyle \Delta _{n}\to X}
1044:Projective and acyclic complexes
58:; this leads to the idea of the
2868:category of triangulable spaces
2704:{\displaystyle \Sigma ^{-1}(C)}
2633:{\displaystyle \Sigma ^{-1}(C)}
2336:if the augmented chain complex
1902:be an augmented endofunctor on
3684:Theorems in algebraic topology
3589:Lectures on Algebraic Topology
3577:
3549:
3533:
3488:
3482:
3466:
3460:
3287:
3252:
3246:
3200:
3067:
3048:
2974:
2698:
2692:
2656:
2627:
2621:
2588:
2582:
2569:
2566:
2560:
2368:
2365:
2359:
2346:
2255:
2178:
2168:
2109:
2103:
2080:
1938:
1455:{\displaystyle {\mathcal {A}}}
1397:
1390:
1363:
1357:
1336:{\displaystyle {\mathcal {A}}}
1244:{\displaystyle {\mathcal {K}}}
1182:
1175:
1142:
1125:{\displaystyle K_{0}\to L_{0}}
1109:
874:
839:
833:
820:
817:
811:
763:
734:
728:
715:
712:
706:
567:
564:
558:
552:
522:
510:
217:
211:
201:
118:
112:
87:{\displaystyle {\mathcal {K}}}
1:
3527:
1852:{\displaystyle \Sigma ^{-1}C}
852:are natural transformations,
579:{\displaystyle H_{k}(V(M))=0}
276:{\displaystyle F_{i}=V_{i}=0}
43:was developed by topologists
31:can be used to show that two
3323:. It can be shown that both
1714:, then the total complex of
1641:{\displaystyle L\in \Gamma }
1615:{\displaystyle K\in \Gamma }
1010:In particular the chain map
747:induces a natural chain map
54:It can be used to prove the
7:
3183:-simplex as there are maps
3116:{\displaystyle \Delta _{n}}
10:
3700:
2841:
883:{\displaystyle f,g:F\to V}
3403:-acyclic (the proof that
2838:is homotopy equivalence.
2601:extends, in the category
1957:(the identity functor on
1158:, unique up to homotopy.
3303:and this induces one on
2983:{\displaystyle E:X\to X}
2956:simplicial chain complex
2818:is an isomorphism, then
2665:{\displaystyle f:K\to L}
2430:be an acyclic class and
1438:of chain complexes over
1030:is unique up to natural
772:{\displaystyle f:F\to V}
66:Statement of the theorem
56:Eilenberg–Zilber theorem
3665:(1) (1976) pp.167--168.
3572:See chapter 9, thm 9.12
3562:(1988) Springer-Verlag
3436:{\displaystyle \Gamma }
3296:{\displaystyle EX\to X}
3267:{\displaystyle G(C)=CE}
2443:{\displaystyle \Sigma }
2423:{\displaystyle \Gamma }
2397:{\displaystyle \Gamma }
2148:{\displaystyle \Gamma }
1872:{\displaystyle \Sigma }
1819:{\displaystyle \Sigma }
1799:{\displaystyle \Gamma }
1775:{\displaystyle \Sigma }
1747:{\displaystyle \Gamma }
1707:{\displaystyle \Gamma }
1664:{\displaystyle \Gamma }
1526:{\displaystyle \Gamma }
1483:{\displaystyle \Gamma }
1431:{\displaystyle \Gamma }
1132:extends to a chain map
372:{\displaystyle k\geq 0}
3660:Proc. Amer. Math. Soc.
3515:
3495:
3437:
3417:
3397:
3377:
3357:
3337:
3317:
3297:
3268:
3230:
3210:
3177:
3157:
3137:
3117:
3086:
3004:
2984:
2948:
2932:singular chain complex
2924:
2904:
2884:
2860:
2832:
2812:
2785:
2765:
2745:
2725:
2705:
2672:and this is unique in
2666:
2634:
2595:
2524:
2504:
2484:
2464:
2444:
2424:
2398:
2378:
2326:
2306:
2282:
2149:
2126:
2035:
2011:
1991:
1971:
1951:
1916:
1896:
1873:
1853:
1820:
1800:
1776:
1748:
1728:
1708:
1688:
1665:
1642:
1616:
1590:
1570:
1547:
1527:
1507:
1484:
1456:
1432:
1412:
1370:
1337:
1305:
1285:
1265:
1245:
1218:
1197:
1152:
1151:{\displaystyle K\to L}
1126:
1086:
1062:
1024:
1001:
981:
961:
924:
884:
846:
773:
741:
682:natural transformation
666:
606:
605:{\displaystyle k>0}
580:
529:
497:
477:
451:
431:
400:
373:
347:
303:
302:{\displaystyle i<0}
277:
224:
169:
145:
125:
88:
29:acyclic models theorem
23:, a discipline within
3639:(2002) CRM monograph
3516:
3496:
3438:
3418:
3398:
3378:
3358:
3338:
3318:
3298:
3269:
3231:
3211:
3178:
3158:
3138:
3118:
3087:
3005:
2985:
2949:
2925:
2905:
2885:
2861:
2833:
2813:
2811:{\displaystyle f_{0}}
2786:
2766:
2746:
2726:
2706:
2667:
2635:
2596:
2525:
2505:
2485:
2465:
2445:
2425:
2399:
2379:
2327:
2307:
2283:
2150:
2127:
2036:
2012:
1992:
1972:
1952:
1917:
1897:
1874:
1854:
1821:
1801:
1777:
1749:
1729:
1709:
1689:
1666:
1643:
1617:
1591:
1571:
1548:
1528:
1508:
1485:
1457:
1433:
1413:
1371:
1338:
1306:
1286:
1266:
1246:
1219:
1198:
1153:
1127:
1087:
1063:
1025:
1002:
982:
962:
925:
885:
847:
774:
742:
667:
607:
581:
530:
528:{\displaystyle (k+1)}
498:
478:
452:
432:
401:
399:{\displaystyle F_{k}}
374:
348:
304:
278:
225:
170:
146:
126:
89:
3505:
3447:
3427:
3407:
3387:
3367:
3347:
3327:
3307:
3278:
3240:
3220:
3187:
3167:
3147:
3127:
3100:
3017:
2994:
2962:
2938:
2914:
2894:
2874:
2850:
2822:
2795:
2775:
2755:
2735:
2715:
2676:
2644:
2605:
2534:
2514:
2494:
2474:
2454:
2434:
2414:
2388:
2340:
2316:
2296:
2162:
2139:
2048:
2041:, the chain complex
2025:
2001:
1981:
1961:
1926:
1906:
1886:
1863:
1830:
1810:
1790:
1766:
1738:
1718:
1698:
1678:
1655:
1626:
1600:
1580:
1560:
1537:
1517:
1497:
1474:
1470:The 0 complex is in
1442:
1422:
1380:
1347:
1323:
1295:
1275:
1255:
1231:
1208:
1165:
1136:
1096:
1076:
1052:
1014:
991:
971:
934:
894:
856:
786:
751:
687:
616:
590:
539:
507:
487:
467:
441:
410:
383:
357:
316:
287:
241:
179:
159:
135:
102:
74:
3679:Homological algebra
2791:-acyclic, and that
2751:-presentable, that
3511:
3491:
3433:
3413:
3393:
3373:
3353:
3333:
3313:
3293:
3264:
3226:
3206:
3173:
3153:
3133:
3113:
3082:
3071:
3035:
3000:
2980:
2944:
2920:
2900:
2880:
2856:
2828:
2808:
2781:
2761:
2741:
2721:
2701:
2662:
2630:
2591:
2520:
2500:
2480:
2460:
2440:
2420:
2394:
2374:
2322:
2302:
2278:
2145:
2122:
2031:
2007:
1987:
1967:
1947:
1912:
1892:
1869:
1849:
1816:
1796:
1772:
1744:
1724:
1704:
1684:
1661:
1638:
1612:
1596:are homotopic and
1586:
1566:
1543:
1523:
1503:
1480:
1462:will be called an
1452:
1428:
1408:
1366:
1333:
1301:
1281:
1261:
1241:
1214:
1193:
1148:
1122:
1082:
1058:
1020:
997:
977:
957:
920:
880:
842:
769:
737:
662:
602:
576:
525:
493:
473:
447:
427:
396:
369:
343:
299:
273:
232:covariant functors
220:
165:
141:
121:
84:
21:algebraic topology
3514:{\displaystyle X}
3416:{\displaystyle L}
3396:{\displaystyle G}
3383:-presentable and
3376:{\displaystyle G}
3356:{\displaystyle L}
3336:{\displaystyle K}
3316:{\displaystyle G}
3229:{\displaystyle G}
3176:{\displaystyle n}
3156:{\displaystyle X}
3136:{\displaystyle n}
3045:
3036:
3020:
3003:{\displaystyle X}
2947:{\displaystyle L}
2923:{\displaystyle K}
2903:{\displaystyle X}
2883:{\displaystyle C}
2859:{\displaystyle X}
2831:{\displaystyle f}
2784:{\displaystyle G}
2764:{\displaystyle K}
2744:{\displaystyle G}
2724:{\displaystyle L}
2523:{\displaystyle G}
2503:{\displaystyle L}
2490:-presentable and
2483:{\displaystyle G}
2463:{\displaystyle K}
2325:{\displaystyle G}
2305:{\displaystyle L}
2034:{\displaystyle n}
2010:{\displaystyle G}
1990:{\displaystyle K}
1970:{\displaystyle C}
1915:{\displaystyle C}
1895:{\displaystyle G}
1727:{\displaystyle D}
1687:{\displaystyle D}
1651:Every complex in
1589:{\displaystyle L}
1569:{\displaystyle K}
1556:If the complexes
1546:{\displaystyle C}
1506:{\displaystyle C}
1304:{\displaystyle V}
1284:{\displaystyle V}
1264:{\displaystyle F}
1217:{\displaystyle V}
1085:{\displaystyle L}
1061:{\displaystyle K}
1023:{\displaystyle f}
1000:{\displaystyle g}
980:{\displaystyle f}
496:{\displaystyle k}
476:{\displaystyle V}
450:{\displaystyle F}
168:{\displaystyle R}
144:{\displaystyle R}
33:homology theories
3691:
3652:
3633:
3624:
3617:
3606:
3605:
3581:
3575:
3556:Joseph J. Rotman
3553:
3547:
3537:
3520:
3518:
3517:
3512:
3500:
3498:
3497:
3492:
3481:
3480:
3459:
3458:
3442:
3440:
3439:
3434:
3422:
3420:
3419:
3414:
3402:
3400:
3399:
3394:
3382:
3380:
3379:
3374:
3362:
3360:
3359:
3354:
3342:
3340:
3339:
3334:
3322:
3320:
3319:
3314:
3302:
3300:
3299:
3294:
3273:
3271:
3270:
3265:
3235:
3233:
3232:
3227:
3215:
3213:
3212:
3207:
3199:
3198:
3182:
3180:
3179:
3174:
3162:
3160:
3159:
3154:
3142:
3140:
3139:
3134:
3122:
3120:
3119:
3114:
3112:
3111:
3091:
3089:
3088:
3083:
3081:
3080:
3070:
3060:
3059:
3047:
3046:
3043:
3034:
3009:
3007:
3006:
3001:
2989:
2987:
2986:
2981:
2953:
2951:
2950:
2945:
2929:
2927:
2926:
2921:
2909:
2907:
2906:
2901:
2889:
2887:
2886:
2881:
2865:
2863:
2862:
2857:
2837:
2835:
2834:
2829:
2817:
2815:
2814:
2809:
2807:
2806:
2790:
2788:
2787:
2782:
2770:
2768:
2767:
2762:
2750:
2748:
2747:
2742:
2730:
2728:
2727:
2722:
2710:
2708:
2707:
2702:
2691:
2690:
2671:
2669:
2668:
2663:
2639:
2637:
2636:
2631:
2620:
2619:
2600:
2598:
2597:
2592:
2581:
2580:
2559:
2558:
2546:
2545:
2529:
2527:
2526:
2521:
2509:
2507:
2506:
2501:
2489:
2487:
2486:
2481:
2469:
2467:
2466:
2461:
2449:
2447:
2446:
2441:
2429:
2427:
2426:
2421:
2403:
2401:
2400:
2395:
2383:
2381:
2380:
2375:
2358:
2357:
2331:
2329:
2328:
2323:
2311:
2309:
2308:
2303:
2287:
2285:
2284:
2279:
2277:
2276:
2267:
2266:
2254:
2253:
2238:
2237:
2225:
2224:
2206:
2205:
2196:
2195:
2186:
2185:
2154:
2152:
2151:
2146:
2131:
2129:
2128:
2123:
2121:
2120:
2102:
2101:
2092:
2091:
2079:
2078:
2063:
2062:
2040:
2038:
2037:
2032:
2016:
2014:
2013:
2008:
1996:
1994:
1993:
1988:
1976:
1974:
1973:
1968:
1956:
1954:
1953:
1948:
1921:
1919:
1918:
1913:
1901:
1899:
1898:
1893:
1878:
1876:
1875:
1870:
1858:
1856:
1855:
1850:
1845:
1844:
1825:
1823:
1822:
1817:
1805:
1803:
1802:
1797:
1781:
1779:
1778:
1773:
1753:
1751:
1750:
1745:
1733:
1731:
1730:
1725:
1713:
1711:
1710:
1705:
1693:
1691:
1690:
1685:
1670:
1668:
1667:
1662:
1647:
1645:
1644:
1639:
1621:
1619:
1618:
1613:
1595:
1593:
1592:
1587:
1575:
1573:
1572:
1567:
1552:
1550:
1549:
1544:
1532:
1530:
1529:
1524:
1512:
1510:
1509:
1504:
1489:
1487:
1486:
1481:
1461:
1459:
1458:
1453:
1451:
1450:
1437:
1435:
1434:
1429:
1417:
1415:
1414:
1409:
1407:
1406:
1405:
1389:
1388:
1375:
1373:
1372:
1367:
1356:
1355:
1342:
1340:
1339:
1334:
1332:
1331:
1310:
1308:
1307:
1302:
1290:
1288:
1287:
1282:
1270:
1268:
1267:
1262:
1250:
1248:
1247:
1242:
1240:
1239:
1223:
1221:
1220:
1215:
1202:
1200:
1199:
1194:
1192:
1191:
1190:
1174:
1173:
1157:
1155:
1154:
1149:
1131:
1129:
1128:
1123:
1121:
1120:
1108:
1107:
1091:
1089:
1088:
1083:
1070:abelian category
1067:
1065:
1064:
1059:
1029:
1027:
1026:
1021:
1006:
1004:
1003:
998:
986:
984:
983:
978:
966:
964:
963:
958:
956:
955:
950:
949:
929:
927:
926:
921:
919:
918:
906:
905:
889:
887:
886:
881:
851:
849:
848:
843:
832:
831:
810:
809:
778:
776:
775:
770:
746:
744:
743:
738:
727:
726:
705:
704:
671:
669:
668:
663:
661:
660:
649:
648:
638:
637:
632:
631:
611:
609:
608:
603:
585:
583:
582:
577:
551:
550:
534:
532:
531:
526:
502:
500:
499:
494:
482:
480:
479:
474:
456:
454:
453:
448:
436:
434:
433:
428:
426:
425:
420:
419:
405:
403:
402:
397:
395:
394:
378:
376:
375:
370:
352:
350:
349:
344:
342:
341:
332:
331:
326:
325:
308:
306:
305:
300:
282:
280:
279:
274:
266:
265:
253:
252:
229:
227:
226:
221:
210:
209:
200:
199:
174:
172:
171:
166:
150:
148:
147:
142:
130:
128:
127:
122:
111:
110:
94:be an arbitrary
93:
91:
90:
85:
83:
82:
49:Saunders MacLane
45:Samuel Eilenberg
3699:
3698:
3694:
3693:
3692:
3690:
3689:
3688:
3669:
3668:
3655:
3634:
3627:
3618:
3609:
3603:
3593:Springer-Verlag
3582:
3578:
3554:
3550:
3538:
3534:
3530:
3506:
3503:
3502:
3476:
3472:
3454:
3450:
3448:
3445:
3444:
3428:
3425:
3424:
3408:
3405:
3404:
3388:
3385:
3384:
3368:
3365:
3364:
3348:
3345:
3344:
3328:
3325:
3324:
3308:
3305:
3304:
3279:
3276:
3275:
3241:
3238:
3237:
3221:
3218:
3217:
3194:
3190:
3188:
3185:
3184:
3168:
3165:
3164:
3148:
3145:
3144:
3128:
3125:
3124:
3107:
3103:
3101:
3098:
3097:
3076:
3072:
3055:
3051:
3042:
3041:
3040:
3024:
3018:
3015:
3014:
2995:
2992:
2991:
2963:
2960:
2959:
2939:
2936:
2935:
2915:
2912:
2911:
2895:
2892:
2891:
2875:
2872:
2871:
2851:
2848:
2847:
2844:
2823:
2820:
2819:
2802:
2798:
2796:
2793:
2792:
2776:
2773:
2772:
2756:
2753:
2752:
2736:
2733:
2732:
2716:
2713:
2712:
2683:
2679:
2677:
2674:
2673:
2645:
2642:
2641:
2612:
2608:
2606:
2603:
2602:
2576:
2572:
2554:
2550:
2541:
2537:
2535:
2532:
2531:
2515:
2512:
2511:
2495:
2492:
2491:
2475:
2472:
2471:
2455:
2452:
2451:
2435:
2432:
2431:
2415:
2412:
2411:
2389:
2386:
2385:
2353:
2349:
2341:
2338:
2337:
2317:
2314:
2313:
2297:
2294:
2293:
2272:
2268:
2262:
2258:
2243:
2239:
2233:
2229:
2214:
2210:
2201:
2197:
2191:
2187:
2181:
2177:
2163:
2160:
2159:
2140:
2137:
2136:
2116:
2112:
2097:
2093:
2087:
2083:
2068:
2064:
2058:
2054:
2049:
2046:
2045:
2026:
2023:
2022:
2002:
1999:
1998:
1982:
1979:
1978:
1962:
1959:
1958:
1927:
1924:
1923:
1907:
1904:
1903:
1887:
1884:
1883:
1864:
1861:
1860:
1837:
1833:
1831:
1828:
1827:
1811:
1808:
1807:
1791:
1788:
1787:
1767:
1764:
1763:
1739:
1736:
1735:
1719:
1716:
1715:
1699:
1696:
1695:
1679:
1676:
1675:
1656:
1653:
1652:
1627:
1624:
1623:
1601:
1598:
1597:
1581:
1578:
1577:
1561:
1558:
1557:
1538:
1535:
1534:
1518:
1515:
1514:
1498:
1495:
1494:
1475:
1472:
1471:
1466:provided that:
1446:
1445:
1443:
1440:
1439:
1423:
1420:
1419:
1401:
1400:
1396:
1384:
1383:
1381:
1378:
1377:
1351:
1350:
1348:
1345:
1344:
1327:
1326:
1324:
1321:
1320:
1317:
1315:Acyclic classes
1296:
1293:
1292:
1276:
1273:
1272:
1256:
1253:
1252:
1235:
1234:
1232:
1229:
1228:
1209:
1206:
1205:
1186:
1185:
1181:
1169:
1168:
1166:
1163:
1162:
1137:
1134:
1133:
1116:
1112:
1103:
1099:
1097:
1094:
1093:
1077:
1074:
1073:
1053:
1050:
1049:
1046:
1041:
1039:Generalizations
1015:
1012:
1011:
992:
989:
988:
972:
969:
968:
951:
945:
944:
943:
935:
932:
931:
930:for all models
914:
910:
901:
897:
895:
892:
891:
857:
854:
853:
827:
823:
805:
801:
787:
784:
783:
752:
749:
748:
722:
718:
700:
696:
688:
685:
684:
650:
644:
643:
642:
633:
627:
626:
625:
617:
614:
613:
591:
588:
587:
546:
542:
540:
537:
536:
508:
505:
504:
488:
485:
484:
468:
465:
464:
442:
439:
438:
421:
415:
414:
413:
411:
408:
407:
406:has a basis in
390:
386:
384:
381:
380:
358:
355:
354:
337:
336:
327:
321:
320:
319:
317:
314:
313:
288:
285:
284:
261:
257:
248:
244:
242:
239:
238:
205:
204:
195:
194:
180:
177:
176:
160:
157:
156:
155:over some ring
136:
133:
132:
106:
105:
103:
100:
99:
78:
77:
75:
72:
71:
68:
17:
12:
11:
5:
3697:
3687:
3686:
3681:
3667:
3666:
3654:
3653:
3649:978-0821828779
3637:Acyclic Models
3625:
3621:Acyclic Models
3607:
3601:
3585:Dold, Albrecht
3576:
3548:
3541:Amer. J. Math.
3531:
3529:
3526:
3510:
3490:
3487:
3484:
3479:
3475:
3471:
3468:
3465:
3462:
3457:
3453:
3432:
3412:
3392:
3372:
3352:
3332:
3312:
3292:
3289:
3286:
3283:
3263:
3260:
3257:
3254:
3251:
3248:
3245:
3236:be defined by
3225:
3205:
3202:
3197:
3193:
3172:
3152:
3132:
3110:
3106:
3094:
3093:
3079:
3075:
3069:
3066:
3063:
3058:
3054:
3050:
3039:
3033:
3030:
3027:
3023:
2999:
2979:
2976:
2973:
2970:
2967:
2943:
2919:
2899:
2879:
2855:
2843:
2840:
2827:
2805:
2801:
2780:
2760:
2740:
2720:
2700:
2697:
2694:
2689:
2686:
2682:
2661:
2658:
2655:
2652:
2649:
2629:
2626:
2623:
2618:
2615:
2611:
2590:
2587:
2584:
2579:
2575:
2571:
2568:
2565:
2562:
2557:
2553:
2549:
2544:
2540:
2519:
2499:
2479:
2459:
2439:
2419:
2393:
2373:
2370:
2367:
2364:
2361:
2356:
2352:
2348:
2345:
2321:
2301:
2290:
2289:
2275:
2271:
2265:
2261:
2257:
2252:
2249:
2246:
2242:
2236:
2232:
2228:
2223:
2220:
2217:
2213:
2209:
2204:
2200:
2194:
2190:
2184:
2180:
2176:
2173:
2170:
2167:
2144:
2133:
2132:
2119:
2115:
2111:
2108:
2105:
2100:
2096:
2090:
2086:
2082:
2077:
2074:
2071:
2067:
2061:
2057:
2053:
2030:
2006:
1986:
1966:
1946:
1943:
1940:
1937:
1934:
1931:
1911:
1891:
1868:
1848:
1843:
1840:
1836:
1815:
1795:
1771:
1756:
1755:
1743:
1723:
1703:
1683:
1672:
1660:
1649:
1637:
1634:
1631:
1611:
1608:
1605:
1585:
1565:
1554:
1542:
1522:
1502:
1491:
1479:
1449:
1427:
1404:
1399:
1395:
1392:
1387:
1365:
1362:
1359:
1354:
1330:
1316:
1313:
1300:
1280:
1260:
1238:
1213:
1189:
1184:
1180:
1177:
1172:
1147:
1144:
1141:
1119:
1115:
1111:
1106:
1102:
1081:
1057:
1045:
1042:
1040:
1037:
1036:
1035:
1032:chain homotopy
1019:
1008:
996:
976:
954:
948:
942:
939:
917:
913:
909:
904:
900:
879:
876:
873:
870:
867:
864:
861:
841:
838:
835:
830:
826:
822:
819:
816:
813:
808:
804:
800:
797:
794:
791:
780:
768:
765:
762:
759:
756:
736:
733:
730:
725:
721:
717:
714:
711:
708:
703:
699:
695:
692:
674:
673:
659:
656:
653:
647:
641:
636:
630:
624:
621:
601:
598:
595:
575:
572:
569:
566:
563:
560:
557:
554:
549:
545:
524:
521:
518:
515:
512:
492:
472:
462:
446:
424:
418:
393:
389:
368:
365:
362:
340:
335:
330:
324:
310:
298:
295:
292:
272:
269:
264:
260:
256:
251:
247:
219:
216:
213:
208:
203:
198:
193:
190:
187:
184:
164:
140:
120:
117:
114:
109:
81:
67:
64:
60:model category
15:
9:
6:
4:
3:
2:
3696:
3685:
3682:
3680:
3677:
3676:
3674:
3664:
3661:
3657:
3656:
3650:
3646:
3642:
3638:
3632:
3630:
3622:
3616:
3614:
3612:
3604:
3602:3-540-10369-4
3598:
3594:
3590:
3586:
3580:
3573:
3569:
3568:0-387-96678-1
3565:
3561:
3557:
3552:
3545:
3542:
3536:
3532:
3525:
3522:
3508:
3485:
3477:
3473:
3469:
3463:
3455:
3451:
3410:
3390:
3370:
3350:
3330:
3310:
3290:
3284:
3281:
3261:
3258:
3255:
3249:
3243:
3223:
3203:
3195:
3170:
3150:
3130:
3108:
3077:
3064:
3061:
3056:
3037:
3031:
3028:
3025:
3021:
3013:
3012:
3011:
2997:
2977:
2971:
2968:
2965:
2958:functor. Let
2957:
2941:
2933:
2917:
2897:
2877:
2869:
2853:
2839:
2825:
2803:
2799:
2778:
2758:
2738:
2718:
2695:
2687:
2684:
2659:
2653:
2650:
2647:
2624:
2616:
2613:
2585:
2577:
2573:
2563:
2555:
2551:
2547:
2542:
2538:
2517:
2497:
2477:
2457:
2409:
2405:
2371:
2362:
2354:
2350:
2343:
2335:
2319:
2299:
2273:
2269:
2263:
2259:
2250:
2247:
2244:
2240:
2234:
2230:
2226:
2221:
2218:
2215:
2211:
2207:
2202:
2198:
2192:
2188:
2182:
2174:
2171:
2165:
2158:
2157:
2156:
2117:
2113:
2106:
2098:
2094:
2088:
2084:
2075:
2072:
2069:
2065:
2059:
2055:
2051:
2044:
2043:
2042:
2028:
2020:
2004:
1984:
1964:
1944:
1941:
1935:
1932:
1929:
1909:
1889:
1880:
1846:
1841:
1838:
1785:
1760:
1721:
1681:
1673:
1650:
1632:
1629:
1606:
1603:
1583:
1563:
1555:
1540:
1500:
1492:
1469:
1468:
1467:
1465:
1464:acyclic class
1393:
1360:
1312:
1298:
1278:
1258:
1225:
1211:
1178:
1159:
1145:
1139:
1117:
1113:
1104:
1100:
1079:
1071:
1055:
1033:
1017:
1009:
994:
974:
952:
940:
937:
915:
911:
907:
902:
898:
877:
871:
868:
865:
862:
859:
836:
828:
824:
814:
806:
802:
798:
795:
792:
789:
781:
766:
760:
757:
754:
731:
723:
719:
709:
701:
697:
693:
690:
683:
679:
678:
677:
657:
654:
651:
639:
634:
622:
619:
599:
596:
593:
573:
570:
561:
555:
547:
543:
519:
516:
513:
490:
470:
463:
460:
444:
422:
391:
387:
366:
363:
360:
333:
328:
311:
296:
293:
290:
270:
267:
262:
258:
254:
249:
245:
237:
236:
235:
233:
214:
191:
188:
185:
182:
162:
154:
138:
115:
97:
63:
61:
57:
52:
50:
46:
42:
38:
34:
30:
26:
22:
3662:
3659:
3640:
3636:
3588:
3579:
3571:
3559:
3551:
3546:, pp.189–199
3543:
3540:
3535:
3523:
3095:
2934:functor and
2845:
2407:
2406:
2333:
2291:
2134:
2021:if for each
2018:
1881:
1784:mapping cone
1761:
1757:
1493:The complex
1463:
1318:
1311:is acyclic.
1226:
1160:
1047:
675:
459:free functor
69:
53:
28:
18:
3216:. Then let
2384:belongs to
2135:belongs to
2019:presentable
1806:. Although
1786:belongs to
1734:belongs to
1671:is acyclic.
1513:belongs to
1418:). A class
234:such that:
25:mathematics
3673:Categories
3619:M. Barr, "
3528:References
3010:the space
379:such that
312:There are
37:isomorphic
3635:M. Barr,
3623:" (1999).
3470:≃
3431:Γ
3363:are both
3288:→
3201:→
3192:Δ
3105:Δ
3074:Δ
3053:Δ
3038:∑
3029:≥
3022:∑
2975:→
2685:−
2681:Σ
2657:→
2614:−
2610:Σ
2570:→
2438:Σ
2418:Γ
2392:Γ
2369:→
2347:→
2256:→
2219:−
2208:ϵ
2172:−
2166:∑
2143:Γ
2110:→
2107:⋯
2104:→
2081:→
2052:⋯
1939:→
1930:ϵ
1867:Σ
1839:−
1835:Σ
1814:Σ
1794:Γ
1770:Σ
1742:Γ
1702:Γ
1659:Γ
1636:Γ
1633:∈
1610:Γ
1607:∈
1521:Γ
1478:Γ
1426:Γ
1143:→
1110:→
941:∈
912:ψ
899:φ
875:→
821:→
796:ψ
790:φ
764:→
716:→
691:φ
640:∪
623:∈
364:≥
334:⊆
202:→
3587:(1980),
612:and all
586:for all
96:category
3123:is the
2954:be the
2930:be the
2866:be the
2842:Example
2408:Theorem
2334:acyclic
1622:, then
153:modules
41:theorem
3647:
3599:
3566:
3096:Here,
2910:. Let
2410:. Let
680:Every
503:- and
175:. Let
39:. The
27:, the
1553:does.
457:is a
437:, so
3645:ISBN
3597:ISBN
3564:ISBN
3343:and
2870:and
1882:Let
1762:Let
1576:and
1072:and
987:and
597:>
353:for
294:<
283:for
98:and
70:Let
47:and
35:are
3044:Hom
2771:is
2731:is
2510:is
2470:is
2312:is
1997:is
1674:If
1376:or
782:If
483:is
230:be
19:In
3675::
3663:59
3641:17
3628:^
3610:^
3595:,
3558:,
3544:75
3521:.
2404:.
1879:.
62:.
3651:.
3574:)
3570:(
3509:X
3489:)
3486:L
3483:(
3478:0
3474:H
3467:)
3464:K
3461:(
3456:0
3452:H
3411:L
3391:G
3371:G
3351:L
3331:K
3311:G
3291:X
3285:X
3282:E
3262:E
3259:C
3256:=
3253:)
3250:C
3247:(
3244:G
3224:G
3204:X
3196:n
3171:n
3151:X
3131:n
3109:n
3092:.
3078:n
3068:)
3065:X
3062:,
3057:n
3049:(
3032:0
3026:n
2998:X
2978:X
2972:X
2969::
2966:E
2942:L
2918:K
2898:X
2878:C
2854:X
2826:f
2804:0
2800:f
2779:G
2759:K
2739:G
2719:L
2699:)
2696:C
2693:(
2688:1
2660:L
2654:K
2651::
2648:f
2628:)
2625:C
2622:(
2617:1
2589:)
2586:L
2583:(
2578:0
2574:H
2567:)
2564:K
2561:(
2556:0
2552:H
2548::
2543:0
2539:f
2518:G
2498:L
2478:G
2458:K
2372:0
2366:)
2363:L
2360:(
2355:0
2351:H
2344:L
2332:-
2320:G
2300:L
2288:.
2274:m
2270:G
2264:n
2260:K
2251:1
2248:+
2245:m
2241:G
2235:n
2231:K
2227::
2222:i
2216:m
2212:G
2203:i
2199:G
2193:n
2189:K
2183:i
2179:)
2175:1
2169:(
2118:n
2114:K
2099:m
2095:G
2089:n
2085:K
2076:1
2073:+
2070:m
2066:G
2060:n
2056:K
2029:n
2017:-
2005:G
1985:K
1965:C
1945:d
1942:I
1936:G
1933::
1910:C
1890:G
1847:C
1842:1
1754:.
1722:D
1682:D
1648:.
1630:L
1604:K
1584:L
1564:K
1541:C
1501:C
1490:.
1448:A
1403:K
1398:)
1394:R
1391:(
1386:C
1364:)
1361:R
1358:(
1353:C
1329:A
1299:V
1279:V
1259:F
1237:K
1212:V
1188:K
1183:)
1179:R
1176:(
1171:C
1146:L
1140:K
1118:0
1114:L
1105:0
1101:K
1080:L
1056:K
1034:.
1018:f
1007:.
995:g
975:f
953:0
947:M
938:M
916:M
908:=
903:M
878:V
872:F
869::
866:g
863:,
860:f
840:)
837:V
834:(
829:0
825:H
818:)
815:F
812:(
807:0
803:H
799::
793:,
779:.
767:V
761:F
758::
755:f
735:)
732:V
729:(
724:0
720:H
713:)
710:F
707:(
702:0
698:H
694::
672:.
658:1
655:+
652:k
646:M
635:k
629:M
620:M
600:0
594:k
574:0
571:=
568:)
565:)
562:M
559:(
556:V
553:(
548:k
544:H
523:)
520:1
517:+
514:k
511:(
491:k
471:V
461:.
445:F
423:k
417:M
392:k
388:F
367:0
361:k
339:K
329:k
323:M
309:.
297:0
291:i
271:0
268:=
263:i
259:V
255:=
250:i
246:F
218:)
215:R
212:(
207:C
197:K
192::
189:V
186:,
183:F
163:R
151:-
139:R
119:)
116:R
113:(
108:C
80:K
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