5045:
2797:
5292:
5312:
5302:
4570:
In the literature, one finds the terms "directed limit", "direct inductive limit", "directed colimit", "direct colimit" and "inductive limit" for the concept of direct limit defined above. The term "inductive limit" is ambiguous however, as some authors use it for the general concept of colimit.
4349:
and form the colimit of this functor. One can show that a category has all directed limits if and only if it has all filtered colimits, and a functor defined on such a category commutes with all direct limits if and only if it commutes with all filtered colimits.
3387:
3845:
4025:
1546:
2185:
1989:
4251:
1829:. Intuitively, two elements in the disjoint union are equivalent if and only if they "eventually become equal" in the direct system. An equivalent formulation that highlights the duality to the
1049:
2948:
Unlike for algebraic objects, not every direct system in an arbitrary category has a direct limit. If it does, however, the direct limit is unique in a strong sense: given another direct limit
2431:
3921:
4296:
2513:
3311:
1827:
4521:
4486:
3192:
2916:
2588:
2273:
2079:
1366:
1286:
233:
2708:
2669:
2630:
2340:
2786:
3472:
1891:
1201:
525:
4093:
919:
2740:
3237:) that enlarges matrices by putting a 1 in the lower right corner and zeros elsewhere in the last row and column. The direct limit of this system is the general linear group of
2864:
2030:
1670:
2119:
1406:
968:
791:
478:
441:
4549:
4451:
4423:
4399:
4375:
4347:
4323:
4166:
4120:
4057:
2385:
2297:
2220:
1629:
1589:
3054:
1233:
187:
2943:
1109:
4194:
2539:
1917:
1748:
1722:
1466:
1137:
1075:
2457:
1696:
3146:
3000:
1437:
708:
3525:
2361:
993:
4214:
4140:
3499:
3431:
3407:
3316:
3119:
3096:
3020:
1310:
751:
728:
3735:
3926:
651:
is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be
3249:) can be thought of as an infinite invertible matrix that differs from the infinite identity matrix in only finitely many entries. The group GL(
630:
1475:
2124:
1928:
4689:
4562:. As above, inverse limits can be viewed as limits of certain functors and are closely related to limits over cofiltered categories.
4655:
4974:
4222:
87:
998:
2390:
3861:
4267:
5336:
4611:
2462:
623:
575:
3274:
4426:
1753:
793:. This notation suppresses the system of homomorphisms; however, the limit depends on the system of homomorphisms.
4491:
4456:
3151:
2872:
2544:
2229:
2035:
1322:
1242:
197:
2674:
2635:
2596:
2306:
2745:
3702:
3439:
1836:
1144:
492:
4682:
4660:
4066:
892:
2713:
4886:
4841:
616:
483:
5315:
5255:
3528:
333:
4964:
2823:
1994:
5305:
5091:
4955:
4863:
2032:
are defined such that these maps become homomorphisms. Formally, the direct limit of the direct system
1634:
93:
2084:
1371:
928:
756:
454:
417:
108:
5341:
5264:
4908:
4846:
4769:
4530:
4432:
4404:
4380:
4356:
4328:
4304:
4145:
4101:
4038:
2366:
2278:
2201:
1594:
1554:
5295:
5251:
4856:
4675:
3029:
1206:
568:
371:
321:
170:
4851:
4833:
4254:
813:
797:
380:
114:
73:
5058:
4824:
4804:
4727:
4580:
4555:
2921:
2196:
1527:
837:
805:
664:
537:
388:
339:
120:
1083:
4940:
4779:
4173:
2518:
1896:
1833:
is that an element is equivalent to all its images under the maps of the direct system, i.e.
1727:
1701:
1450:
1114:
1054:
2436:
1675:
4752:
4747:
4643:
3851:
3478:
3206:
3124:
2978:
1445:
1415:
869:
865:
833:
686:
261:
135:
3382:{\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} \rightarrow \mathbb {Z} /p^{n+1}\mathbb {Z} }
8:
5096:
5044:
4970:
4774:
3608:
3543:
3504:
2803:
2345:
977:
873:
857:
853:
829:
825:
652:
543:
351:
302:
247:
141:
127:
55:
23:
4950:
4945:
4927:
4809:
4784:
4199:
4125:
3840:{\displaystyle \mathrm {Hom} (\varinjlim X_{i},Y)=\varprojlim \mathrm {Hom} (X_{i},Y).}
3686:
3484:
3416:
3392:
3254:
3104:
3081:
3005:
2223:
1295:
883:
861:
736:
713:
672:
656:
556:
42:
5259:
5196:
5184:
5086:
5011:
5006:
4960:
4742:
4737:
4650:
4607:
4299:
4261:
4217:
3547:
849:
676:
597:
394:
159:
100:
5220:
5106:
5081:
5016:
5001:
4996:
4935:
4764:
4732:
4631:
3099:
603:
589:
403:
345:
308:
81:
67:
4020:{\displaystyle 0\to \varinjlim A_{i}\to \varinjlim B_{i}\to \varinjlim C_{i}\to 0}
5132:
4698:
4639:
4601:
1440:
801:
365:
315:
153:
3434:
683:
in the category) between those smaller objects. The direct limit of the objects
5169:
5164:
5148:
5111:
5101:
5021:
4257:
of this functor is the same as the direct limit of the original direct system.
4168:
4096:
3858:. This means that if you start with a directed system of short exact sequences
3706:
3410:
3269:
1409:
409:
5330:
5159:
4991:
4868:
4794:
4559:
3855:
3726:
3068:
3023:
1830:
809:
550:
446:
61:
4814:
3265:
3026:
by inclusion. If the collection is directed, its direct limit is the union
1991:
sending each element to its equivalence class. The algebraic operations on
922:
878:
731:
668:
660:
582:
357:
253:
5174:
5154:
5026:
4896:
2956:
644:
562:
273:
147:
29:
2796:
5206:
5144:
4757:
4524:
3713:
3121:. The direct limit of any corresponding direct system is isomorphic to
3072:
327:
4425:(consider for example the category of finite sets, or the category of
3850:
An important property is that taking direct limits in the category of
3606:
is the restriction map. The direct limit of this system is called the
3477:
There is a (non-obvious) injective ring homomorphism from the ring of
5200:
4891:
3527:
variables. Forming the direct limit of this direct system yields the
971:
287:
192:
5269:
4901:
4799:
3057:
1541:{\displaystyle \varinjlim A_{i}=\bigsqcup _{i}A_{i}{\bigg /}\sim .}
680:
281:
267:
4667:
5239:
5229:
4878:
4789:
4060:
3061:
2180:{\displaystyle \phi _{j}\colon A_{j}\rightarrow \varinjlim A_{i}}
1984:{\displaystyle \phi _{j}\colon A_{j}\rightarrow \varinjlim A_{i}}
165:
49:
848:
In this section objects are understood to consist of underlying
5234:
5116:
667:. The way they are put together is specified by a system of
4030:
836:, and then the general definition, which can be used in any
3923:
and form direct limits, you obtain a short exact sequence
2190:
4246:{\displaystyle {\mathcal {I}}\rightarrow {\mathcal {C}}}
3409:. The direct limit of this system consists of all the
4533:
4494:
4459:
4435:
4407:
4383:
4359:
4331:
4307:
4270:
4225:
4202:
4176:
4148:
4128:
4104:
4069:
4041:
3929:
3864:
3738:
3507:
3487:
3442:
3419:
3395:
3319:
3277:
3154:
3127:
3107:
3084:
3032:
3008:
2981:
2924:
2875:
2826:
2748:
2716:
2677:
2638:
2599:
2547:
2521:
2465:
2439:
2393:
2369:
2348:
2309:
2281:
2232:
2204:
2127:
2087:
2038:
1997:
1931:
1899:
1839:
1756:
1730:
1704:
1678:
1637:
1597:
1557:
1478:
1453:
1418:
1408:
and is defined as follows. Its underlying set is the
1374:
1325:
1298:
1245:
1209:
1147:
1117:
1086:
1057:
1001:
980:
931:
895:
843:
759:
739:
716:
689:
495:
457:
420:
200:
173:
1044:{\displaystyle f_{ij}\colon A_{i}\rightarrow A_{j}}
796:Direct limits are a special case of the concept of
4543:
4515:
4488:in which all direct limits exist; the objects of
4480:
4445:
4417:
4393:
4369:
4341:
4317:
4290:
4260:A notion closely related to direct limits are the
4245:
4208:
4188:
4160:
4134:
4114:
4087:
4051:
4019:
3915:
3839:
3519:
3501:variables to the ring of symmetric polynomials in
3493:
3466:
3425:
3401:
3381:
3305:
3186:
3140:
3113:
3090:
3048:
3014:
2994:
2937:
2910:
2858:
2780:
2734:
2702:
2663:
2624:
2582:
2533:
2507:
2451:
2426:{\displaystyle \phi _{i}\colon X_{i}\rightarrow X}
2425:
2379:
2355:
2334:
2291:
2267:
2214:
2179:
2113:
2073:
2024:
1983:
1911:
1885:
1821:
1742:
1716:
1690:
1664:
1623:
1583:
1540:
1460:
1431:
1400:
1360:
1304:
1280:
1227:
1195:
1131:
1103:
1069:
1043:
987:
962:
913:
785:
745:
722:
702:
519:
472:
435:
227:
181:
3916:{\displaystyle 0\to A_{i}\to B_{i}\to C_{i}\to 0}
5328:
4291:{\displaystyle {\mathcal {J}}\to {\mathcal {C}}}
3056:. The same is true for a directed collection of
2195:The direct limit can be defined in an arbitrary
2508:{\displaystyle \phi _{i}=\phi _{j}\circ f_{ij}}
2275:be a direct system of objects and morphisms in
4059:admits an alternative description in terms of
3306:{\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }
3060:of a given group, or a directed collection of
4683:
4663:, vol. 5 (2nd ed.), Springer-Verlag
4603:Locally Presentable and Accessible Categories
4599:
3709:on the underlying set-theoretic direct limit.
3268:. Consider the direct system composed of the
1822:{\displaystyle f_{ik}(x_{i})=f_{jk}(x_{j})\,}
882:are understood in the corresponding setting (
624:
4516:{\displaystyle {\text{Ind}}({\mathcal {C}})}
4481:{\displaystyle {\text{Ind}}({\mathcal {C}})}
4082:
4070:
3187:{\displaystyle \phi _{m}:X_{m}\rightarrow X}
2966:that commutes with the canonical morphisms.
2911:{\displaystyle \langle X_{i},f_{ij}\rangle }
2905:
2876:
2697:
2678:
2658:
2639:
2619:
2600:
2583:{\displaystyle \langle X_{i},f_{ij}\rangle }
2577:
2548:
2329:
2310:
2268:{\displaystyle \langle X_{i},f_{ij}\rangle }
2262:
2233:
2074:{\displaystyle \langle A_{i},f_{ij}\rangle }
2068:
2039:
1361:{\displaystyle \langle A_{i},f_{ij}\rangle }
1355:
1326:
1281:{\displaystyle \langle A_{i},f_{ij}\rangle }
1275:
1246:
957:
932:
908:
896:
228:{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }
4035:We note that a direct system in a category
2703:{\displaystyle \langle Y,\psi _{i}\rangle }
2664:{\displaystyle \langle X,\phi _{i}\rangle }
2625:{\displaystyle \langle X,\phi _{i}\rangle }
2335:{\displaystyle \langle X,\phi _{i}\rangle }
5311:
5301:
5057:
4690:
4676:
4638:, Translated from French, Paris: Hermann,
4606:. Cambridge University Press. p. 15.
3565:form a directed set ordered by inclusion (
2781:{\displaystyle u\circ \phi _{i}=\psi _{i}}
2121:together with the canonical homomorphisms
631:
617:
16:Special case of colimit in category theory
4264:. Here we start with a covariant functor
4031:Related constructions and generalizations
3467:{\displaystyle \mathbb {Z} (p^{\infty })}
3444:
3375:
3349:
3341:
3321:
3299:
3279:
2352:
2021:
1886:{\displaystyle x_{i}\sim \,f_{ij}(x_{i})}
1853:
1818:
1651:
1457:
1196:{\displaystyle f_{ik}=f_{jk}\circ f_{ij}}
1128:
1100:
984:
520:{\displaystyle \mathbb {Z} (p^{\infty })}
497:
460:
423:
221:
208:
175:
4656:Categories for the Working Mathematician
4649:
4630:
4636:Elements of mathematics. Theory of sets
4088:{\displaystyle \langle I,\leq \rangle }
3581:). The corresponding direct system is (
914:{\displaystyle \langle I,\leq \rangle }
5329:
2735:{\displaystyle u\colon X\rightarrow Y}
2541:. A direct limit of the direct system
2191:Direct limits in an arbitrary category
824:We will first give the definition for
5056:
4709:
4671:
4429:). In this case, we can always embed
819:
88:Free product of associative algebras
4697:
3201:be a field. For a positive integer
13:
4558:of the direct limit is called the
4536:
4505:
4470:
4438:
4410:
4401:that don't have a direct limit in
4386:
4362:
4334:
4310:
4283:
4273:
4238:
4228:
4107:
4044:
3808:
3805:
3802:
3746:
3743:
3740:
3456:
3225:. We have a group homomorphism GL(
2859:{\displaystyle X=\varinjlim X_{i}}
2817:The direct limit is often denoted
2372:
2284:
2207:
2025:{\displaystyle \varinjlim A_{i}\,}
844:Direct limits of algebraic objects
509:
14:
5353:
4427:finitely generated abelian groups
4377:, there may be direct systems in
3716:is an inductive limit of schemes.
1922:One obtains from this definition
1665:{\displaystyle x_{i}\sim \,x_{j}}
576:Noncommutative algebraic geometry
5310:
5300:
5291:
5290:
5043:
4710:
4600:Adamek, J.; Rosicky, J. (1994).
2795:
2671:is a target and for each target
2114:{\displaystyle \varinjlim A_{i}}
1401:{\displaystyle \varinjlim A_{i}}
963:{\displaystyle \{A_{i}:i\in I\}}
786:{\displaystyle \varinjlim A_{i}}
473:{\displaystyle \mathbb {Q} _{p}}
436:{\displaystyle \mathbb {Z} _{p}}
4122:whose objects are the elements
1077:with the following properties:
663:or in general objects from any
4593:
4565:
4544:{\displaystyle {\mathcal {C}}}
4510:
4500:
4475:
4465:
4446:{\displaystyle {\mathcal {C}}}
4418:{\displaystyle {\mathcal {C}}}
4394:{\displaystyle {\mathcal {C}}}
4370:{\displaystyle {\mathcal {C}}}
4342:{\displaystyle {\mathcal {C}}}
4318:{\displaystyle {\mathcal {J}}}
4278:
4233:
4161:{\displaystyle i\rightarrow j}
4152:
4115:{\displaystyle {\mathcal {I}}}
4052:{\displaystyle {\mathcal {C}}}
4011:
3985:
3959:
3933:
3907:
3894:
3881:
3868:
3831:
3812:
3782:
3750:
3703:category of topological spaces
3461:
3448:
3345:
3178:
2726:
2417:
2380:{\displaystyle {\mathcal {C}}}
2292:{\displaystyle {\mathcal {C}}}
2215:{\displaystyle {\mathcal {C}}}
2151:
1955:
1880:
1867:
1815:
1802:
1783:
1770:
1624:{\displaystyle x_{j}\in A_{j}}
1584:{\displaystyle x_{i}\in A_{i}}
1028:
812:, which are a special case of
514:
501:
1:
4661:Graduate Texts in Mathematics
4624:
3720:
3389:induced by multiplication by
3221:- matrices with entries from
3075:is defined as a direct limit.
3049:{\displaystyle \bigcup M_{i}}
2710:, there is a unique morphism
1672:if and only if there is some
1228:{\displaystyle i\leq j\leq k}
4353:Given an arbitrary category
3725:Direct limits are linked to
3561:. The open neighborhoods of
3253:) is of vital importance in
2918:and the canonical morphisms
2592:universally repelling target
182:{\displaystyle \mathbb {Z} }
7:
4985:Constructions on categories
4574:
3529:ring of symmetric functions
3213:) consisting of invertible
3148:and the canonical morphism
3098:be any directed set with a
2969:
876:), etc. With this in mind,
334:Unique factorization domain
10:
5358:
5092:Higher-dimensional algebra
4142:and there is a morphisms
1051:be a homomorphism for all
94:Tensor product of algebras
5286:
5219:
5183:
5131:
5124:
5075:
5065:
5052:
5041:
4984:
4926:
4877:
4832:
4823:
4720:
4716:
4705:
3705:are given by placing the
3637:, the canonical morphism
2938:{\displaystyle \phi _{i}}
5337:Limits (category theory)
4586:
3654:associates to a section
3629:. For each neighborhood
2975:A collection of subsets
2792:. The following diagram
1104:{\displaystyle f_{ii}\,}
372:Formal power series ring
322:Integrally closed domain
4902:Cokernels and quotients
4825:Universal constructions
4581:Direct limits of groups
4196:. A direct system over
4189:{\displaystyle i\leq j}
4095:can be considered as a
3413:of order some power of
2869:with the direct system
2534:{\displaystyle i\leq j}
2433:are morphisms for each
2081:consists of the object
1912:{\displaystyle i\leq j}
1743:{\displaystyle j\leq k}
1717:{\displaystyle i\leq k}
1461:{\displaystyle \sim \,}
1132:{\displaystyle A_{i}\,}
1070:{\displaystyle i\leq j}
970:be a family of objects
381:Algebraic number theory
74:Total ring of fractions
5059:Higher category theory
4805:Natural transformation
4545:
4517:
4482:
4447:
4419:
4395:
4371:
4343:
4319:
4292:
4247:
4216:is then the same as a
4210:
4190:
4162:
4136:
4116:
4089:
4053:
4021:
3917:
3841:
3521:
3495:
3468:
3427:
3403:
3383:
3313:and the homomorphisms
3307:
3188:
3142:
3115:
3092:
3050:
3016:
2996:
2939:
2912:
2860:
2782:
2736:
2704:
2665:
2626:
2584:
2535:
2509:
2453:
2452:{\displaystyle i\in I}
2427:
2381:
2357:
2336:
2299:(as defined above). A
2293:
2269:
2216:
2181:
2115:
2075:
2026:
1985:
1913:
1887:
1823:
1744:
1718:
1692:
1691:{\displaystyle k\in I}
1666:
1625:
1585:
1542:
1462:
1433:
1402:
1362:
1306:
1282:
1229:
1197:
1133:
1105:
1071:
1045:
989:
964:
915:
852:equipped with a given
787:
747:
724:
704:
538:Noncommutative algebra
521:
474:
437:
389:Algebraic number field
340:Principal ideal domain
229:
183:
121:Frobenius endomorphism
4546:
4518:
4483:
4448:
4420:
4396:
4372:
4344:
4320:
4293:
4248:
4211:
4191:
4163:
4137:
4117:
4090:
4054:
4022:
3918:
3842:
3701:Direct limits in the
3522:
3496:
3479:symmetric polynomials
3469:
3428:
3404:
3384:
3308:
3189:
3143:
3141:{\displaystyle X_{m}}
3116:
3093:
3064:of a given ring, etc.
3051:
3017:
2997:
2995:{\displaystyle M_{i}}
2940:
2913:
2861:
2783:
2737:
2705:
2666:
2627:
2585:
2536:
2510:
2454:
2428:
2382:
2358:
2337:
2294:
2270:
2217:
2182:
2116:
2076:
2027:
1986:
1914:
1888:
1824:
1745:
1719:
1693:
1667:
1626:
1586:
1543:
1463:
1434:
1432:{\displaystyle A_{i}}
1403:
1363:
1319:of the direct system
1307:
1283:
1230:
1198:
1134:
1106:
1072:
1046:
990:
965:
916:
868:(over a fixed ring),
788:
748:
725:
705:
703:{\displaystyle A_{i}}
522:
475:
438:
230:
184:
4928:Algebraic categories
4531:
4492:
4457:
4433:
4405:
4381:
4357:
4329:
4305:
4268:
4223:
4200:
4174:
4146:
4126:
4102:
4067:
4039:
3927:
3862:
3736:
3505:
3485:
3440:
3433:, and is called the
3417:
3393:
3317:
3275:
3245:). An element of GL(
3207:general linear group
3152:
3125:
3105:
3082:
3030:
3006:
2979:
2922:
2873:
2824:
2746:
2714:
2675:
2636:
2597:
2545:
2519:
2463:
2437:
2391:
2367:
2346:
2307:
2279:
2230:
2202:
2125:
2085:
2036:
1995:
1929:
1897:
1837:
1754:
1728:
1702:
1676:
1635:
1595:
1555:
1476:
1451:
1446:equivalence relation
1416:
1372:
1323:
1296:
1243:
1207:
1145:
1115:
1084:
1055:
999:
978:
929:
893:
826:algebraic structures
816:in category theory.
804:. Direct limits are
757:
737:
714:
687:
544:Noncommutative rings
493:
455:
418:
262:Non-associative ring
198:
171:
128:Algebraic structures
5097:Homotopy hypothesis
4775:Commutative diagram
4063:. Any directed set
3520:{\displaystyle n+1}
2356:{\displaystyle X\,}
1924:canonical functions
1111:is the identity on
988:{\displaystyle I\,}
884:group homomorphisms
854:algebraic structure
303:Commutative algebra
142:Associative algebra
24:Algebraic structure
4810:Universal property
4651:Mac Lane, Saunders
4541:
4513:
4478:
4443:
4415:
4391:
4367:
4339:
4315:
4288:
4243:
4206:
4186:
4158:
4132:
4112:
4085:
4049:
4017:
3996:
3970:
3944:
3913:
3837:
3796:
3761:
3517:
3491:
3464:
3423:
3399:
3379:
3303:
3255:algebraic K-theory
3194:is an isomorphism.
3184:
3138:
3111:
3088:
3046:
3012:
2992:
2945:being understood.
2935:
2908:
2856:
2841:
2778:
2732:
2700:
2661:
2632:in the sense that
2622:
2580:
2531:
2505:
2449:
2423:
2377:
2353:
2332:
2289:
2265:
2224:universal property
2212:
2177:
2162:
2111:
2096:
2071:
2022:
2006:
1981:
1966:
1909:
1883:
1819:
1740:
1714:
1688:
1662:
1621:
1581:
1538:
1514:
1487:
1458:
1429:
1398:
1383:
1358:
1302:
1278:
1225:
1193:
1129:
1101:
1067:
1041:
985:
960:
911:
783:
768:
743:
720:
700:
673:group homomorphism
557:Semiprimitive ring
517:
470:
433:
241:Related structures
225:
179:
115:Inner automorphism
101:Ring homomorphisms
5324:
5323:
5282:
5281:
5278:
5277:
5260:monoidal category
5215:
5214:
5087:Enriched category
5039:
5038:
5035:
5034:
5012:Quotient category
5007:Opposite category
4922:
4921:
4632:Bourbaki, Nicolas
4498:
4463:
4325:to some category
4300:filtered category
4262:filtered colimits
4218:covariant functor
4209:{\displaystyle I}
4135:{\displaystyle I}
3989:
3963:
3937:
3789:
3754:
3548:topological space
3494:{\displaystyle n}
3426:{\displaystyle p}
3402:{\displaystyle p}
3114:{\displaystyle m}
3091:{\displaystyle X}
3024:partially ordered
3015:{\displaystyle M}
2952:′ there exists a
2834:
2155:
2089:
1999:
1959:
1505:
1480:
1376:
1305:{\displaystyle I}
820:Formal definition
761:
746:{\displaystyle I}
730:ranges over some
723:{\displaystyle i}
677:ring homomorphism
641:
640:
598:Geometric algebra
309:Commutative rings
160:Category of rings
5349:
5342:Abstract algebra
5314:
5313:
5304:
5303:
5294:
5293:
5129:
5128:
5107:Simplex category
5082:Categorification
5073:
5072:
5054:
5053:
5047:
5017:Product category
5002:Kleisli category
4997:Functor category
4842:Terminal objects
4830:
4829:
4765:Adjoint functors
4718:
4717:
4707:
4706:
4692:
4685:
4678:
4669:
4668:
4664:
4646:
4618:
4617:
4597:
4556:categorical dual
4550:
4548:
4547:
4542:
4540:
4539:
4522:
4520:
4519:
4514:
4509:
4508:
4499:
4496:
4487:
4485:
4484:
4479:
4474:
4473:
4464:
4461:
4453:into a category
4452:
4450:
4449:
4444:
4442:
4441:
4424:
4422:
4421:
4416:
4414:
4413:
4400:
4398:
4397:
4392:
4390:
4389:
4376:
4374:
4373:
4368:
4366:
4365:
4348:
4346:
4345:
4340:
4338:
4337:
4324:
4322:
4321:
4316:
4314:
4313:
4297:
4295:
4294:
4289:
4287:
4286:
4277:
4276:
4252:
4250:
4249:
4244:
4242:
4241:
4232:
4231:
4215:
4213:
4212:
4207:
4195:
4193:
4192:
4187:
4167:
4165:
4164:
4159:
4141:
4139:
4138:
4133:
4121:
4119:
4118:
4113:
4111:
4110:
4094:
4092:
4091:
4086:
4058:
4056:
4055:
4050:
4048:
4047:
4026:
4024:
4023:
4018:
4010:
4009:
3997:
3984:
3983:
3971:
3958:
3957:
3945:
3922:
3920:
3919:
3914:
3906:
3905:
3893:
3892:
3880:
3879:
3846:
3844:
3843:
3838:
3824:
3823:
3811:
3797:
3775:
3774:
3762:
3749:
3526:
3524:
3523:
3518:
3500:
3498:
3497:
3492:
3473:
3471:
3470:
3465:
3460:
3459:
3447:
3432:
3430:
3429:
3424:
3408:
3406:
3405:
3400:
3388:
3386:
3385:
3380:
3378:
3373:
3372:
3357:
3352:
3344:
3339:
3338:
3329:
3324:
3312:
3310:
3309:
3304:
3302:
3297:
3296:
3287:
3282:
3241:, written as GL(
3193:
3191:
3190:
3185:
3177:
3176:
3164:
3163:
3147:
3145:
3144:
3139:
3137:
3136:
3120:
3118:
3117:
3112:
3100:greatest element
3097:
3095:
3094:
3089:
3055:
3053:
3052:
3047:
3045:
3044:
3021:
3019:
3018:
3013:
3001:
2999:
2998:
2993:
2991:
2990:
2944:
2942:
2941:
2936:
2934:
2933:
2917:
2915:
2914:
2909:
2904:
2903:
2888:
2887:
2865:
2863:
2862:
2857:
2855:
2854:
2842:
2799:
2787:
2785:
2784:
2779:
2777:
2776:
2764:
2763:
2741:
2739:
2738:
2733:
2709:
2707:
2706:
2701:
2696:
2695:
2670:
2668:
2667:
2662:
2657:
2656:
2631:
2629:
2628:
2623:
2618:
2617:
2589:
2587:
2586:
2581:
2576:
2575:
2560:
2559:
2540:
2538:
2537:
2532:
2514:
2512:
2511:
2506:
2504:
2503:
2488:
2487:
2475:
2474:
2458:
2456:
2455:
2450:
2432:
2430:
2429:
2424:
2416:
2415:
2403:
2402:
2386:
2384:
2383:
2378:
2376:
2375:
2363:is an object in
2362:
2360:
2359:
2354:
2341:
2339:
2338:
2333:
2328:
2327:
2298:
2296:
2295:
2290:
2288:
2287:
2274:
2272:
2271:
2266:
2261:
2260:
2245:
2244:
2221:
2219:
2218:
2213:
2211:
2210:
2186:
2184:
2183:
2178:
2176:
2175:
2163:
2150:
2149:
2137:
2136:
2120:
2118:
2117:
2112:
2110:
2109:
2097:
2080:
2078:
2077:
2072:
2067:
2066:
2051:
2050:
2031:
2029:
2028:
2023:
2020:
2019:
2007:
1990:
1988:
1987:
1982:
1980:
1979:
1967:
1954:
1953:
1941:
1940:
1918:
1916:
1915:
1910:
1892:
1890:
1889:
1884:
1879:
1878:
1866:
1865:
1849:
1848:
1828:
1826:
1825:
1820:
1814:
1813:
1801:
1800:
1782:
1781:
1769:
1768:
1749:
1747:
1746:
1741:
1723:
1721:
1720:
1715:
1697:
1695:
1694:
1689:
1671:
1669:
1668:
1663:
1661:
1660:
1647:
1646:
1630:
1628:
1627:
1622:
1620:
1619:
1607:
1606:
1590:
1588:
1587:
1582:
1580:
1579:
1567:
1566:
1547:
1545:
1544:
1539:
1531:
1530:
1524:
1523:
1513:
1501:
1500:
1488:
1468:
1467:
1465:
1464:
1459:
1438:
1436:
1435:
1430:
1428:
1427:
1407:
1405:
1404:
1399:
1397:
1396:
1384:
1367:
1365:
1364:
1359:
1354:
1353:
1338:
1337:
1311:
1309:
1308:
1303:
1287:
1285:
1284:
1279:
1274:
1273:
1258:
1257:
1234:
1232:
1231:
1226:
1202:
1200:
1199:
1194:
1192:
1191:
1176:
1175:
1160:
1159:
1138:
1136:
1135:
1130:
1127:
1126:
1110:
1108:
1107:
1102:
1099:
1098:
1076:
1074:
1073:
1068:
1050:
1048:
1047:
1042:
1040:
1039:
1027:
1026:
1014:
1013:
994:
992:
991:
986:
969:
967:
966:
961:
944:
943:
920:
918:
917:
912:
792:
790:
789:
784:
782:
781:
769:
753:, is denoted by
752:
750:
749:
744:
729:
727:
726:
721:
709:
707:
706:
701:
699:
698:
679:, or in general
633:
626:
619:
604:Operator algebra
590:Clifford algebra
526:
524:
523:
518:
513:
512:
500:
479:
477:
476:
471:
469:
468:
463:
442:
440:
439:
434:
432:
431:
426:
404:Ring of integers
398:
395:Integers modulo
346:Euclidean domain
234:
232:
231:
226:
224:
216:
211:
188:
186:
185:
180:
178:
82:Product of rings
68:Fractional ideal
27:
19:
18:
5357:
5356:
5352:
5351:
5350:
5348:
5347:
5346:
5327:
5326:
5325:
5320:
5274:
5244:
5211:
5188:
5179:
5136:
5120:
5071:
5061:
5048:
5031:
4980:
4918:
4887:Initial objects
4873:
4819:
4712:
4701:
4699:Category theory
4696:
4627:
4622:
4621:
4614:
4598:
4594:
4589:
4577:
4568:
4535:
4534:
4532:
4529:
4528:
4504:
4503:
4495:
4493:
4490:
4489:
4469:
4468:
4460:
4458:
4455:
4454:
4437:
4436:
4434:
4431:
4430:
4409:
4408:
4406:
4403:
4402:
4385:
4384:
4382:
4379:
4378:
4361:
4360:
4358:
4355:
4354:
4333:
4332:
4330:
4327:
4326:
4309:
4308:
4306:
4303:
4302:
4282:
4281:
4272:
4271:
4269:
4266:
4265:
4237:
4236:
4227:
4226:
4224:
4221:
4220:
4201:
4198:
4197:
4175:
4172:
4171:
4147:
4144:
4143:
4127:
4124:
4123:
4106:
4105:
4103:
4100:
4099:
4068:
4065:
4064:
4043:
4042:
4040:
4037:
4036:
4033:
4005:
4001:
3988:
3979:
3975:
3962:
3953:
3949:
3936:
3928:
3925:
3924:
3901:
3897:
3888:
3884:
3875:
3871:
3863:
3860:
3859:
3819:
3815:
3801:
3788:
3770:
3766:
3753:
3739:
3737:
3734:
3733:
3723:
3683:
3674:
3653:
3628:
3601:
3573:if and only if
3506:
3503:
3502:
3486:
3483:
3482:
3455:
3451:
3443:
3441:
3438:
3437:
3418:
3415:
3414:
3394:
3391:
3390:
3374:
3362:
3358:
3353:
3348:
3340:
3334:
3330:
3325:
3320:
3318:
3315:
3314:
3298:
3292:
3288:
3283:
3278:
3276:
3273:
3272:
3205:, consider the
3172:
3168:
3159:
3155:
3153:
3150:
3149:
3132:
3128:
3126:
3123:
3122:
3106:
3103:
3102:
3083:
3080:
3079:
3040:
3036:
3031:
3028:
3027:
3007:
3004:
3003:
2986:
2982:
2980:
2977:
2976:
2972:
2929:
2925:
2923:
2920:
2919:
2896:
2892:
2883:
2879:
2874:
2871:
2870:
2850:
2846:
2833:
2825:
2822:
2821:
2800:
2772:
2768:
2759:
2755:
2747:
2744:
2743:
2715:
2712:
2711:
2691:
2687:
2676:
2673:
2672:
2652:
2648:
2637:
2634:
2633:
2613:
2609:
2598:
2595:
2594:
2568:
2564:
2555:
2551:
2546:
2543:
2542:
2520:
2517:
2516:
2496:
2492:
2483:
2479:
2470:
2466:
2464:
2461:
2460:
2438:
2435:
2434:
2411:
2407:
2398:
2394:
2392:
2389:
2388:
2371:
2370:
2368:
2365:
2364:
2347:
2344:
2343:
2323:
2319:
2308:
2305:
2304:
2283:
2282:
2280:
2277:
2276:
2253:
2249:
2240:
2236:
2231:
2228:
2227:
2206:
2205:
2203:
2200:
2199:
2193:
2171:
2167:
2154:
2145:
2141:
2132:
2128:
2126:
2123:
2122:
2105:
2101:
2088:
2086:
2083:
2082:
2059:
2055:
2046:
2042:
2037:
2034:
2033:
2015:
2011:
1998:
1996:
1993:
1992:
1975:
1971:
1958:
1949:
1945:
1936:
1932:
1930:
1927:
1926:
1898:
1895:
1894:
1874:
1870:
1858:
1854:
1844:
1840:
1838:
1835:
1834:
1809:
1805:
1793:
1789:
1777:
1773:
1761:
1757:
1755:
1752:
1751:
1729:
1726:
1725:
1703:
1700:
1699:
1677:
1674:
1673:
1656:
1652:
1642:
1638:
1636:
1633:
1632:
1615:
1611:
1602:
1598:
1596:
1593:
1592:
1575:
1571:
1562:
1558:
1556:
1553:
1552:
1526:
1525:
1519:
1515:
1509:
1496:
1492:
1479:
1477:
1474:
1473:
1452:
1449:
1448:
1444:
1423:
1419:
1417:
1414:
1413:
1392:
1388:
1375:
1373:
1370:
1369:
1346:
1342:
1333:
1329:
1324:
1321:
1320:
1297:
1294:
1293:
1266:
1262:
1253:
1249:
1244:
1241:
1240:
1208:
1205:
1204:
1184:
1180:
1168:
1164:
1152:
1148:
1146:
1143:
1142:
1122:
1118:
1116:
1113:
1112:
1091:
1087:
1085:
1082:
1081:
1056:
1053:
1052:
1035:
1031:
1022:
1018:
1006:
1002:
1000:
997:
996:
979:
976:
975:
939:
935:
930:
927:
926:
894:
891:
890:
846:
822:
802:category theory
777:
773:
760:
758:
755:
754:
738:
735:
734:
715:
712:
711:
694:
690:
688:
685:
684:
637:
608:
607:
540:
530:
529:
508:
504:
496:
494:
491:
490:
464:
459:
458:
456:
453:
452:
427:
422:
421:
419:
416:
415:
396:
366:Polynomial ring
316:Integral domain
305:
295:
294:
220:
212:
207:
199:
196:
195:
174:
172:
169:
168:
154:Involutive ring
39:
28:
22:
17:
12:
11:
5:
5355:
5345:
5344:
5339:
5322:
5321:
5319:
5318:
5308:
5298:
5287:
5284:
5283:
5280:
5279:
5276:
5275:
5273:
5272:
5267:
5262:
5248:
5242:
5237:
5232:
5226:
5224:
5217:
5216:
5213:
5212:
5210:
5209:
5204:
5193:
5191:
5186:
5181:
5180:
5178:
5177:
5172:
5167:
5162:
5157:
5152:
5141:
5139:
5134:
5126:
5122:
5121:
5119:
5114:
5112:String diagram
5109:
5104:
5102:Model category
5099:
5094:
5089:
5084:
5079:
5077:
5070:
5069:
5066:
5063:
5062:
5050:
5049:
5042:
5040:
5037:
5036:
5033:
5032:
5030:
5029:
5024:
5022:Comma category
5019:
5014:
5009:
5004:
4999:
4994:
4988:
4986:
4982:
4981:
4979:
4978:
4968:
4958:
4956:Abelian groups
4953:
4948:
4943:
4938:
4932:
4930:
4924:
4923:
4920:
4919:
4917:
4916:
4911:
4906:
4905:
4904:
4894:
4889:
4883:
4881:
4875:
4874:
4872:
4871:
4866:
4861:
4860:
4859:
4849:
4844:
4838:
4836:
4827:
4821:
4820:
4818:
4817:
4812:
4807:
4802:
4797:
4792:
4787:
4782:
4777:
4772:
4767:
4762:
4761:
4760:
4755:
4750:
4745:
4740:
4735:
4724:
4722:
4714:
4713:
4703:
4702:
4695:
4694:
4687:
4680:
4672:
4666:
4665:
4647:
4626:
4623:
4620:
4619:
4612:
4591:
4590:
4588:
4585:
4584:
4583:
4576:
4573:
4567:
4564:
4538:
4512:
4507:
4502:
4477:
4472:
4467:
4440:
4412:
4388:
4364:
4336:
4312:
4285:
4280:
4275:
4240:
4235:
4230:
4205:
4185:
4182:
4179:
4169:if and only if
4157:
4154:
4151:
4131:
4109:
4097:small category
4084:
4081:
4078:
4075:
4072:
4046:
4032:
4029:
4016:
4013:
4008:
4004:
4000:
3995:
3992:
3987:
3982:
3978:
3974:
3969:
3966:
3961:
3956:
3952:
3948:
3943:
3940:
3935:
3932:
3912:
3909:
3904:
3900:
3896:
3891:
3887:
3883:
3878:
3874:
3870:
3867:
3848:
3847:
3836:
3833:
3830:
3827:
3822:
3818:
3814:
3810:
3807:
3804:
3800:
3795:
3792:
3787:
3784:
3781:
3778:
3773:
3769:
3765:
3760:
3757:
3752:
3748:
3745:
3742:
3727:inverse limits
3722:
3719:
3718:
3717:
3710:
3707:final topology
3699:
3679:
3670:
3649:
3624:
3593:
3553:. Fix a point
3532:
3516:
3513:
3510:
3490:
3475:
3463:
3458:
3454:
3450:
3446:
3422:
3411:roots of unity
3398:
3377:
3371:
3368:
3365:
3361:
3356:
3351:
3347:
3343:
3337:
3333:
3328:
3323:
3301:
3295:
3291:
3286:
3281:
3258:
3195:
3183:
3180:
3175:
3171:
3167:
3162:
3158:
3135:
3131:
3110:
3087:
3076:
3065:
3043:
3039:
3035:
3011:
2989:
2985:
2971:
2968:
2932:
2928:
2907:
2902:
2899:
2895:
2891:
2886:
2882:
2878:
2867:
2866:
2853:
2849:
2845:
2840:
2837:
2832:
2829:
2794:
2775:
2771:
2767:
2762:
2758:
2754:
2751:
2731:
2728:
2725:
2722:
2719:
2699:
2694:
2690:
2686:
2683:
2680:
2660:
2655:
2651:
2647:
2644:
2641:
2621:
2616:
2612:
2608:
2605:
2602:
2579:
2574:
2571:
2567:
2563:
2558:
2554:
2550:
2530:
2527:
2524:
2502:
2499:
2495:
2491:
2486:
2482:
2478:
2473:
2469:
2448:
2445:
2442:
2422:
2419:
2414:
2410:
2406:
2401:
2397:
2374:
2351:
2331:
2326:
2322:
2318:
2315:
2312:
2286:
2264:
2259:
2256:
2252:
2248:
2243:
2239:
2235:
2222:by means of a
2209:
2192:
2189:
2174:
2170:
2166:
2161:
2158:
2153:
2148:
2144:
2140:
2135:
2131:
2108:
2104:
2100:
2095:
2092:
2070:
2065:
2062:
2058:
2054:
2049:
2045:
2041:
2018:
2014:
2010:
2005:
2002:
1978:
1974:
1970:
1965:
1962:
1957:
1952:
1948:
1944:
1939:
1935:
1908:
1905:
1902:
1882:
1877:
1873:
1869:
1864:
1861:
1857:
1852:
1847:
1843:
1817:
1812:
1808:
1804:
1799:
1796:
1792:
1788:
1785:
1780:
1776:
1772:
1767:
1764:
1760:
1739:
1736:
1733:
1713:
1710:
1707:
1687:
1684:
1681:
1659:
1655:
1650:
1645:
1641:
1618:
1614:
1610:
1605:
1601:
1578:
1574:
1570:
1565:
1561:
1549:
1548:
1537:
1534:
1529:
1522:
1518:
1512:
1508:
1504:
1499:
1495:
1491:
1486:
1483:
1456:
1426:
1422:
1410:disjoint union
1395:
1391:
1387:
1382:
1379:
1368:is denoted by
1357:
1352:
1349:
1345:
1341:
1336:
1332:
1328:
1301:
1277:
1272:
1269:
1265:
1261:
1256:
1252:
1248:
1239:Then the pair
1237:
1236:
1224:
1221:
1218:
1215:
1212:
1190:
1187:
1183:
1179:
1174:
1171:
1167:
1163:
1158:
1155:
1151:
1140:
1125:
1121:
1097:
1094:
1090:
1066:
1063:
1060:
1038:
1034:
1030:
1025:
1021:
1017:
1012:
1009:
1005:
983:
959:
956:
953:
950:
947:
942:
938:
934:
910:
907:
904:
901:
898:
872:(over a fixed
845:
842:
821:
818:
810:inverse limits
780:
776:
772:
767:
764:
742:
719:
697:
693:
639:
638:
636:
635:
628:
621:
613:
610:
609:
601:
600:
572:
571:
565:
559:
553:
541:
536:
535:
532:
531:
528:
527:
516:
511:
507:
503:
499:
480:
467:
462:
443:
430:
425:
413:-adic integers
406:
400:
391:
377:
376:
375:
374:
368:
362:
361:
360:
348:
342:
336:
330:
324:
306:
301:
300:
297:
296:
293:
292:
291:
290:
278:
277:
276:
270:
258:
257:
256:
238:
237:
236:
235:
223:
219:
215:
210:
206:
203:
189:
177:
156:
150:
144:
138:
124:
123:
117:
111:
97:
96:
90:
84:
78:
77:
76:
70:
58:
52:
40:
38:Basic concepts
37:
36:
33:
32:
15:
9:
6:
4:
3:
2:
5354:
5343:
5340:
5338:
5335:
5334:
5332:
5317:
5309:
5307:
5299:
5297:
5289:
5288:
5285:
5271:
5268:
5266:
5263:
5261:
5257:
5253:
5249:
5247:
5245:
5238:
5236:
5233:
5231:
5228:
5227:
5225:
5222:
5218:
5208:
5205:
5202:
5198:
5195:
5194:
5192:
5190:
5182:
5176:
5173:
5171:
5168:
5166:
5163:
5161:
5160:Tetracategory
5158:
5156:
5153:
5150:
5149:pseudofunctor
5146:
5143:
5142:
5140:
5138:
5130:
5127:
5123:
5118:
5115:
5113:
5110:
5108:
5105:
5103:
5100:
5098:
5095:
5093:
5090:
5088:
5085:
5083:
5080:
5078:
5074:
5068:
5067:
5064:
5060:
5055:
5051:
5046:
5028:
5025:
5023:
5020:
5018:
5015:
5013:
5010:
5008:
5005:
5003:
5000:
4998:
4995:
4993:
4992:Free category
4990:
4989:
4987:
4983:
4976:
4975:Vector spaces
4972:
4969:
4966:
4962:
4959:
4957:
4954:
4952:
4949:
4947:
4944:
4942:
4939:
4937:
4934:
4933:
4931:
4929:
4925:
4915:
4912:
4910:
4907:
4903:
4900:
4899:
4898:
4895:
4893:
4890:
4888:
4885:
4884:
4882:
4880:
4876:
4870:
4869:Inverse limit
4867:
4865:
4862:
4858:
4855:
4854:
4853:
4850:
4848:
4845:
4843:
4840:
4839:
4837:
4835:
4831:
4828:
4826:
4822:
4816:
4813:
4811:
4808:
4806:
4803:
4801:
4798:
4796:
4795:Kan extension
4793:
4791:
4788:
4786:
4783:
4781:
4778:
4776:
4773:
4771:
4768:
4766:
4763:
4759:
4756:
4754:
4751:
4749:
4746:
4744:
4741:
4739:
4736:
4734:
4731:
4730:
4729:
4726:
4725:
4723:
4719:
4715:
4708:
4704:
4700:
4693:
4688:
4686:
4681:
4679:
4674:
4673:
4670:
4662:
4658:
4657:
4652:
4648:
4645:
4641:
4637:
4633:
4629:
4628:
4615:
4613:9780521422611
4609:
4605:
4604:
4596:
4592:
4582:
4579:
4578:
4572:
4563:
4561:
4560:inverse limit
4557:
4552:
4526:
4428:
4351:
4301:
4263:
4258:
4256:
4219:
4203:
4183:
4180:
4177:
4170:
4155:
4149:
4129:
4098:
4079:
4076:
4073:
4062:
4028:
4014:
4006:
4002:
3998:
3993:
3990:
3980:
3976:
3972:
3967:
3964:
3954:
3950:
3946:
3941:
3938:
3930:
3910:
3902:
3898:
3889:
3885:
3876:
3872:
3865:
3857:
3856:exact functor
3853:
3834:
3828:
3825:
3820:
3816:
3798:
3793:
3790:
3785:
3779:
3776:
3771:
3767:
3763:
3758:
3755:
3732:
3731:
3730:
3728:
3715:
3711:
3708:
3704:
3700:
3697:
3693:
3689:
3688:
3682:
3678:
3675:of the stalk
3673:
3669:
3665:
3661:
3657:
3652:
3648:
3644:
3640:
3636:
3632:
3627:
3623:
3619:
3615:
3611:
3610:
3605:
3600:
3596:
3592:
3588:
3584:
3580:
3576:
3572:
3568:
3564:
3560:
3556:
3552:
3549:
3545:
3541:
3537:
3533:
3530:
3514:
3511:
3508:
3488:
3480:
3476:
3452:
3436:
3420:
3412:
3396:
3369:
3366:
3363:
3359:
3354:
3335:
3331:
3326:
3293:
3289:
3284:
3271:
3270:factor groups
3267:
3263:
3259:
3256:
3252:
3248:
3244:
3240:
3236:
3232:
3228:
3224:
3220:
3216:
3212:
3208:
3204:
3200:
3196:
3181:
3173:
3169:
3165:
3160:
3156:
3133:
3129:
3108:
3101:
3085:
3077:
3074:
3070:
3069:weak topology
3066:
3063:
3059:
3041:
3037:
3033:
3025:
3009:
2987:
2983:
2974:
2973:
2967:
2965:
2961:
2958:
2955:
2951:
2946:
2930:
2926:
2900:
2897:
2893:
2889:
2884:
2880:
2851:
2847:
2843:
2838:
2835:
2830:
2827:
2820:
2819:
2818:
2815:
2813:
2809:
2805:
2798:
2793:
2791:
2773:
2769:
2765:
2760:
2756:
2752:
2749:
2729:
2723:
2720:
2717:
2692:
2688:
2684:
2681:
2653:
2649:
2645:
2642:
2614:
2610:
2606:
2603:
2593:
2572:
2569:
2565:
2561:
2556:
2552:
2528:
2525:
2522:
2500:
2497:
2493:
2489:
2484:
2480:
2476:
2471:
2467:
2446:
2443:
2440:
2420:
2412:
2408:
2404:
2399:
2395:
2349:
2324:
2320:
2316:
2313:
2302:
2257:
2254:
2250:
2246:
2241:
2237:
2225:
2198:
2188:
2172:
2168:
2164:
2159:
2156:
2146:
2142:
2138:
2133:
2129:
2106:
2102:
2098:
2093:
2090:
2063:
2060:
2056:
2052:
2047:
2043:
2016:
2012:
2008:
2003:
2000:
1976:
1972:
1968:
1963:
1960:
1950:
1946:
1942:
1937:
1933:
1925:
1920:
1906:
1903:
1900:
1875:
1871:
1862:
1859:
1855:
1850:
1845:
1841:
1832:
1831:inverse limit
1810:
1806:
1797:
1794:
1790:
1786:
1778:
1774:
1765:
1762:
1758:
1737:
1734:
1731:
1711:
1708:
1705:
1685:
1682:
1679:
1657:
1653:
1648:
1643:
1639:
1616:
1612:
1608:
1603:
1599:
1576:
1572:
1568:
1563:
1559:
1535:
1532:
1520:
1516:
1510:
1506:
1502:
1497:
1493:
1489:
1484:
1481:
1472:
1471:
1470:
1454:
1447:
1442:
1424:
1420:
1411:
1393:
1389:
1385:
1380:
1377:
1350:
1347:
1343:
1339:
1334:
1330:
1318:
1313:
1299:
1291:
1290:direct system
1270:
1267:
1263:
1259:
1254:
1250:
1222:
1219:
1216:
1213:
1210:
1188:
1185:
1181:
1177:
1172:
1169:
1165:
1161:
1156:
1153:
1149:
1141:
1123:
1119:
1095:
1092:
1088:
1080:
1079:
1078:
1064:
1061:
1058:
1036:
1032:
1023:
1019:
1015:
1010:
1007:
1003:
981:
973:
954:
951:
948:
945:
940:
936:
924:
905:
902:
899:
887:
885:
881:
880:
879:homomorphisms
875:
871:
867:
863:
859:
855:
851:
841:
839:
835:
831:
827:
817:
815:
811:
807:
803:
799:
794:
778:
774:
770:
765:
762:
740:
733:
717:
695:
691:
682:
678:
674:
670:
669:homomorphisms
666:
662:
661:vector spaces
658:
654:
650:
646:
634:
629:
627:
622:
620:
615:
614:
612:
611:
606:
605:
599:
595:
594:
593:
592:
591:
586:
585:
584:
579:
578:
577:
570:
566:
564:
560:
558:
554:
552:
551:Division ring
548:
547:
546:
545:
539:
534:
533:
505:
489:
487:
481:
465:
451:
450:-adic numbers
449:
444:
428:
414:
412:
407:
405:
401:
399:
392:
390:
386:
385:
384:
383:
382:
373:
369:
367:
363:
359:
355:
354:
353:
349:
347:
343:
341:
337:
335:
331:
329:
325:
323:
319:
318:
317:
313:
312:
311:
310:
304:
299:
298:
289:
285:
284:
283:
279:
275:
271:
269:
265:
264:
263:
259:
255:
251:
250:
249:
245:
244:
243:
242:
217:
213:
204:
201:
194:
193:Terminal ring
190:
167:
163:
162:
161:
157:
155:
151:
149:
145:
143:
139:
137:
133:
132:
131:
130:
129:
122:
118:
116:
112:
110:
106:
105:
104:
103:
102:
95:
91:
89:
85:
83:
79:
75:
71:
69:
65:
64:
63:
62:Quotient ring
59:
57:
53:
51:
47:
46:
45:
44:
35:
34:
31:
26:→ Ring theory
25:
21:
20:
5240:
5221:Categorified
5125:n-categories
5076:Key concepts
4914:Direct limit
4913:
4897:Coequalizers
4815:Yoneda lemma
4721:Key concepts
4711:Key concepts
4654:
4635:
4602:
4595:
4569:
4553:
4352:
4259:
4034:
3849:
3724:
3695:
3691:
3685:
3680:
3676:
3671:
3667:
3663:
3659:
3655:
3650:
3646:
3642:
3638:
3634:
3630:
3625:
3621:
3617:
3613:
3607:
3603:
3598:
3594:
3590:
3586:
3582:
3578:
3574:
3570:
3566:
3562:
3558:
3554:
3550:
3539:
3535:
3435:Prüfer group
3266:prime number
3261:
3250:
3246:
3242:
3238:
3234:
3230:
3226:
3222:
3218:
3214:
3210:
3202:
3198:
2963:
2959:
2953:
2949:
2947:
2868:
2816:
2811:
2807:
2801:
2789:
2591:
2300:
2194:
1923:
1921:
1550:
1317:direct limit
1316:
1314:
1289:
1288:is called a
1238:
923:directed set
888:
877:
847:
823:
795:
732:directed set
649:direct limit
648:
642:
602:
588:
587:
583:Free algebra
581:
580:
574:
573:
542:
485:
447:
410:
379:
378:
358:Finite field
307:
254:Finite field
240:
239:
166:Initial ring
126:
125:
99:
98:
41:
5189:-categories
5165:Kan complex
5155:Tricategory
5137:-categories
5027:Subcategory
4785:Exponential
4753:Preadditive
4748:Pre-abelian
4566:Terminology
4525:ind-objects
4523:are called
3684:called the
3666:an element
2957:isomorphism
645:mathematics
563:Simple ring
274:Jordan ring
148:Graded ring
30:Ring theory
5331:Categories
5207:3-category
5197:2-category
5170:∞-groupoid
5145:Bicategory
4892:Coproducts
4852:Equalizers
4758:Bicategory
4625:References
3721:Properties
3714:ind-scheme
3620:, denoted
3073:CW complex
2802:will then
2742:such that
2515:whenever
2459:such that
2303:is a pair
1750:such that
1443:a certain
856:, such as
569:Commutator
328:GCD domain
5256:Symmetric
5201:2-functor
4941:Relations
4864:Pullbacks
4279:→
4234:→
4181:≤
4153:→
4083:⟩
4080:≤
4071:⟨
4012:→
3999:
3994:→
3986:→
3973:
3968:→
3960:→
3947:
3942:→
3934:→
3908:→
3895:→
3882:→
3869:→
3799:
3794:←
3764:
3759:→
3577:contains
3457:∞
3346:→
3179:→
3157:ϕ
3058:subgroups
3034:⋃
3002:of a set
2927:ϕ
2906:⟩
2877:⟨
2844:
2839:→
2788:for each
2770:ψ
2757:ϕ
2753:∘
2727:→
2721::
2698:⟩
2689:ψ
2679:⟨
2659:⟩
2650:ϕ
2640:⟨
2620:⟩
2611:ϕ
2601:⟨
2578:⟩
2549:⟨
2526:≤
2490:∘
2481:ϕ
2468:ϕ
2444:∈
2418:→
2405::
2396:ϕ
2330:⟩
2321:ϕ
2311:⟨
2263:⟩
2234:⟨
2165:
2160:→
2152:→
2139::
2130:ϕ
2099:
2094:→
2069:⟩
2040:⟨
2009:
2004:→
1969:
1964:→
1956:→
1943::
1934:ϕ
1904:≤
1893:whenever
1851:∼
1735:≤
1709:≤
1683:∈
1649:∼
1609:∈
1569:∈
1551:Here, if
1533:∼
1507:⨆
1490:
1485:→
1455:∼
1386:
1381:→
1356:⟩
1327:⟨
1276:⟩
1247:⟨
1220:≤
1214:≤
1178:∘
1062:≤
1029:→
1016::
952:∈
909:⟩
906:≤
897:⟨
886:, etc.).
771:
766:→
681:morphisms
510:∞
288:Semifield
5316:Glossary
5296:Category
5270:n-monoid
5223:concepts
4879:Colimits
4847:Products
4800:Morphism
4743:Concrete
4738:Additive
4728:Category
4653:(1998),
4634:(1968),
4575:See also
4061:functors
3602:) where
3542:-valued
3062:subrings
2970:Examples
2806:for all
2197:category
1203:for all
870:algebras
838:category
710:, where
665:category
282:Semiring
268:Lie ring
50:Subrings
5306:Outline
5265:n-group
5230:2-group
5185:Strict
5175:∞-topos
4971:Modules
4909:Pushout
4857:Kernels
4790:Functor
4733:Abelian
4644:0237342
4298:from a
4255:colimit
3852:modules
3229:) → GL(
3022:can be
2804:commute
1631:, then
1412:of the
972:indexed
866:modules
834:modules
798:colimit
484:Prüfer
86:•
5252:Traced
5235:2-ring
4965:Fields
4951:Groups
4946:Magmas
4834:Limits
4642:
4610:
4253:. The
3854:is an
2954:unique
2342:where
2301:target
2226:. Let
1441:modulo
925:. Let
858:groups
830:groups
814:limits
653:groups
136:Module
109:Kernel
5246:-ring
5133:Weak
5117:Topos
4961:Rings
4587:Notes
3662:over
3609:stalk
3546:on a
3544:sheaf
3538:be a
3264:be a
3071:of a
2590:is a
1698:with
1292:over
1139:, and
995:and
921:be a
874:field
862:rings
828:like
657:rings
488:-ring
352:Field
248:Field
56:Ideal
43:Rings
4936:Sets
4608:ISBN
4554:The
4551:.
4527:of
3729:via
3687:germ
3645:) →
3534:Let
3260:Let
3197:Let
3078:Let
3067:The
2962:′ →
2387:and
1724:and
1591:and
1315:The
889:Let
850:sets
832:and
806:dual
647:, a
4780:End
4770:CCC
4497:Ind
4462:Ind
3991:lim
3965:lim
3939:lim
3791:lim
3756:lim
3712:An
3694:at
3690:of
3658:of
3633:of
3616:at
3612:of
3589:),
3557:in
3481:in
3233:+1;
3227:n;K
3211:n;K
3209:GL(
2836:lim
2814:.
2157:lim
2091:lim
2001:lim
1961:lim
1482:lim
1439:'s
1378:lim
974:by
808:to
800:in
763:lim
643:In
5333::
5258:)
5254:)(
4659:,
4640:MR
4027:.
3569:≤
3217:x
2810:,
2187:.
1919:.
1469::
1312:.
864:,
860:,
840:.
675:,
659:,
655:,
596:•
567:•
561:•
555:•
549:•
482:•
445:•
408:•
402:•
393:•
387:•
370:•
364:•
356:•
350:•
344:•
338:•
332:•
326:•
320:•
314:•
286:•
280:•
272:•
266:•
260:•
252:•
246:•
191:•
164:•
158:•
152:•
146:•
140:•
134:•
119:•
113:•
107:•
92:•
80:•
72:•
66:•
60:•
54:•
48:•
5250:(
5243:n
5241:E
5203:)
5199:(
5187:n
5151:)
5147:(
5135:n
4977:)
4973:(
4967:)
4963:(
4691:e
4684:t
4677:v
4616:.
4537:C
4511:)
4506:C
4501:(
4476:)
4471:C
4466:(
4439:C
4411:C
4387:C
4363:C
4335:C
4311:J
4284:C
4274:J
4239:C
4229:I
4204:I
4184:j
4178:i
4156:j
4150:i
4130:I
4108:I
4077:,
4074:I
4045:C
4015:0
4007:i
4003:C
3981:i
3977:B
3955:i
3951:A
3931:0
3911:0
3903:i
3899:C
3890:i
3886:B
3877:i
3873:A
3866:0
3835:.
3832:)
3829:Y
3826:,
3821:i
3817:X
3813:(
3809:m
3806:o
3803:H
3786:=
3783:)
3780:Y
3777:,
3772:i
3768:X
3751:(
3747:m
3744:o
3741:H
3698:.
3696:x
3692:s
3681:x
3677:F
3672:x
3668:s
3664:U
3660:F
3656:s
3651:x
3647:F
3643:U
3641:(
3639:F
3635:x
3631:U
3626:x
3622:F
3618:x
3614:F
3604:r
3599:V
3597:,
3595:U
3591:r
3587:U
3585:(
3583:F
3579:V
3575:U
3571:V
3567:U
3563:x
3559:X
3555:x
3551:X
3540:C
3536:F
3531:.
3515:1
3512:+
3509:n
3489:n
3474:.
3462:)
3453:p
3449:(
3445:Z
3421:p
3397:p
3376:Z
3370:1
3367:+
3364:n
3360:p
3355:/
3350:Z
3342:Z
3336:n
3332:p
3327:/
3322:Z
3300:Z
3294:n
3290:p
3285:/
3280:Z
3262:p
3257:.
3251:K
3247:K
3243:K
3239:K
3235:K
3231:n
3223:K
3219:n
3215:n
3203:n
3199:K
3182:X
3174:m
3170:X
3166::
3161:m
3134:m
3130:X
3109:m
3086:X
3042:i
3038:M
3010:M
2988:i
2984:M
2964:X
2960:X
2950:X
2931:i
2901:j
2898:i
2894:f
2890:,
2885:i
2881:X
2852:i
2848:X
2831:=
2828:X
2812:j
2808:i
2790:i
2774:i
2766:=
2761:i
2750:u
2730:Y
2724:X
2718:u
2693:i
2685:,
2682:Y
2654:i
2646:,
2643:X
2615:i
2607:,
2604:X
2573:j
2570:i
2566:f
2562:,
2557:i
2553:X
2529:j
2523:i
2501:j
2498:i
2494:f
2485:j
2477:=
2472:i
2447:I
2441:i
2421:X
2413:i
2409:X
2400:i
2373:C
2350:X
2325:i
2317:,
2314:X
2285:C
2258:j
2255:i
2251:f
2247:,
2242:i
2238:X
2208:C
2173:i
2169:A
2147:j
2143:A
2134:j
2107:i
2103:A
2064:j
2061:i
2057:f
2053:,
2048:i
2044:A
2017:i
2013:A
1977:i
1973:A
1951:j
1947:A
1938:j
1907:j
1901:i
1881:)
1876:i
1872:x
1868:(
1863:j
1860:i
1856:f
1846:i
1842:x
1816:)
1811:j
1807:x
1803:(
1798:k
1795:j
1791:f
1787:=
1784:)
1779:i
1775:x
1771:(
1766:k
1763:i
1759:f
1738:k
1732:j
1712:k
1706:i
1686:I
1680:k
1658:j
1654:x
1644:i
1640:x
1617:j
1613:A
1604:j
1600:x
1577:i
1573:A
1564:i
1560:x
1536:.
1528:/
1521:i
1517:A
1511:i
1503:=
1498:i
1494:A
1425:i
1421:A
1394:i
1390:A
1351:j
1348:i
1344:f
1340:,
1335:i
1331:A
1300:I
1271:j
1268:i
1264:f
1260:,
1255:i
1251:A
1235:.
1223:k
1217:j
1211:i
1189:j
1186:i
1182:f
1173:k
1170:j
1166:f
1162:=
1157:k
1154:i
1150:f
1124:i
1120:A
1096:i
1093:i
1089:f
1065:j
1059:i
1037:j
1033:A
1024:i
1020:A
1011:j
1008:i
1004:f
982:I
958:}
955:I
949:i
946::
941:i
937:A
933:{
903:,
900:I
779:i
775:A
741:I
718:i
696:i
692:A
671:(
632:e
625:t
618:v
515:)
506:p
502:(
498:Z
486:p
466:p
461:Q
448:p
429:p
424:Z
411:p
397:n
222:Z
218:1
214:/
209:Z
205:=
202:0
176:Z
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