4238:
4485:
1239:
4505:
4495:
868:
2854:
2344:, every inverse system has an inverse limit, which can be constructed in an elementary manner as a subset of the product of the sets forming the inverse system. The inverse limit of any inverse system of non-empty finite sets is non-empty. This is a generalization of
680:
2998:
The name "Mittag-Leffler" for this condition was given by
Bourbaki in their chapter on uniform structures for a similar result about inverse limits of complete Hausdorff uniform spaces. Mittag-Leffler used a similar argument in the proof of
3551:
a commutative ring; it is not necessarily true in an arbitrary abelian category (see Roos' "Derived functors of inverse limits revisited" for examples of abelian categories in which lim, on diagrams indexed by a countable set, is nonzero
2586:
2711:
389:
3215:
2726:
3412:
635:
485:
3321:
1913:
1711:
2462:
1804:
1594:
2989:
1532:
2196:
1466:
215:
3698:
3482:
1856:
1654:
2254:
2132:
1302:
3063:
2066:
863:{\displaystyle A=\varprojlim _{i\in I}{A_{i}}=\left\{\left.{\vec {a}}\in \prod _{i\in I}A_{i}\;\right|\;a_{i}=f_{ij}(a_{j}){\text{ for all }}i\leq j{\text{ in }}I\right\}.}
2521:
3517:
1354:
1095:
2026:
125:
1942:
1740:
241:
2306:
1180:
535:
274:
998:
671:
301:
1359:
In some categories, the inverse limit of certain inverse systems does not exist. If it does, however, it is unique in a strong sense: given any two inverse limits
1968:
1988:
964:
944:
891:
505:
2533:
1806:
with the index set being the natural numbers with the usual order, and the morphisms being "take remainder". That is, one considers sequences of real numbers
2651:
308:
3733:
3459:
3576:
of category theory. The terminology is somewhat confusing: inverse limits are a class of limits, while direct limits are a class of colimits.
3146:
2849:{\displaystyle 0\rightarrow \varprojlim A_{i}\rightarrow \varprojlim B_{i}\rightarrow \varprojlim C_{i}\rightarrow \varprojlim {}^{1}A_{i}}
3882:
3680:
4167:
3352:
547:
397:
3835:
3658:
3629:
3601:
3267:
2378:
is the inverse limit of the set of finite strings, and is thus endowed with the limit topology. As the original spaces are
1861:
1659:
3022:
a system of finite-dimensional vector spaces or finite abelian groups or modules of finite length or
Artinian modules.
4529:
3689:
2421:
1767:
1557:
2916:
1758:
2360:
43:
although their existence depends on the category that is considered. They are a special case of the concept of
3875:
3335:
1604:
with the usual order, and the morphisms being "take remainder". That is, one considers sequences of integers
3475:≠ 0. Roos has since shown (in "Derived functors of inverse limits revisited") that his result is correct if
1495:
4079:
4034:
2353:
2137:
1432:
164:
1809:
1607:
4508:
4448:
3794:
3736:(2002), "A counterexample to a 1961 "theorem" in homological algebra (with appendix by Pierre Deligne)",
3000:
2201:
2079:
1261:
4157:
3466:
constructed an example of such a system in a category satisfying (AB4) (in addition to (AB4*)) with lim
3029:
4498:
4284:
4148:
4056:
4534:
4457:
4101:
4039:
3962:
3246:
2375:
4488:
4444:
4049:
3868:
3738:
2034:
2499:
4044:
4026:
3703:
3573:
3495:
44:
1993:
1310:
1051:
4251:
4017:
3997:
3920:
3565:
2349:
1469:
1041:
98:
40:
4133:
3972:
1918:
1716:
1419:
220:
2281:
1155:
510:
249:
3945:
3940:
3845:
3815:
3782:
3759:
3726:
2717:
2487:
of abelian groups, the Mittag-Leffler condition is a condition on the transition morphisms
2383:
1021:
1017:
976:
649:
279:
1858:
such that each element of the sequence "projects" down to the previous ones, namely, that
1656:
such that each element of the sequence "projects" down to the previous ones, namely, that
8:
4289:
4237:
4163:
3967:
2581:{\displaystyle \varprojlim {}^{1}:\operatorname {Ab} ^{I}\rightarrow \operatorname {Ab} }
2069:
1947:
1244:
88:
32:
3006:
The following situations are examples where the Mittag-Leffler condition is satisfied:
2345:
4143:
4138:
4120:
4002:
3977:
3830:. Cambridge Studies in Advanced Mathematics. Vol. 38. Cambridge University Press.
1973:
1597:
1045:
1013:
967:
949:
929:
876:
490:
92:
4452:
4389:
4377:
4279:
4204:
4199:
4153:
3935:
3930:
3849:
3831:
3717:
3685:
3675:
3664:
3654:
3635:
3625:
3597:
2379:
1009:
1001:
4413:
4299:
4274:
4209:
4194:
4189:
4128:
3957:
3925:
3803:
3747:
3712:
3646:
3617:
2524:
2409:
2364:
2341:
2309:
2259:
1747:
4325:
3891:
3841:
3811:
3778:
3755:
3722:
3596:
John Rhodes & Benjamin
Steinberg. The q-theory of Finite Semigroups. p. 133.
3254:
3230:
2706:{\displaystyle 0\rightarrow A_{i}\rightarrow B_{i}\rightarrow C_{i}\rightarrow 0}
1029:
4362:
4357:
4341:
4304:
4294:
4214:
3823:
3463:
3455:
1601:
1408:
1397:
642:
158:
3807:
2363:, every inverse system has an inverse limit. It is constructed by placing the
384:{\displaystyle f_{ik}=f_{ij}\circ f_{jk}\quad {\text{for all }}i\leq j\leq k.}
4523:
4352:
4184:
3987:
3789:
3766:
3339:
2468:
2134:, indexed by the natural numbers as usually ordered, with the morphisms from
1548:
1535:
51:
3853:
3668:
3639:
2352:, viewing the finite sets as compact discrete spaces, and then applying the
4106:
4007:
3569:
3524:
3331:
1751:
1480:
can be considered a trivial inverse system, where all objects are equal to
1025:
128:
55:
3751:
4367:
4347:
4219:
4089:
3490:
3454:
the set of non-negative integers (such inverse systems are often called "
1371:
20:
4399:
4337:
3950:
3485:
has shown (in "The cohomological dimension of a directed set") that if
2871:
If the ranges of the morphisms of an inverse system of abelian groups (
2394:
31:) is a construction that allows one to "glue together" several related
3479:
has a set of generators (in addition to satisfying (AB3) and (AB4*)).
4393:
4084:
3210:{\displaystyle \varprojlim {}^{1}A_{i}=\mathbf {Z} _{p}/\mathbf {Z} }
2476:
2367:
on the underlying set-theoretic inverse limit. This is known as the
1005:
36:
4462:
4094:
3992:
1098:
638:
3860:
1238:
4432:
4422:
4071:
3982:
3070:
1389:
67:
1212:) must be universal in the sense that for any other such pair (
39:
between the objects. Thus, inverse limits can be defined in any
4427:
2387:
54:, that is by reversing the arrows, an inverse limit becomes a
4309:
3450:) an inverse system with surjective transition morphisms and
131:
3417:
It was thought for almost 40 years that Roos had proved (in
1040:
The inverse limit can be defined abstractly in an arbitrary
737:
3769:(1961), "Sur les foncteurs dérivés de lim. Applications",
3407:{\displaystyle \varprojlim {}^{n}\cong R^{n}\varprojlim .}
2262:
are defined as inverse limits of (discrete) finite groups.
966:. The inverse limit and the natural projections satisfy a
630:{\displaystyle ((A_{i})_{i\in I},(f_{ij})_{i\leq j\in I})}
480:{\displaystyle ((A_{i})_{i\in I},(f_{ij})_{i\leq j\in I})}
487:
is called an inverse system of groups and morphisms over
3792:(2006), "Derived functors of inverse limits revisited",
2403:
161:
of groups and suppose we have a family of homomorphisms
3559:
1032:. The inverse limit will also belong to that category.
3257:
of the inverse limit functor can thus be defined. The
2205:
2141:
2083:
2038:
1889:
1687:
1313:
1054:
3498:
3355:
3316:{\displaystyle R^{n}\varprojlim :C^{I}\rightarrow C.}
3270:
3149:
3032:
2919:
2729:
2654:
2536:
2502:
2424:
2284:
2204:
2140:
2082:
2037:
1996:
1976:
1950:
1921:
1864:
1812:
1770:
1719:
1662:
1610:
1560:
1498:
1435:
1264:
1158:
979:
952:
932:
879:
683:
652:
550:
513:
493:
400:
311:
282:
252:
223:
167:
101:
3572:(or inductive limit). More general concepts are the
2076:
can be thought of as the inverse limit of the rings
1908:{\displaystyle x_{i}\equiv x_{j}{\mbox{ mod }}p^{i}}
1706:{\displaystyle n_{i}\equiv n_{j}{\mbox{ mod }}p^{i}}
3592:
3590:
3588:
2645:) are three inverse systems of abelian groups, and
1746:-adic integers is the one implied here, namely the
537:are called the transition morphisms of the system.
3511:
3406:
3315:
3209:
3057:
2983:
2848:
2705:
2580:
2515:
2456:
2300:
2248:
2190:
2126:
2060:
2020:
1982:
1962:
1936:
1907:
1850:
1798:
1734:
1705:
1648:
1588:
1526:
1460:
1348:
1296:
1174:
1089:
992:
958:
938:
885:
862:
665:
629:
529:
499:
479:
383:
295:
268:
235:
209:
119:
1534:The inverse limit, if it exists, is defined as a
1384:Inverse systems and inverse limits in a category
973:This same construction may be carried out if the
4521:
3585:
243:(note the order) with the following properties:
35:, the precise gluing process being specified by
3420:Sur les foncteurs dérivés de lim. Applications.
2457:{\displaystyle \varprojlim :C^{I}\rightarrow C}
1764:is the inverse limit of the topological groups
2475:is ordered (not simply partially ordered) and
1799:{\displaystyle \mathbb {R} /p^{n}\mathbb {Z} }
1589:{\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }
3876:
1388:admit an alternative description in terms of
3418:
926:th component of the direct product for each
2984:{\displaystyle f_{kj}(A_{j})=f_{ki}(A_{i})}
2866:
4504:
4494:
4250:
3883:
3869:
3245:is an arbitrary abelian category that has
787:
781:
3716:
1792:
1772:
1582:
1562:
3697:
3681:Categories for the Working Mathematician
3674:
3645:
3616:
1488:. This defines a "trivial functor" from
1468:be the category of these functors (with
2991:one says that the system satisfies the
2348:in graph theory and may be proved with
4522:
3828:An introduction to homological algebra
3822:
3732:
3701:(1972), "Rings with several objects",
1527:{\displaystyle C^{I^{\mathrm {op} }}.}
1400:where the morphisms consist of arrows
4249:
3902:
3864:
3543:-indexed diagrams in the category of
2591:(pronounced "lim one") such that if (
2404:Derived functors of the inverse limit
2191:{\displaystyle \textstyle R/t^{n+j}R}
1461:{\displaystyle C^{I^{\mathrm {op} }}}
1367:of an inverse system, there exists a
1255:. The inverse limit is often denoted
1035:
210:{\displaystyle f_{ij}:A_{j}\to A_{i}}
3788:
3765:
3560:Related concepts and generalizations
3346:to series of functors lim such that
3261:th right derived functor is denoted
1851:{\displaystyle (x_{1},x_{2},\dots )}
1649:{\displaystyle (n_{1},n_{2},\dots )}
1381:commuting with the projection maps.
1097:be an inverse system of objects and
78:
73:
3890:
2386:. This is one way of realizing the
2249:{\displaystyle \textstyle R/t^{n}R}
2127:{\displaystyle \textstyle R/t^{n}R}
1418:. An inverse system is then just a
1297:{\displaystyle X=\varprojlim X_{i}}
1028:are morphisms in the corresponding
1024:(over a fixed ring), etc., and the
83:We start with the definition of an
13:
3500:
3236:
3065:is non-zero is obtained by taking
3058:{\displaystyle \varprojlim {}^{1}}
1554:is the inverse limit of the rings
1513:
1510:
1484:and all arrow are the identity of
1450:
1447:
14:
4546:
1944:Its elements are exactly of form
1222:) there exists a unique morphism
4503:
4493:
4484:
4483:
4236:
3903:
3203:
3187:
3010:a system in which the morphisms
2356:characterization of compactness.
2256:given by the natural projection.
1237:
1105:(same definition as above). The
3458:sequences"). However, in 2002,
3342:generalized the functor lim on
1600:) with the index set being the
970:described in the next section.
357:
16:Construction in category theory
3651:General topology: Chapters 1-4
3304:
2978:
2965:
2946:
2933:
2811:
2785:
2759:
2733:
2697:
2684:
2671:
2658:
2572:
2496:that ensures the exactness of
2448:
2361:category of topological spaces
2315:. Then the natural projection
2242:
2236:
2215:
2209:
2184:
2178:
2151:
2145:
2120:
2114:
2093:
2087:
2054:
2051:
2045:
2042:
2015:
2003:
1845:
1813:
1643:
1611:
1343:
1314:
1084:
1055:
827:
814:
746:
624:
603:
586:
568:
554:
551:
474:
453:
436:
418:
404:
401:
194:
114:
102:
1:
3610:
2061:{\displaystyle \textstyle R]}
3718:10.1016/0001-8708(72)90002-3
2516:{\displaystyle \varprojlim }
2415:, the inverse limit functor
2354:finite intersection property
1742:The natural topology on the
1392:. Any partially ordered set
1109:of this system is an object
7:
4178:Constructions on categories
3512:{\displaystyle \aleph _{d}}
1541:
1349:{\textstyle (X_{i},f_{ij})}
1090:{\textstyle (X_{i},f_{ij})}
10:
4551:
4285:Higher-dimensional algebra
3684:(2nd ed.), Springer,
2021:{\displaystyle r\in [0,1)}
87:(or projective system) of
4479:
4412:
4376:
4324:
4317:
4268:
4258:
4245:
4234:
4177:
4119:
4070:
4025:
4016:
3913:
3909:
3898:
3808:10.1112/S0024610705022416
3568:of an inverse limit is a
3539:+ 2. This applies to the
2720:of inverse systems, then
1990:is a p-adic integer, and
1538:of this trivial functor.
1472:as morphisms). An object
134:(not all authors require
120:{\displaystyle (I,\leq )}
4530:Limits (category theory)
3739:Inventiones Mathematicae
3579:
3001:Mittag-Leffler's theorem
2993:Mittag-Leffler condition
2867:Mittag-Leffler condition
2859:is an exact sequence in
2072:over a commutative ring
1307:with the inverse system
1117:together with morphisms
4095:Cokernels and quotients
4018:Universal constructions
3704:Advances in Mathematics
3069:to be the non-negative
1937:{\displaystyle i<j.}
1735:{\displaystyle i<j.}
1470:natural transformations
1396:can be considered as a
236:{\displaystyle i\leq j}
4252:Higher category theory
3998:Natural transformation
3771:C. R. Acad. Sci. Paris
3513:
3419:
3408:
3317:
3211:
3059:
2985:
2850:
2707:
2582:
2527:constructed a functor
2517:
2458:
2397:(as infinite strings).
2302:
2301:{\displaystyle f_{ij}}
2269:of an inverse system (
2250:
2192:
2128:
2062:
2022:
1984:
1964:
1938:
1909:
1852:
1800:
1736:
1707:
1650:
1590:
1528:
1462:
1350:
1298:
1234:such that the diagram
1176:
1175:{\displaystyle f_{ij}}
1091:
994:
960:
940:
887:
864:
667:
631:
544:of the inverse system
531:
530:{\displaystyle f_{ij}}
501:
481:
385:
297:
270:
269:{\displaystyle f_{ii}}
237:
211:
138:to be directed). Let (
121:
3752:10.1007/s002220100197
3514:
3409:
3318:
3212:
3060:
2986:
2893:, that is, for every
2851:
2708:
2583:
2518:
2459:
2382:, the limit space is
2303:
2251:
2193:
2129:
2063:
2023:
1985:
1965:
1939:
1910:
1853:
1801:
1737:
1708:
1651:
1591:
1529:
1463:
1420:contravariant functor
1351:
1299:
1177:
1092:
1020:(over a fixed ring),
995:
993:{\displaystyle A_{i}}
961:
941:
888:
865:
668:
666:{\displaystyle A_{i}}
632:
532:
502:
482:
386:
298:
296:{\displaystyle A_{i}}
271:
238:
212:
122:
47:in category theory.
4121:Algebraic categories
3795:J. London Math. Soc.
3556: > 1).
3531:lim is zero for all
3496:
3353:
3268:
3147:
3030:
2917:
2727:
2718:short exact sequence
2652:
2534:
2500:
2422:
2384:totally disconnected
2374:The set of infinite
2282:
2202:
2138:
2080:
2035:
1994:
1974:
1948:
1919:
1862:
1810:
1768:
1717:
1660:
1608:
1558:
1496:
1433:
1311:
1262:
1156:
1052:
977:
950:
930:
893:comes equipped with
877:
681:
650:
548:
511:
507:, and the morphisms
491:
398:
309:
280:
250:
221:
165:
99:
4290:Homotopy hypothesis
3968:Commutative diagram
3574:limits and colimits
3241:More generally, if
2350:Tychonoff's theorem
2070:formal power series
2028:is the "remainder".
1963:{\displaystyle n+r}
920:which pick out the
895:natural projections
832: for all
276:is the identity on
4003:Universal property
3824:Weibel, Charles A.
3678:(September 1998),
3676:Mac Lane, Saunders
3509:
3404:
3399:
3364:
3326:In the case where
3313:
3289:
3207:
3158:
3055:
3041:
2981:
2905:such that for all
2846:
2822:
2796:
2770:
2744:
2703:
2578:
2545:
2513:
2511:
2454:
2433:
2337:is an isomorphism.
2298:
2265:Let the index set
2246:
2245:
2188:
2187:
2124:
2123:
2058:
2057:
2018:
1980:
1960:
1934:
1905:
1893:
1848:
1796:
1732:
1703:
1691:
1646:
1598:modular arithmetic
1586:
1524:
1458:
1356:being understood.
1346:
1294:
1279:
1172:
1087:
1046:universal property
1036:General definition
1010:topological spaces
990:
968:universal property
956:
936:
883:
873:The inverse limit
860:
770:
712:
699:
663:
627:
527:
497:
477:
381:
293:
266:
233:
207:
117:
50:By working in the
4517:
4516:
4475:
4474:
4471:
4470:
4453:monoidal category
4408:
4407:
4280:Enriched category
4232:
4231:
4228:
4227:
4205:Quotient category
4200:Opposite category
4115:
4114:
3837:978-0-521-55987-4
3660:978-3-540-64241-1
3647:Bourbaki, Nicolas
3631:978-3-540-64243-5
3618:Bourbaki, Nicolas
3602:978-0-387-09780-0
3525:infinite cardinal
3392:
3357:
3282:
3247:enough injectives
3151:
3034:
3026:An example where
2815:
2789:
2763:
2737:
2538:
2504:
2426:
2260:Pro-finite groups
1983:{\displaystyle n}
1892:
1754:as the open sets.
1690:
1272:
959:{\displaystyle I}
939:{\displaystyle i}
886:{\displaystyle A}
847:
833:
755:
749:
692:
690:
500:{\displaystyle I}
361:
79:Algebraic objects
74:Formal definition
27:(also called the
4542:
4535:Abstract algebra
4507:
4506:
4497:
4496:
4487:
4486:
4322:
4321:
4300:Simplex category
4275:Categorification
4266:
4265:
4247:
4246:
4240:
4210:Product category
4195:Kleisli category
4190:Functor category
4035:Terminal objects
4023:
4022:
3958:Adjoint functors
3911:
3910:
3900:
3899:
3885:
3878:
3871:
3862:
3861:
3857:
3818:
3785:
3762:
3729:
3720:
3694:
3671:
3642:
3605:
3594:
3566:categorical dual
3518:
3516:
3515:
3510:
3508:
3507:
3422:
3413:
3411:
3410:
3405:
3400:
3390:
3389:
3377:
3376:
3371:
3365:
3322:
3320:
3319:
3314:
3303:
3302:
3290:
3280:
3279:
3255:derived functors
3253:, and the right
3216:
3214:
3213:
3208:
3206:
3201:
3196:
3195:
3190:
3181:
3180:
3171:
3170:
3165:
3159:
3064:
3062:
3061:
3056:
3054:
3053:
3048:
3042:
2990:
2988:
2987:
2982:
2977:
2976:
2964:
2963:
2945:
2944:
2932:
2931:
2855:
2853:
2852:
2847:
2845:
2844:
2835:
2834:
2829:
2823:
2810:
2809:
2797:
2784:
2783:
2771:
2758:
2757:
2745:
2712:
2710:
2709:
2704:
2696:
2695:
2683:
2682:
2670:
2669:
2587:
2585:
2584:
2579:
2571:
2570:
2558:
2557:
2552:
2546:
2523:. Specifically,
2522:
2520:
2519:
2514:
2512:
2483:is the category
2463:
2461:
2460:
2455:
2447:
2446:
2434:
2410:abelian category
2365:initial topology
2342:category of sets
2318:
2310:greatest element
2307:
2305:
2304:
2299:
2297:
2296:
2255:
2253:
2252:
2247:
2232:
2231:
2222:
2197:
2195:
2194:
2189:
2174:
2173:
2158:
2133:
2131:
2130:
2125:
2110:
2109:
2100:
2067:
2065:
2064:
2059:
2027:
2025:
2024:
2019:
1989:
1987:
1986:
1981:
1969:
1967:
1966:
1961:
1943:
1941:
1940:
1935:
1914:
1912:
1911:
1906:
1904:
1903:
1894:
1890:
1887:
1886:
1874:
1873:
1857:
1855:
1854:
1849:
1838:
1837:
1825:
1824:
1805:
1803:
1802:
1797:
1795:
1790:
1789:
1780:
1775:
1748:product topology
1741:
1739:
1738:
1733:
1712:
1710:
1709:
1704:
1702:
1701:
1692:
1688:
1685:
1684:
1672:
1671:
1655:
1653:
1652:
1647:
1636:
1635:
1623:
1622:
1595:
1593:
1592:
1587:
1585:
1580:
1579:
1570:
1565:
1533:
1531:
1530:
1525:
1520:
1519:
1518:
1517:
1516:
1467:
1465:
1464:
1459:
1457:
1456:
1455:
1454:
1453:
1355:
1353:
1352:
1347:
1342:
1341:
1326:
1325:
1303:
1301:
1300:
1295:
1293:
1292:
1280:
1241:
1206:
1185:
1181:
1179:
1178:
1173:
1171:
1170:
1146:
1120:
1096:
1094:
1093:
1088:
1083:
1082:
1067:
1066:
999:
997:
996:
991:
989:
988:
965:
963:
962:
957:
945:
943:
942:
937:
925:
919:
900:
892:
890:
889:
884:
869:
867:
866:
861:
856:
852:
848:
845:
834:
831:
826:
825:
813:
812:
797:
796:
786:
782:
780:
779:
769:
751:
750:
742:
727:
726:
725:
711:
700:
672:
670:
669:
664:
662:
661:
637:as a particular
636:
634:
633:
628:
623:
622:
601:
600:
582:
581:
566:
565:
536:
534:
533:
528:
526:
525:
506:
504:
503:
498:
486:
484:
483:
478:
473:
472:
451:
450:
432:
431:
416:
415:
390:
388:
387:
382:
362:
359:
356:
355:
340:
339:
324:
323:
302:
300:
299:
294:
292:
291:
275:
273:
272:
267:
265:
264:
242:
240:
239:
234:
216:
214:
213:
208:
206:
205:
193:
192:
180:
179:
126:
124:
123:
118:
29:projective limit
4550:
4549:
4545:
4544:
4543:
4541:
4540:
4539:
4520:
4519:
4518:
4513:
4467:
4437:
4404:
4381:
4372:
4329:
4313:
4264:
4254:
4241:
4224:
4173:
4111:
4080:Initial objects
4066:
4012:
3905:
3894:
3892:Category theory
3889:
3838:
3821:Section 3.5 of
3699:Mitchell, Barry
3692:
3661:
3632:
3613:
3608:
3595:
3586:
3582:
3562:
3547:-modules, with
3503:
3499:
3497:
3494:
3493:
3474:
3449:
3440:
3431:
3391:
3385:
3381:
3372:
3370:
3369:
3356:
3354:
3351:
3350:
3298:
3294:
3281:
3275:
3271:
3269:
3266:
3265:
3249:, then so does
3239:
3237:Further results
3231:p-adic integers
3228:
3202:
3197:
3191:
3186:
3185:
3176:
3172:
3166:
3164:
3163:
3150:
3148:
3145:
3144:
3128:
3119:
3110:
3097:
3081:
3049:
3047:
3046:
3033:
3031:
3028:
3027:
3018:
2972:
2968:
2956:
2952:
2940:
2936:
2924:
2920:
2918:
2915:
2914:
2888:
2879:
2869:
2840:
2836:
2830:
2828:
2827:
2814:
2805:
2801:
2788:
2779:
2775:
2762:
2753:
2749:
2736:
2728:
2725:
2724:
2691:
2687:
2678:
2674:
2665:
2661:
2653:
2650:
2649:
2644:
2635:
2626:
2617:
2608:
2599:
2566:
2562:
2553:
2551:
2550:
2537:
2535:
2532:
2531:
2503:
2501:
2498:
2497:
2495:
2442:
2438:
2425:
2423:
2420:
2419:
2406:
2336:
2323:
2316:
2289:
2285:
2283:
2280:
2279:
2277:
2227:
2223:
2218:
2203:
2200:
2199:
2163:
2159:
2154:
2139:
2136:
2135:
2105:
2101:
2096:
2081:
2078:
2077:
2036:
2033:
2032:
1995:
1992:
1991:
1975:
1972:
1971:
1949:
1946:
1945:
1920:
1917:
1916:
1899:
1895:
1891: mod
1888:
1882:
1878:
1869:
1865:
1863:
1860:
1859:
1833:
1829:
1820:
1816:
1811:
1808:
1807:
1791:
1785:
1781:
1776:
1771:
1769:
1766:
1765:
1718:
1715:
1714:
1697:
1693:
1689: mod
1686:
1680:
1676:
1667:
1663:
1661:
1658:
1657:
1631:
1627:
1618:
1614:
1609:
1606:
1605:
1602:natural numbers
1581:
1575:
1571:
1566:
1561:
1559:
1556:
1555:
1544:
1509:
1508:
1504:
1503:
1499:
1497:
1494:
1493:
1446:
1445:
1441:
1440:
1436:
1434:
1431:
1430:
1334:
1330:
1321:
1317:
1312:
1309:
1308:
1288:
1284:
1271:
1263:
1260:
1259:
1242:
1221:
1211:
1204:
1190:
1183:
1163:
1159:
1157:
1154:
1153:
1151:
1144:
1138:
1125:
1118:
1075:
1071:
1062:
1058:
1053:
1050:
1049:
1038:
984:
980:
978:
975:
974:
951:
948:
947:
931:
928:
927:
921:
918:
905:
898:
897:
878:
875:
874:
844:
830:
821:
817:
805:
801:
792:
788:
775:
771:
759:
741:
740:
739:
736:
735:
731:
721:
717:
716:
701:
691:
682:
679:
678:
657:
653:
651:
648:
647:
606:
602:
593:
589:
571:
567:
561:
557:
549:
546:
545:
518:
514:
512:
509:
508:
492:
489:
488:
456:
452:
443:
439:
421:
417:
411:
407:
399:
396:
395:
358:
348:
344:
332:
328:
316:
312:
310:
307:
306:
287:
283:
281:
278:
277:
257:
253:
251:
248:
247:
222:
219:
218:
201:
197:
188:
184:
172:
168:
166:
163:
162:
156:
146:
100:
97:
96:
81:
76:
60:inductive limit
17:
12:
11:
5:
4548:
4538:
4537:
4532:
4515:
4514:
4512:
4511:
4501:
4491:
4480:
4477:
4476:
4473:
4472:
4469:
4468:
4466:
4465:
4460:
4455:
4441:
4435:
4430:
4425:
4419:
4417:
4410:
4409:
4406:
4405:
4403:
4402:
4397:
4386:
4384:
4379:
4374:
4373:
4371:
4370:
4365:
4360:
4355:
4350:
4345:
4334:
4332:
4327:
4319:
4315:
4314:
4312:
4307:
4305:String diagram
4302:
4297:
4295:Model category
4292:
4287:
4282:
4277:
4272:
4270:
4263:
4262:
4259:
4256:
4255:
4243:
4242:
4235:
4233:
4230:
4229:
4226:
4225:
4223:
4222:
4217:
4215:Comma category
4212:
4207:
4202:
4197:
4192:
4187:
4181:
4179:
4175:
4174:
4172:
4171:
4161:
4151:
4149:Abelian groups
4146:
4141:
4136:
4131:
4125:
4123:
4117:
4116:
4113:
4112:
4110:
4109:
4104:
4099:
4098:
4097:
4087:
4082:
4076:
4074:
4068:
4067:
4065:
4064:
4059:
4054:
4053:
4052:
4042:
4037:
4031:
4029:
4020:
4014:
4013:
4011:
4010:
4005:
4000:
3995:
3990:
3985:
3980:
3975:
3970:
3965:
3960:
3955:
3954:
3953:
3948:
3943:
3938:
3933:
3928:
3917:
3915:
3907:
3906:
3896:
3895:
3888:
3887:
3880:
3873:
3865:
3859:
3858:
3836:
3819:
3790:Roos, Jan-Erik
3786:
3767:Roos, Jan-Erik
3763:
3746:(2): 397–420,
3730:
3695:
3690:
3672:
3659:
3643:
3630:
3612:
3609:
3607:
3606:
3583:
3581:
3578:
3561:
3558:
3506:
3502:
3483:Barry Mitchell
3470:
3464:Pierre Deligne
3456:Mittag-Leffler
3445:
3436:
3427:
3415:
3414:
3403:
3398:
3395:
3388:
3384:
3380:
3375:
3368:
3363:
3360:
3324:
3323:
3312:
3309:
3306:
3301:
3297:
3293:
3288:
3285:
3278:
3274:
3238:
3235:
3224:
3218:
3217:
3205:
3200:
3194:
3189:
3184:
3179:
3175:
3169:
3162:
3157:
3154:
3124:
3115:
3106:
3093:
3077:
3052:
3045:
3040:
3037:
3024:
3023:
3020:
3019:are surjective
3014:
2980:
2975:
2971:
2967:
2962:
2959:
2955:
2951:
2948:
2943:
2939:
2935:
2930:
2927:
2923:
2884:
2875:
2868:
2865:
2857:
2856:
2843:
2839:
2833:
2826:
2821:
2818:
2813:
2808:
2804:
2800:
2795:
2792:
2787:
2782:
2778:
2774:
2769:
2766:
2761:
2756:
2752:
2748:
2743:
2740:
2735:
2732:
2714:
2713:
2702:
2699:
2694:
2690:
2686:
2681:
2677:
2673:
2668:
2664:
2660:
2657:
2640:
2631:
2622:
2613:
2604:
2595:
2589:
2588:
2577:
2574:
2569:
2565:
2561:
2556:
2549:
2544:
2541:
2510:
2507:
2491:
2465:
2464:
2453:
2450:
2445:
2441:
2437:
2432:
2429:
2405:
2402:
2401:
2400:
2399:
2398:
2369:limit topology
2357:
2338:
2332:
2319:
2295:
2292:
2288:
2273:
2263:
2257:
2244:
2241:
2238:
2235:
2230:
2226:
2221:
2217:
2214:
2211:
2208:
2186:
2183:
2180:
2177:
2172:
2169:
2166:
2162:
2157:
2153:
2150:
2147:
2144:
2122:
2119:
2116:
2113:
2108:
2104:
2099:
2095:
2092:
2089:
2086:
2056:
2053:
2050:
2047:
2044:
2041:
2029:
2017:
2014:
2011:
2008:
2005:
2002:
1999:
1979:
1959:
1956:
1953:
1933:
1930:
1927:
1924:
1902:
1898:
1885:
1881:
1877:
1872:
1868:
1847:
1844:
1841:
1836:
1832:
1828:
1823:
1819:
1815:
1794:
1788:
1784:
1779:
1774:
1762:-adic solenoid
1755:
1731:
1728:
1725:
1722:
1700:
1696:
1683:
1679:
1675:
1670:
1666:
1645:
1642:
1639:
1634:
1630:
1626:
1621:
1617:
1613:
1584:
1578:
1574:
1569:
1564:
1552:-adic integers
1543:
1540:
1523:
1515:
1512:
1507:
1502:
1452:
1449:
1444:
1439:
1409:if and only if
1398:small category
1345:
1340:
1337:
1333:
1329:
1324:
1320:
1316:
1305:
1304:
1291:
1287:
1283:
1278:
1275:
1270:
1267:
1236:
1217:
1207:
1186:
1169:
1166:
1162:
1147:
1134:
1121:
1101:in a category
1086:
1081:
1078:
1074:
1070:
1065:
1061:
1057:
1044:by means of a
1037:
1034:
987:
983:
955:
935:
914:
901:
882:
871:
870:
859:
855:
851:
846: in
843:
840:
837:
829:
824:
820:
816:
811:
808:
804:
800:
795:
791:
785:
778:
774:
768:
765:
762:
758:
754:
748:
745:
738:
734:
730:
724:
720:
715:
710:
707:
704:
698:
695:
689:
686:
660:
656:
643:direct product
626:
621:
618:
615:
612:
609:
605:
599:
596:
592:
588:
585:
580:
577:
574:
570:
564:
560:
556:
553:
540:We define the
524:
521:
517:
496:
476:
471:
468:
465:
462:
459:
455:
449:
446:
442:
438:
435:
430:
427:
424:
420:
414:
410:
406:
403:
394:Then the pair
392:
391:
380:
377:
374:
371:
368:
365:
354:
351:
347:
343:
338:
335:
331:
327:
322:
319:
315:
304:
290:
286:
263:
260:
256:
232:
229:
226:
204:
200:
196:
191:
187:
183:
178:
175:
171:
148:
142:
116:
113:
110:
107:
104:
85:inverse system
80:
77:
75:
72:
15:
9:
6:
4:
3:
2:
4547:
4536:
4533:
4531:
4528:
4527:
4525:
4510:
4502:
4500:
4492:
4490:
4482:
4481:
4478:
4464:
4461:
4459:
4456:
4454:
4450:
4446:
4442:
4440:
4438:
4431:
4429:
4426:
4424:
4421:
4420:
4418:
4415:
4411:
4401:
4398:
4395:
4391:
4388:
4387:
4385:
4383:
4375:
4369:
4366:
4364:
4361:
4359:
4356:
4354:
4353:Tetracategory
4351:
4349:
4346:
4343:
4342:pseudofunctor
4339:
4336:
4335:
4333:
4331:
4323:
4320:
4316:
4311:
4308:
4306:
4303:
4301:
4298:
4296:
4293:
4291:
4288:
4286:
4283:
4281:
4278:
4276:
4273:
4271:
4267:
4261:
4260:
4257:
4253:
4248:
4244:
4239:
4221:
4218:
4216:
4213:
4211:
4208:
4206:
4203:
4201:
4198:
4196:
4193:
4191:
4188:
4186:
4185:Free category
4183:
4182:
4180:
4176:
4169:
4168:Vector spaces
4165:
4162:
4159:
4155:
4152:
4150:
4147:
4145:
4142:
4140:
4137:
4135:
4132:
4130:
4127:
4126:
4124:
4122:
4118:
4108:
4105:
4103:
4100:
4096:
4093:
4092:
4091:
4088:
4086:
4083:
4081:
4078:
4077:
4075:
4073:
4069:
4063:
4062:Inverse limit
4060:
4058:
4055:
4051:
4048:
4047:
4046:
4043:
4041:
4038:
4036:
4033:
4032:
4030:
4028:
4024:
4021:
4019:
4015:
4009:
4006:
4004:
4001:
3999:
3996:
3994:
3991:
3989:
3988:Kan extension
3986:
3984:
3981:
3979:
3976:
3974:
3971:
3969:
3966:
3964:
3961:
3959:
3956:
3952:
3949:
3947:
3944:
3942:
3939:
3937:
3934:
3932:
3929:
3927:
3924:
3923:
3922:
3919:
3918:
3916:
3912:
3908:
3901:
3897:
3893:
3886:
3881:
3879:
3874:
3872:
3867:
3866:
3863:
3855:
3851:
3847:
3843:
3839:
3833:
3829:
3825:
3820:
3817:
3813:
3809:
3805:
3801:
3797:
3796:
3791:
3787:
3784:
3780:
3777:: 3702–3704,
3776:
3772:
3768:
3764:
3761:
3757:
3753:
3749:
3745:
3741:
3740:
3735:
3734:Neeman, Amnon
3731:
3728:
3724:
3719:
3714:
3710:
3706:
3705:
3700:
3696:
3693:
3691:0-387-98403-8
3687:
3683:
3682:
3677:
3673:
3670:
3666:
3662:
3656:
3652:
3648:
3644:
3641:
3637:
3633:
3627:
3623:
3619:
3615:
3614:
3603:
3599:
3593:
3591:
3589:
3584:
3577:
3575:
3571:
3567:
3557:
3555:
3550:
3546:
3542:
3538:
3534:
3530:
3526:
3522:
3504:
3492:
3488:
3484:
3480:
3478:
3473:
3469:
3465:
3461:
3457:
3453:
3448:
3444:
3439:
3435:
3430:
3426:
3421:
3401:
3396:
3393:
3386:
3382:
3378:
3373:
3366:
3361:
3358:
3349:
3348:
3347:
3345:
3341:
3340:Jan-Erik Roos
3337:
3333:
3329:
3310:
3307:
3299:
3295:
3291:
3286:
3283:
3276:
3272:
3264:
3263:
3262:
3260:
3256:
3252:
3248:
3244:
3234:
3232:
3227:
3223:
3198:
3192:
3182:
3177:
3173:
3167:
3160:
3155:
3152:
3143:
3142:
3141:
3139:
3136:
3132:
3127:
3123:
3118:
3114:
3109:
3105:
3101:
3096:
3092:
3088:
3085:
3080:
3076:
3072:
3068:
3050:
3043:
3038:
3035:
3021:
3017:
3013:
3009:
3008:
3007:
3004:
3002:
2996:
2994:
2973:
2969:
2960:
2957:
2953:
2949:
2941:
2937:
2928:
2925:
2921:
2912:
2908:
2904:
2900:
2897:there exists
2896:
2892:
2887:
2883:
2878:
2874:
2864:
2862:
2841:
2837:
2831:
2824:
2819:
2816:
2806:
2802:
2798:
2793:
2790:
2780:
2776:
2772:
2767:
2764:
2754:
2750:
2746:
2741:
2738:
2730:
2723:
2722:
2721:
2719:
2700:
2692:
2688:
2679:
2675:
2666:
2662:
2655:
2648:
2647:
2646:
2643:
2639:
2634:
2630:
2625:
2621:
2616:
2612:
2607:
2603:
2598:
2594:
2575:
2567:
2563:
2559:
2554:
2547:
2542:
2539:
2530:
2529:
2528:
2526:
2508:
2505:
2494:
2490:
2486:
2482:
2478:
2474:
2470:
2451:
2443:
2439:
2435:
2430:
2427:
2418:
2417:
2416:
2414:
2411:
2396:
2392:
2391:-adic numbers
2390:
2385:
2381:
2377:
2373:
2372:
2370:
2366:
2362:
2358:
2355:
2351:
2347:
2346:Kőnig's lemma
2343:
2339:
2335:
2331:
2327:
2322:
2314:
2311:
2293:
2290:
2286:
2276:
2272:
2268:
2264:
2261:
2258:
2239:
2233:
2228:
2224:
2219:
2212:
2206:
2181:
2175:
2170:
2167:
2164:
2160:
2155:
2148:
2142:
2117:
2111:
2106:
2102:
2097:
2090:
2084:
2075:
2071:
2048:
2039:
2030:
2012:
2009:
2006:
2000:
1997:
1977:
1957:
1954:
1951:
1931:
1928:
1925:
1922:
1900:
1896:
1883:
1879:
1875:
1870:
1866:
1842:
1839:
1834:
1830:
1826:
1821:
1817:
1786:
1782:
1777:
1763:
1761:
1756:
1753:
1752:cylinder sets
1749:
1745:
1729:
1726:
1723:
1720:
1698:
1694:
1681:
1677:
1673:
1668:
1664:
1640:
1637:
1632:
1628:
1624:
1619:
1615:
1603:
1599:
1576:
1572:
1567:
1553:
1551:
1546:
1545:
1539:
1537:
1536:right adjoint
1521:
1505:
1500:
1491:
1487:
1483:
1479:
1475:
1471:
1442:
1437:
1428:
1424:
1421:
1417:
1413:
1410:
1407:
1403:
1399:
1395:
1391:
1387:
1382:
1380:
1376:
1373:
1370:
1366:
1362:
1357:
1338:
1335:
1331:
1327:
1322:
1318:
1289:
1285:
1281:
1276:
1273:
1268:
1265:
1258:
1257:
1256:
1254:
1250:
1246:
1240:
1235:
1233:
1229:
1225:
1220:
1215:
1210:
1202:
1198:
1194:
1189:
1167:
1164:
1160:
1150:
1143:) satisfying
1142:
1137:
1133:
1129:
1124:
1116:
1112:
1108:
1107:inverse limit
1104:
1100:
1079:
1076:
1072:
1068:
1063:
1059:
1047:
1043:
1033:
1031:
1027:
1026:homomorphisms
1023:
1019:
1015:
1011:
1007:
1003:
985:
981:
971:
969:
953:
933:
924:
917:
913:
909:
904:
896:
880:
857:
853:
849:
841:
838:
835:
822:
818:
809:
806:
802:
798:
793:
789:
783:
776:
772:
766:
763:
760:
756:
752:
743:
732:
728:
722:
718:
713:
708:
705:
702:
696:
693:
687:
684:
677:
676:
675:
673:
658:
654:
644:
640:
619:
616:
613:
610:
607:
597:
594:
590:
583:
578:
575:
572:
562:
558:
543:
542:inverse limit
538:
522:
519:
515:
494:
469:
466:
463:
460:
457:
447:
444:
440:
433:
428:
425:
422:
412:
408:
378:
375:
372:
369:
366:
363:
360:for all
352:
349:
345:
341:
336:
333:
329:
325:
320:
317:
313:
305:
288:
284:
261:
258:
254:
246:
245:
244:
230:
227:
224:
202:
198:
189:
185:
181:
176:
173:
169:
160:
155:
151:
145:
141:
137:
133:
130:
111:
108:
105:
94:
93:homomorphisms
90:
86:
71:
69:
65:
61:
57:
53:
52:dual category
48:
46:
42:
38:
34:
30:
26:
25:inverse limit
22:
4433:
4414:Categorified
4318:n-categories
4269:Key concepts
4107:Direct limit
4090:Coequalizers
4061:
4008:Yoneda lemma
3914:Key concepts
3904:Key concepts
3827:
3802:(1): 65–83,
3799:
3798:, Series 2,
3793:
3774:
3770:
3743:
3737:
3708:
3702:
3679:
3653:, Springer,
3650:
3624:, Springer,
3621:
3570:direct limit
3563:
3553:
3548:
3544:
3540:
3536:
3532:
3528:
3520:
3486:
3481:
3476:
3471:
3467:
3460:Amnon Neeman
3451:
3446:
3442:
3437:
3433:
3428:
3424:
3416:
3343:
3332:Grothendieck
3327:
3325:
3258:
3250:
3242:
3240:
3229:denotes the
3225:
3221:
3219:
3137:
3134:
3130:
3125:
3121:
3116:
3112:
3107:
3103:
3099:
3094:
3090:
3086:
3083:
3078:
3074:
3066:
3025:
3015:
3011:
3005:
2997:
2992:
2910:
2906:
2902:
2898:
2894:
2890:
2885:
2881:
2876:
2872:
2870:
2860:
2858:
2715:
2641:
2637:
2632:
2628:
2623:
2619:
2614:
2610:
2605:
2601:
2596:
2592:
2590:
2492:
2488:
2484:
2480:
2472:
2466:
2412:
2407:
2388:
2368:
2333:
2329:
2325:
2320:
2312:
2274:
2270:
2266:
2073:
1759:
1743:
1549:
1547:The ring of
1489:
1485:
1481:
1477:
1473:
1426:
1422:
1415:
1411:
1405:
1401:
1393:
1385:
1383:
1378:
1374:
1368:
1364:
1360:
1358:
1306:
1252:
1248:
1243:
1231:
1227:
1223:
1218:
1213:
1208:
1200:
1199:. The pair (
1196:
1192:
1187:
1148:
1140:
1135:
1131:
1127:
1122:
1114:
1110:
1106:
1102:
1039:
972:
922:
915:
911:
907:
902:
894:
872:
646:
541:
539:
393:
153:
149:
143:
139:
135:
84:
82:
63:
59:
56:direct limit
49:
28:
24:
18:
4382:-categories
4358:Kan complex
4348:Tricategory
4330:-categories
4220:Subcategory
3978:Exponential
3946:Preadditive
3941:Pre-abelian
3491:cardinality
3423:) that lim
1372:isomorphism
1141:projections
21:mathematics
4524:Categories
4400:3-category
4390:2-category
4363:∞-groupoid
4338:Bicategory
4085:Coproducts
4045:Equalizers
3951:Bicategory
3611:References
3330:satisfies
3073:, letting
2891:stationary
2469:left exact
2395:Cantor set
1377:′ →
1006:semigroups
66:becomes a
4449:Symmetric
4394:2-functor
4134:Relations
4057:Pullbacks
3711:: 1–161,
3622:Algebra I
3552:for
3501:ℵ
3432:= 0 for (
3397:←
3379:≅
3367:
3362:←
3334:'s axiom
3305:→
3287:←
3161:
3156:←
3044:
3039:←
2825:
2820:←
2812:→
2799:
2794:←
2786:→
2773:
2768:←
2760:→
2747:
2742:←
2734:→
2698:→
2685:→
2672:→
2659:→
2573:→
2548:
2543:←
2525:Eilenberg
2509:←
2477:countable
2449:→
2431:←
2308:) have a
2031:The ring
2001:∈
1915:whenever
1876:≡
1843:…
1713:whenever
1674:≡
1641:…
1282:
1277:←
1099:morphisms
839:≤
764:∈
757:∏
753:∈
747:→
714:
706:∈
697:←
617:∈
611:≤
576:∈
467:∈
461:≤
426:∈
373:≤
367:≤
342:∘
228:≤
195:→
112:≤
37:morphisms
4509:Glossary
4489:Category
4463:n-monoid
4416:concepts
4072:Colimits
4040:Products
3993:Morphism
3936:Concrete
3931:Additive
3921:Category
3854:36131259
3826:(1994).
3669:40551485
3649:(1989),
3640:40551484
3620:(1989),
3527:), then
3071:integers
2627:), and (
2393:and the
2380:discrete
1970:, where
1542:Examples
1390:functors
1247:for all
1245:commutes
1191:for all
1139:(called
1042:category
1030:category
1022:algebras
639:subgroup
217:for all
129:directed
62:, and a
41:category
4499:Outline
4458:n-group
4423:2-group
4378:Strict
4368:∞-topos
4164:Modules
4102:Pushout
4050:Kernels
3983:Functor
3926:Abelian
3846:1269324
3816:2197371
3783:0132091
3760:1906154
3727:0294454
3140:. Then
2913: :
2408:For an
2376:strings
2359:In the
2340:In the
1018:modules
1000:'s are
645:of the
641:of the
68:colimit
33:objects
4445:Traced
4428:2-ring
4158:Fields
4144:Groups
4139:Magmas
4027:Limits
3852:
3844:
3834:
3814:
3781:
3758:
3725:
3688:
3667:
3657:
3638:
3628:
3600:
3336:(AB4*)
3220:where
3102:, and
2889:) are
2479:, and
1429:. Let
1369:unique
1048:. Let
159:family
95:. Let
89:groups
23:, the
4439:-ring
4326:Weak
4310:Topos
4154:Rings
3580:Notes
3519:(the
2716:is a
2471:. If
1750:with
1596:(see
1014:rings
157:be a
132:poset
127:be a
64:limit
45:limit
4129:Sets
3850:OCLC
3832:ISBN
3686:ISBN
3665:OCLC
3655:ISBN
3636:OCLC
3626:ISBN
3598:ISBN
3564:The
3489:has
3462:and
2609:), (
1926:<
1757:The
1724:<
1363:and
1002:sets
674:'s:
91:and
3973:End
3963:CCC
3804:doi
3775:252
3748:doi
3744:148
3713:doi
3523:th
3394:lim
3359:lim
3284:lim
3153:lim
3036:lim
2817:lim
2791:lim
2765:lim
2739:lim
2540:lim
2506:lim
2467:is
2428:lim
2198:to
2068:of
1492:to
1476:of
1274:lim
1216:, ψ
1113:in
946:in
694:lim
58:or
19:In
4526::
4451:)
4447:)(
3848:.
3842:MR
3840:.
3812:MR
3810:,
3800:73
3779:MR
3773:,
3756:MR
3754:,
3742:,
3723:MR
3721:,
3707:,
3663:,
3634:,
3587:^
3535:≥
3447:ij
3441:,
3344:Ab
3338:,
3233:.
3129:=
3120:/
3111:=
3098:=
3089:,
3082:=
3016:ij
3003:.
2995:.
2909:≥
2901:≥
2886:ij
2880:,
2863:.
2861:Ab
2642:ij
2636:,
2624:ij
2618:,
2606:ij
2600:,
2576:Ab
2564:Ab
2493:ij
2485:Ab
2371:.
2328:→
2324::
2278:,
1425:→
1414:≤
1404:→
1365:X'
1251:≤
1230:→
1226::
1203:,
1195:≤
1182:∘
1152:=
1130:→
1126::
1016:,
1012:,
1008:,
1004:,
910:→
906::
70:.
4443:(
4436:n
4434:E
4396:)
4392:(
4380:n
4344:)
4340:(
4328:n
4170:)
4166:(
4160:)
4156:(
3884:e
3877:t
3870:v
3856:.
3806::
3750::
3715::
3709:8
3604:.
3554:n
3549:R
3545:R
3541:I
3537:d
3533:n
3529:R
3521:d
3505:d
3487:I
3477:C
3472:i
3468:A
3452:I
3443:f
3438:i
3434:A
3429:i
3425:A
3402:.
3387:n
3383:R
3374:n
3328:C
3311:.
3308:C
3300:I
3296:C
3292::
3277:n
3273:R
3259:n
3251:C
3243:C
3226:p
3222:Z
3204:Z
3199:/
3193:p
3188:Z
3183:=
3178:i
3174:A
3168:1
3138:Z
3135:p
3133:/
3131:Z
3126:i
3122:A
3117:i
3113:B
3108:i
3104:C
3100:Z
3095:i
3091:B
3087:Z
3084:p
3079:i
3075:A
3067:I
3051:1
3012:f
2979:)
2974:i
2970:A
2966:(
2961:i
2958:k
2954:f
2950:=
2947:)
2942:j
2938:A
2934:(
2929:j
2926:k
2922:f
2911:j
2907:i
2903:k
2899:j
2895:k
2882:f
2877:i
2873:A
2842:i
2838:A
2832:1
2807:i
2803:C
2781:i
2777:B
2755:i
2751:A
2731:0
2701:0
2693:i
2689:C
2680:i
2676:B
2667:i
2663:A
2656:0
2638:h
2633:i
2629:C
2620:g
2615:i
2611:B
2602:f
2597:i
2593:A
2568:I
2560::
2555:1
2489:f
2481:C
2473:I
2452:C
2444:I
2440:C
2436::
2413:C
2389:p
2334:m
2330:X
2326:X
2321:m
2317:π
2313:m
2294:j
2291:i
2287:f
2275:i
2271:X
2267:I
2243:]
2240:t
2237:[
2234:R
2229:n
2225:t
2220:/
2216:]
2213:t
2210:[
2207:R
2185:]
2182:t
2179:[
2176:R
2171:j
2168:+
2165:n
2161:t
2156:/
2152:]
2149:t
2146:[
2143:R
2121:]
2118:t
2115:[
2112:R
2107:n
2103:t
2098:/
2094:]
2091:t
2088:[
2085:R
2074:R
2055:]
2052:]
2049:t
2046:[
2043:[
2040:R
2016:)
2013:1
2010:,
2007:0
2004:[
1998:r
1978:n
1958:r
1955:+
1952:n
1932:.
1929:j
1923:i
1901:i
1897:p
1884:j
1880:x
1871:i
1867:x
1846:)
1840:,
1835:2
1831:x
1827:,
1822:1
1818:x
1814:(
1793:Z
1787:n
1783:p
1778:/
1773:R
1760:p
1744:p
1730:.
1727:j
1721:i
1699:i
1695:p
1682:j
1678:n
1669:i
1665:n
1644:)
1638:,
1633:2
1629:n
1625:,
1620:1
1616:n
1612:(
1583:Z
1577:n
1573:p
1568:/
1563:Z
1550:p
1522:.
1514:p
1511:o
1506:I
1501:C
1490:C
1486:X
1482:X
1478:C
1474:X
1451:p
1448:o
1443:I
1438:C
1427:C
1423:I
1416:j
1412:i
1406:j
1402:i
1394:I
1386:C
1379:X
1375:X
1361:X
1344:)
1339:j
1336:i
1332:f
1328:,
1323:i
1319:X
1315:(
1290:i
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