4249:
4496:
1250:
4516:
4506:
879:
2865:
2355:, 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
691:
3009:
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
3562:
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
2597:
2722:
400:
3226:
2737:
3423:
646:
496:
3332:
1924:
1722:
2473:
1815:
1605:
3000:
1543:
2207:
1477:
226:
3709:
3493:
1867:
1665:
2265:
2143:
1313:
3074:
2077:
874:{\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\}.}
2532:
3528:
1365:
1106:
2037:
136:
1953:
1751:
252:
2317:
1191:
546:
285:
1009:
682:
312:
1370:
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
1979:
1999:
975:
955:
902:
516:
2544:
1817:
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
2662:
319:
3744:
3470:
3587:
of category theory. The terminology is somewhat confusing: inverse limits are a class of limits, while direct limits are a class of colimits.
3157:
2860:{\displaystyle 0\rightarrow \varprojlim A_{i}\rightarrow \varprojlim B_{i}\rightarrow \varprojlim C_{i}\rightarrow \varprojlim {}^{1}A_{i}}
3893:
3691:
4178:
3363:
558:
408:
3846:
3669:
3640:
3612:
3278:
2389:
is the inverse limit of the set of finite strings, and is thus endowed with the limit topology. As the original spaces are
1872:
1670:
3033:
a system of finite-dimensional vector spaces or finite abelian groups or modules of finite length or
Artinian modules.
4540:
3700:
2432:
1778:
1568:
2927:
1769:
2371:
54:
although their existence depends on the category that is considered. They are a special case of the concept of
17:
3886:
3346:
1615:
with the usual order, and the morphisms being "take remainder". That is, one considers sequences of integers
3486:≠ 0. Roos has since shown (in "Derived functors of inverse limits revisited") that his result is correct if
1506:
4090:
4045:
2364:
2148:
1443:
175:
1820:
1618:
4519:
4459:
3805:
3747:(2002), "A counterexample to a 1961 "theorem" in homological algebra (with appendix by Pierre Deligne)",
3011:
2212:
2090:
1272:
4168:
3477:
constructed an example of such a system in a category satisfying (AB4) (in addition to (AB4*)) with lim
3040:
4509:
4295:
4159:
4067:
4545:
4468:
4112:
4050:
3973:
3257:
2386:
4499:
4455:
4060:
3879:
3749:
2045:
2510:
4055:
4037:
3714:
3584:
3506:
55:
2004:
1321:
1062:
4262:
4028:
4008:
3931:
3576:
2360:
1480:
1052:
109:
51:
4144:
3983:
1929:
1727:
1430:
231:
2292:
1166:
521:
260:
3956:
3951:
3856:
3826:
3793:
3770:
3737:
2728:
2498:
of abelian groups, the Mittag-Leffler condition is a condition on the transition morphisms
2394:
1032:
1028:
987:
660:
290:
1869:
such that each element of the sequence "projects" down to the previous ones, namely, that
1667:
such that each element of the sequence "projects" down to the previous ones, namely, that
8:
4300:
4248:
4174:
3978:
2592:{\displaystyle \varprojlim {}^{1}:\operatorname {Ab} ^{I}\rightarrow \operatorname {Ab} }
2080:
1958:
1255:
99:
43:
3017:
The following situations are examples where the Mittag-Leffler condition is satisfied:
2356:
4154:
4149:
4131:
4013:
3988:
3841:. Cambridge Studies in Advanced Mathematics. Vol. 38. Cambridge University Press.
1984:
1608:
1056:
1024:
978:
960:
940:
887:
501:
103:
4463:
4400:
4388:
4290:
4215:
4210:
4164:
3946:
3941:
3860:
3842:
3728:
3696:
3686:
3675:
3665:
3646:
3636:
3608:
2390:
1020:
1012:
4424:
4310:
4285:
4220:
4205:
4200:
4139:
3968:
3936:
3814:
3758:
3723:
3657:
3628:
2535:
2420:
2375:
2352:
2320:
2270:
1758:
4336:
3902:
3852:
3822:
3789:
3766:
3733:
3607:
John Rhodes & Benjamin
Steinberg. The q-theory of Finite Semigroups. p. 133.
3265:
3241:
2717:{\displaystyle 0\rightarrow A_{i}\rightarrow B_{i}\rightarrow C_{i}\rightarrow 0}
1040:
4373:
4368:
4352:
4315:
4305:
4225:
3834:
3474:
3466:
1612:
1419:
1408:
653:
169:
3818:
2374:, every inverse system has an inverse limit. It is constructed by placing the
395:{\displaystyle f_{ik}=f_{ij}\circ f_{jk}\quad {\text{for all }}i\leq j\leq k.}
4534:
4363:
4195:
3998:
3800:
3777:
3350:
2479:
2145:, indexed by the natural numbers as usually ordered, with the morphisms from
1559:
1546:
62:
3864:
3679:
3650:
2363:, viewing the finite sets as compact discrete spaces, and then applying the
4117:
4018:
3580:
3535:
3342:
1762:
1491:
can be considered a trivial inverse system, where all objects are equal to
1036:
139:
66:
3762:
4378:
4358:
4230:
4100:
3501:
3465:
the set of non-negative integers (such inverse systems are often called "
1382:
31:
4410:
4348:
3961:
3496:
has shown (in "The cohomological dimension of a directed set") that if
2882:
If the ranges of the morphisms of an inverse system of abelian groups (
2405:
42:) is a construction that allows one to "glue together" several related
3490:
has a set of generators (in addition to satisfying (AB3) and (AB4*)).
4404:
4095:
3221:{\displaystyle \varprojlim {}^{1}A_{i}=\mathbf {Z} _{p}/\mathbf {Z} }
2487:
2378:
on the underlying set-theoretic inverse limit. This is known as the
1016:
47:
4473:
4105:
4003:
1109:
649:
3871:
1249:
4443:
4433:
4082:
3993:
3081:
1400:
78:
1223:) must be universal in the sense that for any other such pair (
50:
between the objects. Thus, inverse limits can be defined in any
4438:
2398:
65:, that is by reversing the arrows, an inverse limit becomes a
4320:
3461:) an inverse system with surjective transition morphisms and
142:
3428:
It was thought for almost 40 years that Roos had proved (in
1051:
The inverse limit can be defined abstractly in an arbitrary
748:
3780:(1961), "Sur les foncteurs dérivés de lim. Applications",
3418:{\displaystyle \varprojlim {}^{n}\cong R^{n}\varprojlim .}
2273:
are defined as inverse limits of (discrete) finite groups.
977:. The inverse limit and the natural projections satisfy a
641:{\displaystyle ((A_{i})_{i\in I},(f_{ij})_{i\leq j\in I})}
491:{\displaystyle ((A_{i})_{i\in I},(f_{ij})_{i\leq j\in I})}
498:
is called an inverse system of groups and morphisms over
3803:(2006), "Derived functors of inverse limits revisited",
2414:
172:
of groups and suppose we have a family of homomorphisms
3570:
1043:. The inverse limit will also belong to that category.
3268:
of the inverse limit functor can thus be defined. The
2216:
2152:
2094:
2049:
1900:
1698:
1324:
1065:
3509:
3366:
3327:{\displaystyle R^{n}\varprojlim :C^{I}\rightarrow C.}
3281:
3160:
3043:
2930:
2740:
2665:
2547:
2513:
2435:
2295:
2215:
2151:
2093:
2048:
2007:
1987:
1961:
1932:
1875:
1823:
1781:
1730:
1673:
1621:
1571:
1509:
1446:
1275:
1169:
990:
963:
943:
890:
694:
663:
561:
524:
504:
411:
322:
293:
263:
234:
178:
112:
3583:(or inductive limit). More general concepts are the
2087:
can be thought of as the inverse limit of the rings
1919:{\displaystyle x_{i}\equiv x_{j}{\mbox{ mod }}p^{i}}
1717:{\displaystyle n_{i}\equiv n_{j}{\mbox{ mod }}p^{i}}
3603:
3601:
3599:
2656:) are three inverse systems of abelian groups, and
1757:-adic integers is the one implied here, namely the
548:are called the transition morphisms of the system.
3522:
3417:
3326:
3220:
3068:
2994:
2859:
2716:
2591:
2526:
2467:
2311:
2259:
2201:
2137:
2071:
2031:
1993:
1973:
1947:
1918:
1861:
1809:
1745:
1716:
1659:
1599:
1537:
1471:
1359:
1307:
1185:
1100:
1003:
969:
949:
896:
873:
676:
640:
540:
510:
490:
394:
306:
279:
246:
220:
130:
1545:The inverse limit, if it exists, is defined as a
1395:Inverse systems and inverse limits in a category
984:This same construction may be carried out if the
4532:
3596:
254:(note the order) with the following properties:
46:, the precise gluing process being specified by
3431:Sur les foncteurs dérivés de lim. Applications.
2468:{\displaystyle \varprojlim :C^{I}\rightarrow C}
1775:is the inverse limit of the topological groups
2486:is ordered (not simply partially ordered) and
1810:{\displaystyle \mathbb {R} /p^{n}\mathbb {Z} }
1600:{\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }
3887:
1399:admit an alternative description in terms of
3429:
937:th component of the direct product for each
2995:{\displaystyle f_{kj}(A_{j})=f_{ki}(A_{i})}
2877:
4515:
4505:
4261:
3894:
3880:
3256:is an arbitrary abelian category that has
798:
792:
3727:
1803:
1783:
1593:
1573:
3708:
3692:Categories for the Working Mathematician
3685:
3656:
3627:
1499:. This defines a "trivial functor" from
1479:be the category of these functors (with
3002:one says that the system satisfies the
2359:in graph theory and may be proved with
14:
4533:
3839:An introduction to homological algebra
3833:
3743:
3712:(1972), "Rings with several objects",
1538:{\displaystyle C^{I^{\mathrm {op} }}.}
1411:where the morphisms consist of arrows
4260:
3913:
3875:
3554:-indexed diagrams in the category of
2602:(pronounced "lim one") such that if (
2415:Derived functors of the inverse limit
2202:{\displaystyle \textstyle R/t^{n+j}R}
1472:{\displaystyle C^{I^{\mathrm {op} }}}
1378:of an inverse system, there exists a
1266:. The inverse limit is often denoted
1046:
221:{\displaystyle f_{ij}:A_{j}\to A_{i}}
3799:
3776:
3571:Related concepts and generalizations
3357:to series of functors lim such that
3272:th right derived functor is denoted
1862:{\displaystyle (x_{1},x_{2},\dots )}
1660:{\displaystyle (n_{1},n_{2},\dots )}
1392:commuting with the projection maps.
1108:be an inverse system of objects and
89:
84:
3901:
2397:. This is one way of realizing the
2260:{\displaystyle \textstyle R/t^{n}R}
2138:{\displaystyle \textstyle R/t^{n}R}
1429:. An inverse system is then just a
1308:{\displaystyle X=\varprojlim X_{i}}
1039:are morphisms in the corresponding
1035:(over a fixed ring), etc., and the
94:We start with the definition of an
24:
3511:
3247:
3076:is non-zero is obtained by taking
3069:{\displaystyle \varprojlim {}^{1}}
1565:is the inverse limit of the rings
1524:
1521:
1495:and all arrow are the identity of
1461:
1458:
25:
4557:
1955:Its elements are exactly of form
1233:) there exists a unique morphism
4514:
4504:
4495:
4494:
4247:
3914:
3214:
3198:
3021:a system in which the morphisms
2367:characterization of compactness.
2267:given by the natural projection.
1248:
1116:(same definition as above). The
3469:sequences"). However, in 2002,
3353:generalized the functor lim on
1611:) with the index set being the
981:described in the next section.
368:
27:Construction in category theory
3662:General topology: Chapters 1-4
3315:
2989:
2976:
2957:
2944:
2822:
2796:
2770:
2744:
2708:
2695:
2682:
2669:
2583:
2507:that ensures the exactness of
2459:
2372:category of topological spaces
2326:. Then the natural projection
2253:
2247:
2226:
2220:
2195:
2189:
2162:
2156:
2131:
2125:
2104:
2098:
2065:
2062:
2056:
2053:
2026:
2014:
1856:
1824:
1654:
1622:
1354:
1325:
1095:
1066:
838:
825:
757:
635:
614:
597:
579:
565:
562:
485:
464:
447:
429:
415:
412:
205:
125:
113:
13:
1:
3621:
2072:{\displaystyle \textstyle R]}
3729:10.1016/0001-8708(72)90002-3
2527:{\displaystyle \varprojlim }
2426:, the inverse limit functor
2365:finite intersection property
1753:The natural topology on the
1403:. Any partially ordered set
1120:of this system is an object
7:
4189:Constructions on categories
3523:{\displaystyle \aleph _{d}}
1552:
1360:{\textstyle (X_{i},f_{ij})}
1101:{\textstyle (X_{i},f_{ij})}
10:
4562:
4296:Higher-dimensional algebra
3695:(2nd ed.), Springer,
2032:{\displaystyle r\in [0,1)}
98:(or projective system) of
4490:
4423:
4387:
4335:
4328:
4279:
4269:
4256:
4245:
4188:
4130:
4081:
4036:
4027:
3924:
3920:
3909:
3819:10.1112/S0024610705022416
3579:of an inverse limit is a
3550:+ 2. This applies to the
2731:of inverse systems, then
2001:is a p-adic integer, and
1549:of this trivial functor.
1483:as morphisms). An object
145:(not all authors require
131:{\displaystyle (I,\leq )}
4541:Limits (category theory)
3750:Inventiones Mathematicae
3590:
3012:Mittag-Leffler's theorem
3004:Mittag-Leffler condition
2878:Mittag-Leffler condition
2870:is an exact sequence in
2083:over a commutative ring
1318:with the inverse system
1128:together with morphisms
4106:Cokernels and quotients
4029:Universal constructions
3715:Advances in Mathematics
3080:to be the non-negative
1948:{\displaystyle i<j.}
1746:{\displaystyle i<j.}
1481:natural transformations
1407:can be considered as a
247:{\displaystyle i\leq j}
4263:Higher category theory
4009:Natural transformation
3782:C. R. Acad. Sci. Paris
3524:
3430:
3419:
3328:
3222:
3070:
2996:
2861:
2718:
2593:
2538:constructed a functor
2528:
2469:
2408:(as infinite strings).
2313:
2312:{\displaystyle f_{ij}}
2280:of an inverse system (
2261:
2203:
2139:
2073:
2033:
1995:
1975:
1949:
1920:
1863:
1811:
1747:
1718:
1661:
1601:
1539:
1473:
1361:
1309:
1245:such that the diagram
1187:
1186:{\displaystyle f_{ij}}
1102:
1005:
971:
951:
898:
875:
678:
642:
555:of the inverse system
542:
541:{\displaystyle f_{ij}}
512:
492:
396:
308:
281:
280:{\displaystyle f_{ii}}
248:
222:
149:to be directed). Let (
132:
3763:10.1007/s002220100197
3525:
3420:
3329:
3223:
3071:
2997:
2904:, that is, for every
2862:
2719:
2594:
2529:
2470:
2393:, the limit space is
2314:
2262:
2204:
2140:
2074:
2034:
1996:
1976:
1950:
1921:
1864:
1812:
1748:
1719:
1662:
1602:
1540:
1474:
1431:contravariant functor
1362:
1310:
1188:
1103:
1031:(over a fixed ring),
1006:
1004:{\displaystyle A_{i}}
972:
952:
899:
876:
679:
677:{\displaystyle A_{i}}
643:
543:
513:
493:
397:
309:
307:{\displaystyle A_{i}}
282:
249:
223:
133:
58:in category theory.
4132:Algebraic categories
3806:J. London Math. Soc.
3567: > 1).
3542:lim is zero for all
3507:
3364:
3279:
3158:
3041:
2928:
2738:
2729:short exact sequence
2663:
2545:
2511:
2433:
2395:totally disconnected
2385:The set of infinite
2293:
2213:
2149:
2091:
2046:
2005:
1985:
1959:
1930:
1873:
1821:
1779:
1728:
1671:
1619:
1569:
1507:
1444:
1322:
1273:
1167:
1063:
988:
961:
941:
904:comes equipped with
888:
692:
661:
559:
522:
518:, and the morphisms
502:
409:
320:
291:
261:
232:
176:
110:
4301:Homotopy hypothesis
3979:Commutative diagram
3585:limits and colimits
3252:More generally, if
2361:Tychonoff's theorem
2081:formal power series
2039:is the "remainder".
1974:{\displaystyle n+r}
931:which pick out the
906:natural projections
843: for all
287:is the identity on
4014:Universal property
3835:Weibel, Charles A.
3689:(September 1998),
3687:Mac Lane, Saunders
3520:
3415:
3410:
3375:
3337:In the case where
3324:
3300:
3218:
3169:
3066:
3052:
2992:
2916:such that for all
2857:
2833:
2807:
2781:
2755:
2714:
2589:
2556:
2524:
2522:
2465:
2444:
2348:is an isomorphism.
2309:
2276:Let the index set
2257:
2256:
2199:
2198:
2135:
2134:
2069:
2068:
2029:
1991:
1971:
1945:
1916:
1904:
1859:
1807:
1743:
1714:
1702:
1657:
1609:modular arithmetic
1597:
1535:
1469:
1367:being understood.
1357:
1305:
1290:
1183:
1098:
1057:universal property
1047:General definition
1021:topological spaces
1001:
979:universal property
967:
947:
894:
884:The inverse limit
871:
781:
723:
710:
674:
638:
538:
508:
488:
392:
304:
277:
244:
218:
128:
61:By working in the
4528:
4527:
4486:
4485:
4482:
4481:
4464:monoidal category
4419:
4418:
4291:Enriched category
4243:
4242:
4239:
4238:
4216:Quotient category
4211:Opposite category
4126:
4125:
3848:978-0-521-55987-4
3671:978-3-540-64241-1
3658:Bourbaki, Nicolas
3642:978-3-540-64243-5
3629:Bourbaki, Nicolas
3613:978-0-387-09780-0
3536:infinite cardinal
3403:
3368:
3293:
3258:enough injectives
3162:
3045:
3037:An example where
2826:
2800:
2774:
2748:
2549:
2515:
2437:
2271:Pro-finite groups
1994:{\displaystyle n}
1903:
1765:as the open sets.
1701:
1283:
970:{\displaystyle I}
950:{\displaystyle i}
897:{\displaystyle A}
858:
844:
766:
760:
703:
701:
511:{\displaystyle I}
372:
90:Algebraic objects
85:Formal definition
38:(also called the
16:(Redirected from
4553:
4546:Abstract algebra
4518:
4517:
4508:
4507:
4498:
4497:
4333:
4332:
4311:Simplex category
4286:Categorification
4277:
4276:
4258:
4257:
4251:
4221:Product category
4206:Kleisli category
4201:Functor category
4046:Terminal objects
4034:
4033:
3969:Adjoint functors
3922:
3921:
3911:
3910:
3896:
3889:
3882:
3873:
3872:
3868:
3829:
3796:
3773:
3740:
3731:
3705:
3682:
3653:
3616:
3605:
3577:categorical dual
3529:
3527:
3526:
3521:
3519:
3518:
3433:
3424:
3422:
3421:
3416:
3411:
3401:
3400:
3388:
3387:
3382:
3376:
3333:
3331:
3330:
3325:
3314:
3313:
3301:
3291:
3290:
3266:derived functors
3264:, and the right
3227:
3225:
3224:
3219:
3217:
3212:
3207:
3206:
3201:
3192:
3191:
3182:
3181:
3176:
3170:
3075:
3073:
3072:
3067:
3065:
3064:
3059:
3053:
3001:
2999:
2998:
2993:
2988:
2987:
2975:
2974:
2956:
2955:
2943:
2942:
2866:
2864:
2863:
2858:
2856:
2855:
2846:
2845:
2840:
2834:
2821:
2820:
2808:
2795:
2794:
2782:
2769:
2768:
2756:
2723:
2721:
2720:
2715:
2707:
2706:
2694:
2693:
2681:
2680:
2598:
2596:
2595:
2590:
2582:
2581:
2569:
2568:
2563:
2557:
2534:. Specifically,
2533:
2531:
2530:
2525:
2523:
2494:is the category
2474:
2472:
2471:
2466:
2458:
2457:
2445:
2421:abelian category
2376:initial topology
2353:category of sets
2329:
2321:greatest element
2318:
2316:
2315:
2310:
2308:
2307:
2266:
2264:
2263:
2258:
2243:
2242:
2233:
2208:
2206:
2205:
2200:
2185:
2184:
2169:
2144:
2142:
2141:
2136:
2121:
2120:
2111:
2078:
2076:
2075:
2070:
2038:
2036:
2035:
2030:
2000:
1998:
1997:
1992:
1980:
1978:
1977:
1972:
1954:
1952:
1951:
1946:
1925:
1923:
1922:
1917:
1915:
1914:
1905:
1901:
1898:
1897:
1885:
1884:
1868:
1866:
1865:
1860:
1849:
1848:
1836:
1835:
1816:
1814:
1813:
1808:
1806:
1801:
1800:
1791:
1786:
1759:product topology
1752:
1750:
1749:
1744:
1723:
1721:
1720:
1715:
1713:
1712:
1703:
1699:
1696:
1695:
1683:
1682:
1666:
1664:
1663:
1658:
1647:
1646:
1634:
1633:
1606:
1604:
1603:
1598:
1596:
1591:
1590:
1581:
1576:
1544:
1542:
1541:
1536:
1531:
1530:
1529:
1528:
1527:
1478:
1476:
1475:
1470:
1468:
1467:
1466:
1465:
1464:
1366:
1364:
1363:
1358:
1353:
1352:
1337:
1336:
1314:
1312:
1311:
1306:
1304:
1303:
1291:
1252:
1217:
1196:
1192:
1190:
1189:
1184:
1182:
1181:
1157:
1131:
1107:
1105:
1104:
1099:
1094:
1093:
1078:
1077:
1010:
1008:
1007:
1002:
1000:
999:
976:
974:
973:
968:
956:
954:
953:
948:
936:
930:
911:
903:
901:
900:
895:
880:
878:
877:
872:
867:
863:
859:
856:
845:
842:
837:
836:
824:
823:
808:
807:
797:
793:
791:
790:
780:
762:
761:
753:
738:
737:
736:
722:
711:
683:
681:
680:
675:
673:
672:
648:as a particular
647:
645:
644:
639:
634:
633:
612:
611:
593:
592:
577:
576:
547:
545:
544:
539:
537:
536:
517:
515:
514:
509:
497:
495:
494:
489:
484:
483:
462:
461:
443:
442:
427:
426:
401:
399:
398:
393:
373:
370:
367:
366:
351:
350:
335:
334:
313:
311:
310:
305:
303:
302:
286:
284:
283:
278:
276:
275:
253:
251:
250:
245:
227:
225:
224:
219:
217:
216:
204:
203:
191:
190:
137:
135:
134:
129:
40:projective limit
21:
4561:
4560:
4556:
4555:
4554:
4552:
4551:
4550:
4531:
4530:
4529:
4524:
4478:
4448:
4415:
4392:
4383:
4340:
4324:
4275:
4265:
4252:
4235:
4184:
4122:
4091:Initial objects
4077:
4023:
3916:
3905:
3903:Category theory
3900:
3849:
3832:Section 3.5 of
3710:Mitchell, Barry
3703:
3672:
3643:
3624:
3619:
3606:
3597:
3593:
3573:
3558:-modules, with
3514:
3510:
3508:
3505:
3504:
3485:
3460:
3451:
3442:
3402:
3396:
3392:
3383:
3381:
3380:
3367:
3365:
3362:
3361:
3309:
3305:
3292:
3286:
3282:
3280:
3277:
3276:
3260:, then so does
3250:
3248:Further results
3242:p-adic integers
3239:
3213:
3208:
3202:
3197:
3196:
3187:
3183:
3177:
3175:
3174:
3161:
3159:
3156:
3155:
3139:
3130:
3121:
3108:
3092:
3060:
3058:
3057:
3044:
3042:
3039:
3038:
3029:
2983:
2979:
2967:
2963:
2951:
2947:
2935:
2931:
2929:
2926:
2925:
2899:
2890:
2880:
2851:
2847:
2841:
2839:
2838:
2825:
2816:
2812:
2799:
2790:
2786:
2773:
2764:
2760:
2747:
2739:
2736:
2735:
2702:
2698:
2689:
2685:
2676:
2672:
2664:
2661:
2660:
2655:
2646:
2637:
2628:
2619:
2610:
2577:
2573:
2564:
2562:
2561:
2548:
2546:
2543:
2542:
2514:
2512:
2509:
2508:
2506:
2453:
2449:
2436:
2434:
2431:
2430:
2417:
2347:
2334:
2327:
2300:
2296:
2294:
2291:
2290:
2288:
2238:
2234:
2229:
2214:
2211:
2210:
2174:
2170:
2165:
2150:
2147:
2146:
2116:
2112:
2107:
2092:
2089:
2088:
2047:
2044:
2043:
2006:
2003:
2002:
1986:
1983:
1982:
1960:
1957:
1956:
1931:
1928:
1927:
1910:
1906:
1902: mod
1899:
1893:
1889:
1880:
1876:
1874:
1871:
1870:
1844:
1840:
1831:
1827:
1822:
1819:
1818:
1802:
1796:
1792:
1787:
1782:
1780:
1777:
1776:
1729:
1726:
1725:
1708:
1704:
1700: mod
1697:
1691:
1687:
1678:
1674:
1672:
1669:
1668:
1642:
1638:
1629:
1625:
1620:
1617:
1616:
1613:natural numbers
1592:
1586:
1582:
1577:
1572:
1570:
1567:
1566:
1555:
1520:
1519:
1515:
1514:
1510:
1508:
1505:
1504:
1457:
1456:
1452:
1451:
1447:
1445:
1442:
1441:
1345:
1341:
1332:
1328:
1323:
1320:
1319:
1299:
1295:
1282:
1274:
1271:
1270:
1253:
1232:
1222:
1215:
1201:
1194:
1174:
1170:
1168:
1165:
1164:
1162:
1155:
1149:
1136:
1129:
1086:
1082:
1073:
1069:
1064:
1061:
1060:
1049:
995:
991:
989:
986:
985:
962:
959:
958:
942:
939:
938:
932:
929:
916:
909:
908:
889:
886:
885:
855:
841:
832:
828:
816:
812:
803:
799:
786:
782:
770:
752:
751:
750:
747:
746:
742:
732:
728:
727:
712:
702:
693:
690:
689:
668:
664:
662:
659:
658:
617:
613:
604:
600:
582:
578:
572:
568:
560:
557:
556:
529:
525:
523:
520:
519:
503:
500:
499:
467:
463:
454:
450:
432:
428:
422:
418:
410:
407:
406:
369:
359:
355:
343:
339:
327:
323:
321:
318:
317:
298:
294:
292:
289:
288:
268:
264:
262:
259:
258:
233:
230:
229:
212:
208:
199:
195:
183:
179:
177:
174:
173:
167:
157:
111:
108:
107:
92:
87:
71:inductive limit
28:
23:
22:
15:
12:
11:
5:
4559:
4549:
4548:
4543:
4526:
4525:
4523:
4522:
4512:
4502:
4491:
4488:
4487:
4484:
4483:
4480:
4479:
4477:
4476:
4471:
4466:
4452:
4446:
4441:
4436:
4430:
4428:
4421:
4420:
4417:
4416:
4414:
4413:
4408:
4397:
4395:
4390:
4385:
4384:
4382:
4381:
4376:
4371:
4366:
4361:
4356:
4345:
4343:
4338:
4330:
4326:
4325:
4323:
4318:
4316:String diagram
4313:
4308:
4306:Model category
4303:
4298:
4293:
4288:
4283:
4281:
4274:
4273:
4270:
4267:
4266:
4254:
4253:
4246:
4244:
4241:
4240:
4237:
4236:
4234:
4233:
4228:
4226:Comma category
4223:
4218:
4213:
4208:
4203:
4198:
4192:
4190:
4186:
4185:
4183:
4182:
4172:
4162:
4160:Abelian groups
4157:
4152:
4147:
4142:
4136:
4134:
4128:
4127:
4124:
4123:
4121:
4120:
4115:
4110:
4109:
4108:
4098:
4093:
4087:
4085:
4079:
4078:
4076:
4075:
4070:
4065:
4064:
4063:
4053:
4048:
4042:
4040:
4031:
4025:
4024:
4022:
4021:
4016:
4011:
4006:
4001:
3996:
3991:
3986:
3981:
3976:
3971:
3966:
3965:
3964:
3959:
3954:
3949:
3944:
3939:
3928:
3926:
3918:
3917:
3907:
3906:
3899:
3898:
3891:
3884:
3876:
3870:
3869:
3847:
3830:
3801:Roos, Jan-Erik
3797:
3778:Roos, Jan-Erik
3774:
3757:(2): 397–420,
3741:
3706:
3701:
3683:
3670:
3654:
3641:
3623:
3620:
3618:
3617:
3594:
3592:
3589:
3572:
3569:
3517:
3513:
3494:Barry Mitchell
3481:
3475:Pierre Deligne
3467:Mittag-Leffler
3456:
3447:
3438:
3426:
3425:
3414:
3409:
3406:
3399:
3395:
3391:
3386:
3379:
3374:
3371:
3335:
3334:
3323:
3320:
3317:
3312:
3308:
3304:
3299:
3296:
3289:
3285:
3249:
3246:
3235:
3229:
3228:
3216:
3211:
3205:
3200:
3195:
3190:
3186:
3180:
3173:
3168:
3165:
3135:
3126:
3117:
3104:
3088:
3063:
3056:
3051:
3048:
3035:
3034:
3031:
3030:are surjective
3025:
2991:
2986:
2982:
2978:
2973:
2970:
2966:
2962:
2959:
2954:
2950:
2946:
2941:
2938:
2934:
2895:
2886:
2879:
2876:
2868:
2867:
2854:
2850:
2844:
2837:
2832:
2829:
2824:
2819:
2815:
2811:
2806:
2803:
2798:
2793:
2789:
2785:
2780:
2777:
2772:
2767:
2763:
2759:
2754:
2751:
2746:
2743:
2725:
2724:
2713:
2710:
2705:
2701:
2697:
2692:
2688:
2684:
2679:
2675:
2671:
2668:
2651:
2642:
2633:
2624:
2615:
2606:
2600:
2599:
2588:
2585:
2580:
2576:
2572:
2567:
2560:
2555:
2552:
2521:
2518:
2502:
2476:
2475:
2464:
2461:
2456:
2452:
2448:
2443:
2440:
2416:
2413:
2412:
2411:
2410:
2409:
2380:limit topology
2368:
2349:
2343:
2330:
2306:
2303:
2299:
2284:
2274:
2268:
2255:
2252:
2249:
2246:
2241:
2237:
2232:
2228:
2225:
2222:
2219:
2197:
2194:
2191:
2188:
2183:
2180:
2177:
2173:
2168:
2164:
2161:
2158:
2155:
2133:
2130:
2127:
2124:
2119:
2115:
2110:
2106:
2103:
2100:
2097:
2067:
2064:
2061:
2058:
2055:
2052:
2040:
2028:
2025:
2022:
2019:
2016:
2013:
2010:
1990:
1970:
1967:
1964:
1944:
1941:
1938:
1935:
1913:
1909:
1896:
1892:
1888:
1883:
1879:
1858:
1855:
1852:
1847:
1843:
1839:
1834:
1830:
1826:
1805:
1799:
1795:
1790:
1785:
1773:-adic solenoid
1766:
1742:
1739:
1736:
1733:
1711:
1707:
1694:
1690:
1686:
1681:
1677:
1656:
1653:
1650:
1645:
1641:
1637:
1632:
1628:
1624:
1595:
1589:
1585:
1580:
1575:
1563:-adic integers
1554:
1551:
1534:
1526:
1523:
1518:
1513:
1463:
1460:
1455:
1450:
1420:if and only if
1409:small category
1356:
1351:
1348:
1344:
1340:
1335:
1331:
1327:
1316:
1315:
1302:
1298:
1294:
1289:
1286:
1281:
1278:
1247:
1228:
1218:
1197:
1180:
1177:
1173:
1158:
1145:
1132:
1112:in a category
1097:
1092:
1089:
1085:
1081:
1076:
1072:
1068:
1055:by means of a
1048:
1045:
998:
994:
966:
946:
925:
912:
893:
882:
881:
870:
866:
862:
857: in
854:
851:
848:
840:
835:
831:
827:
822:
819:
815:
811:
806:
802:
796:
789:
785:
779:
776:
773:
769:
765:
759:
756:
749:
745:
741:
735:
731:
726:
721:
718:
715:
709:
706:
700:
697:
671:
667:
654:direct product
637:
632:
629:
626:
623:
620:
616:
610:
607:
603:
599:
596:
591:
588:
585:
581:
575:
571:
567:
564:
551:We define the
535:
532:
528:
507:
487:
482:
479:
476:
473:
470:
466:
460:
457:
453:
449:
446:
441:
438:
435:
431:
425:
421:
417:
414:
405:Then the pair
403:
402:
391:
388:
385:
382:
379:
376:
365:
362:
358:
354:
349:
346:
342:
338:
333:
330:
326:
315:
301:
297:
274:
271:
267:
243:
240:
237:
215:
211:
207:
202:
198:
194:
189:
186:
182:
159:
153:
127:
124:
121:
118:
115:
96:inverse system
91:
88:
86:
83:
26:
18:Inverse system
9:
6:
4:
3:
2:
4558:
4547:
4544:
4542:
4539:
4538:
4536:
4521:
4513:
4511:
4503:
4501:
4493:
4492:
4489:
4475:
4472:
4470:
4467:
4465:
4461:
4457:
4453:
4451:
4449:
4442:
4440:
4437:
4435:
4432:
4431:
4429:
4426:
4422:
4412:
4409:
4406:
4402:
4399:
4398:
4396:
4394:
4386:
4380:
4377:
4375:
4372:
4370:
4367:
4365:
4364:Tetracategory
4362:
4360:
4357:
4354:
4353:pseudofunctor
4350:
4347:
4346:
4344:
4342:
4334:
4331:
4327:
4322:
4319:
4317:
4314:
4312:
4309:
4307:
4304:
4302:
4299:
4297:
4294:
4292:
4289:
4287:
4284:
4282:
4278:
4272:
4271:
4268:
4264:
4259:
4255:
4250:
4232:
4229:
4227:
4224:
4222:
4219:
4217:
4214:
4212:
4209:
4207:
4204:
4202:
4199:
4197:
4196:Free category
4194:
4193:
4191:
4187:
4180:
4179:Vector spaces
4176:
4173:
4170:
4166:
4163:
4161:
4158:
4156:
4153:
4151:
4148:
4146:
4143:
4141:
4138:
4137:
4135:
4133:
4129:
4119:
4116:
4114:
4111:
4107:
4104:
4103:
4102:
4099:
4097:
4094:
4092:
4089:
4088:
4086:
4084:
4080:
4074:
4073:Inverse limit
4071:
4069:
4066:
4062:
4059:
4058:
4057:
4054:
4052:
4049:
4047:
4044:
4043:
4041:
4039:
4035:
4032:
4030:
4026:
4020:
4017:
4015:
4012:
4010:
4007:
4005:
4002:
4000:
3999:Kan extension
3997:
3995:
3992:
3990:
3987:
3985:
3982:
3980:
3977:
3975:
3972:
3970:
3967:
3963:
3960:
3958:
3955:
3953:
3950:
3948:
3945:
3943:
3940:
3938:
3935:
3934:
3933:
3930:
3929:
3927:
3923:
3919:
3912:
3908:
3904:
3897:
3892:
3890:
3885:
3883:
3878:
3877:
3874:
3866:
3862:
3858:
3854:
3850:
3844:
3840:
3836:
3831:
3828:
3824:
3820:
3816:
3812:
3808:
3807:
3802:
3798:
3795:
3791:
3788:: 3702–3704,
3787:
3783:
3779:
3775:
3772:
3768:
3764:
3760:
3756:
3752:
3751:
3746:
3745:Neeman, Amnon
3742:
3739:
3735:
3730:
3725:
3721:
3717:
3716:
3711:
3707:
3704:
3702:0-387-98403-8
3698:
3694:
3693:
3688:
3684:
3681:
3677:
3673:
3667:
3663:
3659:
3655:
3652:
3648:
3644:
3638:
3634:
3630:
3626:
3625:
3614:
3610:
3604:
3602:
3600:
3595:
3588:
3586:
3582:
3578:
3568:
3566:
3561:
3557:
3553:
3549:
3545:
3541:
3537:
3533:
3515:
3503:
3499:
3495:
3491:
3489:
3484:
3480:
3476:
3472:
3468:
3464:
3459:
3455:
3450:
3446:
3441:
3437:
3432:
3412:
3407:
3404:
3397:
3393:
3389:
3384:
3377:
3372:
3369:
3360:
3359:
3358:
3356:
3352:
3351:Jan-Erik Roos
3348:
3344:
3340:
3321:
3318:
3310:
3306:
3302:
3297:
3294:
3287:
3283:
3275:
3274:
3273:
3271:
3267:
3263:
3259:
3255:
3245:
3243:
3238:
3234:
3209:
3203:
3193:
3188:
3184:
3178:
3171:
3166:
3163:
3154:
3153:
3152:
3150:
3147:
3143:
3138:
3134:
3129:
3125:
3120:
3116:
3112:
3107:
3103:
3099:
3096:
3091:
3087:
3083:
3079:
3061:
3054:
3049:
3046:
3032:
3028:
3024:
3020:
3019:
3018:
3015:
3013:
3007:
3005:
2984:
2980:
2971:
2968:
2964:
2960:
2952:
2948:
2939:
2936:
2932:
2923:
2919:
2915:
2911:
2908:there exists
2907:
2903:
2898:
2894:
2889:
2885:
2875:
2873:
2852:
2848:
2842:
2835:
2830:
2827:
2817:
2813:
2809:
2804:
2801:
2791:
2787:
2783:
2778:
2775:
2765:
2761:
2757:
2752:
2749:
2741:
2734:
2733:
2732:
2730:
2711:
2703:
2699:
2690:
2686:
2677:
2673:
2666:
2659:
2658:
2657:
2654:
2650:
2645:
2641:
2636:
2632:
2627:
2623:
2618:
2614:
2609:
2605:
2586:
2578:
2574:
2570:
2565:
2558:
2553:
2550:
2541:
2540:
2539:
2537:
2519:
2516:
2505:
2501:
2497:
2493:
2489:
2485:
2481:
2462:
2454:
2450:
2446:
2441:
2438:
2429:
2428:
2427:
2425:
2422:
2407:
2403:
2402:-adic numbers
2401:
2396:
2392:
2388:
2384:
2383:
2381:
2377:
2373:
2369:
2366:
2362:
2358:
2357:Kőnig's lemma
2354:
2350:
2346:
2342:
2338:
2333:
2325:
2322:
2304:
2301:
2297:
2287:
2283:
2279:
2275:
2272:
2269:
2250:
2244:
2239:
2235:
2230:
2223:
2217:
2192:
2186:
2181:
2178:
2175:
2171:
2166:
2159:
2153:
2128:
2122:
2117:
2113:
2108:
2101:
2095:
2086:
2082:
2059:
2050:
2041:
2023:
2020:
2017:
2011:
2008:
1988:
1968:
1965:
1962:
1942:
1939:
1936:
1933:
1911:
1907:
1894:
1890:
1886:
1881:
1877:
1853:
1850:
1845:
1841:
1837:
1832:
1828:
1797:
1793:
1788:
1774:
1772:
1767:
1764:
1763:cylinder sets
1760:
1756:
1740:
1737:
1734:
1731:
1709:
1705:
1692:
1688:
1684:
1679:
1675:
1651:
1648:
1643:
1639:
1635:
1630:
1626:
1614:
1610:
1587:
1583:
1578:
1564:
1562:
1557:
1556:
1550:
1548:
1547:right adjoint
1532:
1516:
1511:
1502:
1498:
1494:
1490:
1486:
1482:
1453:
1448:
1439:
1435:
1432:
1428:
1424:
1421:
1418:
1414:
1410:
1406:
1402:
1398:
1393:
1391:
1387:
1384:
1381:
1377:
1373:
1368:
1349:
1346:
1342:
1338:
1333:
1329:
1300:
1296:
1292:
1287:
1284:
1279:
1276:
1269:
1268:
1267:
1265:
1261:
1257:
1251:
1246:
1244:
1240:
1236:
1231:
1226:
1221:
1213:
1209:
1205:
1200:
1178:
1175:
1171:
1161:
1154:) satisfying
1153:
1148:
1144:
1140:
1135:
1127:
1123:
1119:
1118:inverse limit
1115:
1111:
1090:
1087:
1083:
1079:
1074:
1070:
1058:
1054:
1044:
1042:
1038:
1037:homomorphisms
1034:
1030:
1026:
1022:
1018:
1014:
996:
992:
982:
980:
964:
944:
935:
928:
924:
920:
915:
907:
891:
868:
864:
860:
852:
849:
846:
833:
829:
820:
817:
813:
809:
804:
800:
794:
787:
783:
777:
774:
771:
767:
763:
754:
743:
739:
733:
729:
724:
719:
716:
713:
707:
704:
698:
695:
688:
687:
686:
684:
669:
665:
655:
651:
630:
627:
624:
621:
618:
608:
605:
601:
594:
589:
586:
583:
573:
569:
554:
553:inverse limit
549:
533:
530:
526:
505:
480:
477:
474:
471:
468:
458:
455:
451:
444:
439:
436:
433:
423:
419:
389:
386:
383:
380:
377:
374:
371:for all
363:
360:
356:
352:
347:
344:
340:
336:
331:
328:
324:
316:
299:
295:
272:
269:
265:
257:
256:
255:
241:
238:
235:
213:
209:
200:
196:
192:
187:
184:
180:
171:
166:
162:
156:
152:
148:
144:
141:
122:
119:
116:
105:
104:homomorphisms
101:
97:
82:
80:
76:
72:
68:
64:
63:dual category
59:
57:
53:
49:
45:
41:
37:
36:inverse limit
33:
19:
4444:
4425:Categorified
4329:n-categories
4280:Key concepts
4118:Direct limit
4101:Coequalizers
4072:
4019:Yoneda lemma
3925:Key concepts
3915:Key concepts
3838:
3813:(1): 65–83,
3810:
3809:, Series 2,
3804:
3785:
3781:
3754:
3748:
3719:
3713:
3690:
3664:, Springer,
3661:
3635:, Springer,
3632:
3581:direct limit
3574:
3564:
3559:
3555:
3551:
3547:
3543:
3539:
3531:
3497:
3492:
3487:
3482:
3478:
3471:Amnon Neeman
3462:
3457:
3453:
3448:
3444:
3439:
3435:
3427:
3354:
3343:Grothendieck
3338:
3336:
3269:
3261:
3253:
3251:
3240:denotes the
3236:
3232:
3230:
3148:
3145:
3141:
3136:
3132:
3127:
3123:
3118:
3114:
3110:
3105:
3101:
3097:
3094:
3089:
3085:
3077:
3036:
3026:
3022:
3016:
3008:
3003:
2921:
2917:
2913:
2909:
2905:
2901:
2896:
2892:
2887:
2883:
2881:
2871:
2869:
2726:
2652:
2648:
2643:
2639:
2634:
2630:
2625:
2621:
2616:
2612:
2607:
2603:
2601:
2503:
2499:
2495:
2491:
2483:
2477:
2423:
2418:
2399:
2379:
2344:
2340:
2336:
2331:
2323:
2285:
2281:
2277:
2084:
1770:
1754:
1560:
1558:The ring of
1500:
1496:
1492:
1488:
1484:
1437:
1433:
1426:
1422:
1416:
1412:
1404:
1396:
1394:
1389:
1385:
1379:
1375:
1371:
1369:
1317:
1263:
1259:
1254:
1242:
1238:
1234:
1229:
1224:
1219:
1211:
1210:. The pair (
1207:
1203:
1198:
1159:
1151:
1146:
1142:
1138:
1133:
1125:
1121:
1117:
1113:
1050:
983:
933:
926:
922:
918:
913:
905:
883:
657:
552:
550:
404:
164:
160:
154:
150:
146:
95:
93:
74:
70:
67:direct limit
60:
39:
35:
29:
4393:-categories
4369:Kan complex
4359:Tricategory
4341:-categories
4231:Subcategory
3989:Exponential
3957:Preadditive
3952:Pre-abelian
3502:cardinality
3434:) that lim
1383:isomorphism
1152:projections
32:mathematics
4535:Categories
4411:3-category
4401:2-category
4374:∞-groupoid
4349:Bicategory
4096:Coproducts
4056:Equalizers
3962:Bicategory
3622:References
3341:satisfies
3084:, letting
2902:stationary
2480:left exact
2406:Cantor set
1388:′ →
1017:semigroups
77:becomes a
4460:Symmetric
4405:2-functor
4145:Relations
4068:Pullbacks
3722:: 1–161,
3633:Algebra I
3563:for
3512:ℵ
3443:= 0 for (
3408:←
3390:≅
3378:
3373:←
3345:'s axiom
3316:→
3298:←
3172:
3167:←
3055:
3050:←
2836:
2831:←
2823:→
2810:
2805:←
2797:→
2784:
2779:←
2771:→
2758:
2753:←
2745:→
2709:→
2696:→
2683:→
2670:→
2584:→
2559:
2554:←
2536:Eilenberg
2520:←
2488:countable
2460:→
2442:←
2319:) have a
2042:The ring
2012:∈
1926:whenever
1887:≡
1854:…
1724:whenever
1685:≡
1652:…
1293:
1288:←
1110:morphisms
850:≤
775:∈
768:∏
764:∈
758:→
725:
717:∈
708:←
628:∈
622:≤
587:∈
478:∈
472:≤
437:∈
384:≤
378:≤
353:∘
239:≤
206:→
123:≤
48:morphisms
4520:Glossary
4500:Category
4474:n-monoid
4427:concepts
4083:Colimits
4051:Products
4004:Morphism
3947:Concrete
3942:Additive
3932:Category
3865:36131259
3837:(1994).
3680:40551485
3660:(1989),
3651:40551484
3631:(1989),
3538:), then
3082:integers
2638:), and (
2404:and the
2391:discrete
1981:, where
1553:Examples
1401:functors
1258:for all
1256:commutes
1202:for all
1150:(called
1053:category
1041:category
1033:algebras
650:subgroup
228:for all
140:directed
73:, and a
52:category
4510:Outline
4469:n-group
4434:2-group
4389:Strict
4379:∞-topos
4175:Modules
4113:Pushout
4061:Kernels
3994:Functor
3937:Abelian
3857:1269324
3827:2197371
3794:0132091
3771:1906154
3738:0294454
3151:. Then
2924: :
2419:For an
2387:strings
2370:In the
2351:In the
1029:modules
1011:'s are
656:of the
652:of the
79:colimit
44:objects
4456:Traced
4439:2-ring
4169:Fields
4155:Groups
4150:Magmas
4038:Limits
3863:
3855:
3845:
3825:
3792:
3769:
3736:
3699:
3678:
3668:
3649:
3639:
3611:
3347:(AB4*)
3231:where
3113:, and
2900:) are
2490:, and
1440:. Let
1380:unique
1059:. Let
170:family
106:. Let
100:groups
34:, the
4450:-ring
4337:Weak
4321:Topos
4165:Rings
3591:Notes
3530:(the
2727:is a
2482:. If
1761:with
1607:(see
1025:rings
168:be a
143:poset
138:be a
75:limit
56:limit
4140:Sets
3861:OCLC
3843:ISBN
3697:ISBN
3676:OCLC
3666:ISBN
3647:OCLC
3637:ISBN
3609:ISBN
3575:The
3500:has
3473:and
2620:), (
1937:<
1768:The
1735:<
1374:and
1013:sets
685:'s:
102:and
3984:End
3974:CCC
3815:doi
3786:252
3759:doi
3755:148
3724:doi
3534:th
3405:lim
3370:lim
3295:lim
3164:lim
3047:lim
2828:lim
2802:lim
2776:lim
2750:lim
2551:lim
2517:lim
2478:is
2439:lim
2209:to
2079:of
1503:to
1487:of
1285:lim
1227:, ψ
1124:in
957:in
705:lim
69:or
30:In
4537::
4462:)
4458:)(
3859:.
3853:MR
3851:.
3823:MR
3821:,
3811:73
3790:MR
3784:,
3767:MR
3765:,
3753:,
3734:MR
3732:,
3718:,
3674:,
3645:,
3598:^
3546:≥
3458:ij
3452:,
3355:Ab
3349:,
3244:.
3140:=
3131:/
3122:=
3109:=
3100:,
3093:=
3027:ij
3014:.
3006:.
2920:≥
2912:≥
2897:ij
2891:,
2874:.
2872:Ab
2653:ij
2647:,
2635:ij
2629:,
2617:ij
2611:,
2587:Ab
2575:Ab
2504:ij
2496:Ab
2382:.
2339:→
2335::
2289:,
1436:→
1425:≤
1415:→
1376:X'
1262:≤
1241:→
1237::
1214:,
1206:≤
1193:∘
1163:=
1141:→
1137::
1027:,
1023:,
1019:,
1015:,
921:→
917::
81:.
4454:(
4447:n
4445:E
4407:)
4403:(
4391:n
4355:)
4351:(
4339:n
4181:)
4177:(
4171:)
4167:(
3895:e
3888:t
3881:v
3867:.
3817::
3761::
3726::
3720:8
3615:.
3565:n
3560:R
3556:R
3552:I
3548:d
3544:n
3540:R
3532:d
3516:d
3498:I
3488:C
3483:i
3479:A
3463:I
3454:f
3449:i
3445:A
3440:i
3436:A
3413:.
3398:n
3394:R
3385:n
3339:C
3322:.
3319:C
3311:I
3307:C
3303::
3288:n
3284:R
3270:n
3262:C
3254:C
3237:p
3233:Z
3215:Z
3210:/
3204:p
3199:Z
3194:=
3189:i
3185:A
3179:1
3149:Z
3146:p
3144:/
3142:Z
3137:i
3133:A
3128:i
3124:B
3119:i
3115:C
3111:Z
3106:i
3102:B
3098:Z
3095:p
3090:i
3086:A
3078:I
3062:1
3023:f
2990:)
2985:i
2981:A
2977:(
2972:i
2969:k
2965:f
2961:=
2958:)
2953:j
2949:A
2945:(
2940:j
2937:k
2933:f
2922:j
2918:i
2914:k
2910:j
2906:k
2893:f
2888:i
2884:A
2853:i
2849:A
2843:1
2818:i
2814:C
2792:i
2788:B
2766:i
2762:A
2742:0
2712:0
2704:i
2700:C
2691:i
2687:B
2678:i
2674:A
2667:0
2649:h
2644:i
2640:C
2631:g
2626:i
2622:B
2613:f
2608:i
2604:A
2579:I
2571::
2566:1
2500:f
2492:C
2484:I
2463:C
2455:I
2451:C
2447::
2424:C
2400:p
2345:m
2341:X
2337:X
2332:m
2328:π
2324:m
2305:j
2302:i
2298:f
2286:i
2282:X
2278:I
2254:]
2251:t
2248:[
2245:R
2240:n
2236:t
2231:/
2227:]
2224:t
2221:[
2218:R
2196:]
2193:t
2190:[
2187:R
2182:j
2179:+
2176:n
2172:t
2167:/
2163:]
2160:t
2157:[
2154:R
2132:]
2129:t
2126:[
2123:R
2118:n
2114:t
2109:/
2105:]
2102:t
2099:[
2096:R
2085:R
2066:]
2063:]
2060:t
2057:[
2054:[
2051:R
2027:)
2024:1
2021:,
2018:0
2015:[
2009:r
1989:n
1969:r
1966:+
1963:n
1943:.
1940:j
1934:i
1912:i
1908:p
1895:j
1891:x
1882:i
1878:x
1857:)
1851:,
1846:2
1842:x
1838:,
1833:1
1829:x
1825:(
1804:Z
1798:n
1794:p
1789:/
1784:R
1771:p
1755:p
1741:.
1738:j
1732:i
1710:i
1706:p
1693:j
1689:n
1680:i
1676:n
1655:)
1649:,
1644:2
1640:n
1636:,
1631:1
1627:n
1623:(
1594:Z
1588:n
1584:p
1579:/
1574:Z
1561:p
1533:.
1525:p
1522:o
1517:I
1512:C
1501:C
1497:X
1493:X
1489:C
1485:X
1462:p
1459:o
1454:I
1449:C
1438:C
1434:I
1427:j
1423:i
1417:j
1413:i
1405:I
1397:C
1390:X
1386:X
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