38:
2496:. Early category theorists believed that epimorphisms were the correct analogue of surjections in an arbitrary category, similar to how monomorphisms are very nearly an exact analogue of injections. Unfortunately this is incorrect; strong or regular epimorphisms behave much more closely to surjections than ordinary epimorphisms.
2511:
It is a common mistake to believe that epimorphisms are either identical to surjections or that they are a better concept. Unfortunately this is rarely the case; epimorphisms can be very mysterious and have unexpected behavior. It is very difficult, for example, to classify all the epimorphisms of
1420:
274:. Every epimorphism in this algebraic sense is an epimorphism in the sense of category theory, but the converse is not true in all categories. In this article, the term "epimorphism" will be used in the sense of category theory given above. For more on this, see
1511:
1581:
that sends each element to itself. Such a factorization of an arbitrary morphism into an epimorphism followed by a monomorphism can be carried out in all abelian categories and also in all the concrete categories mentioned above in
298:
is an epimorphism. In many concrete categories of interest the converse is also true. For example, in the following categories, the epimorphisms are exactly those morphisms that are surjective on the underlying sets:
1331:
183:
2051:
1795:
2115:
1683:
224:
1442:
2226:
2147:
1910:
1844:
2083:
1981:
2013:
2266:
2170:
1754:
1647:
1949:
1157:
is considered as a category with a single object (composition of morphisms given by multiplication), then the epimorphisms are precisely the right-cancellable elements.
1537:
In many categories it is possible to write every morphism as the composition of an epimorphism followed by a monomorphism. For instance, given a group homomorphism
2246:
2194:
1864:
1815:
1727:
1703:
1112:
1243:
As some of the above examples show, the property of being an epimorphism is not determined by the morphism alone, but also by the category of context. If
1138:
The above differs from the case of monomorphisms where it is more frequently true that monomorphisms are precisely those whose underlying functions are
2794:
1875:
1164:
is considered as a category (objects are the vertices, morphisms are the paths, composition of morphisms is the concatenation of paths), then
2713:
2613:
196:
the concept corresponds exactly to the surjective functions), but they may not exactly coincide in all contexts; for example, the inclusion
1415:{\displaystyle {\begin{matrix}\operatorname {Hom} (Y,Z)&\rightarrow &\operatorname {Hom} (X,Z)\\g&\mapsto &gf\end{matrix}}}
2632:
1530:
is an epimorphism, a consequence of the uniqueness requirement in the definition of coequalizers. It follows in particular that every
2672:
2653:
2592:
105:
2018:
2363:; as noted above, it is a bimorphism, but it is not bijective and therefore not an isomorphism. Similarly, in the category of
2347:
since the inverse map is not continuous at 1, so it is an instance of a bimorphism that is not an isomorphism in the category
2521:
660:
781:
However, there are also many concrete categories of interest where epimorphisms fail to be surjective. A few examples are:
2692:
1763:
1033:
2088:
2687:
1656:
1506:{\displaystyle {\begin{matrix}\operatorname {Hom} (Y,-)&\rightarrow &\operatorname {Hom} (X,-)\end{matrix}}}
2512:
rings. In general, epimorphisms are their own unique concept, related to surjections but fundamentally different.
522:
518:
and group homomorphisms. Also due to
Schreier; the proof given in (Linderholm 1970) establishes this case as well.
199:
2289:
1307:
1271:
1217:
1037:
534:
1534:
is an epimorphism. The converse, namely that every epimorphism be a coequalizer, is not true in all categories.
786:
2199:
2120:
501:
2315:. Every isomorphism is a bimorphism but the converse is not true in general. For example, the map from the
1434:
1026:
291:
242:
235:
227:
1883:
1267:. However the converse need not hold; the smaller category can (and often will) have more epimorphisms.
1018:, similar to the previous example. A similar argument shows that the natural ring homomorphism from any
553:
382:
68:
2459:
2324:
570:
2056:
1954:
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1986:
1824:
669:
581:
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481:
189:
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2155:
1739:
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1154:
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329:
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726:
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1007:
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287:
256:
2736:
2485:
1517:
1019:
681:
304:
252:
193:
1181:
is an epimorphism; indeed only a right-sided inverse is needed: if there exists a morphism
2508:, which are epimorphisms in the modern sense. However, this distinction never caught on.
2379:
2231:
2179:
1849:
1800:
1712:
1688:
694:
605:
368:
45:
2504:, which were maps in a concrete category whose underlying set maps were surjective, and
1874:
1215:
is easily seen to be an epimorphism. A map with such a right-sided inverse is called a
2702:
2364:
1226:
1161:
508:
with one amalgamated subgroup); an elementary proof can be found in (Linderholm 1970).
37:
2783:
2344:
2328:
794:
691:
527:
515:
493:
2603:
30:
This article is about the mathematical function. For the biological phenomenon, see
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31:
17:
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The composition of two epimorphisms is again an epimorphism. If the composition
1178:
1108:
1053:
928:
2579:
2288:
of rings is a homological epimorphism if it is an epimorphism and it induces a
2312:
1310:
between two categories turns epimorphisms into monomorphisms, and vice versa.
584:. This generalizes the two previous examples; to prove that every epimorphism
295:
268:
2755:
1426:
1139:
1123:
472:}. These maps are monotone if {0,1} is given the standard ordering 0 < 1.
1270:
As for most concepts in category theory, epimorphisms are preserved under
1531:
631:
497:
371:
and relation-preserving functions. Here we can use the same proof as for
53:
2643:
2462:, and the equivalence classes are defined to be the quotient objects of
2311:
A morphism that is both a monomorphism and an epimorphism is called a
1150:
943:
as in the previous example. This follows from the observation that
1873:
1222:
979:
is uniquely determined by its value on the element represented by
36:
2771:
2759:
2625:
Handbook of
Categorical Algebra. Volume 1: Basic Category Theory
1313:
The definition of epimorphism may be reformulated to state that
1122:, the epimorphisms are precisely the continuous functions with
178:{\displaystyle g_{1}\circ f=g_{2}\circ f\implies g_{1}=g_{2}.}
2605:
An
Invitation to General Algebra and Universal Constructions
1145:
As for examples of epimorphisms in non-concrete categories:
1006:
is a non-surjective epimorphism; to see this, note that any
804:
is a non-surjective epimorphism. To see this, suppose that
2578:
AdΓ‘mek, JiΕΓ; Herrlich, Horst; Strecker, George E. (1990).
2046:{\displaystyle \beta \circ \varepsilon =\mu \circ \alpha }
375:, equipping {0,1} with the full relation {0,1}×{0,1}.
1259:
that is an epimorphism when considered as a morphism in
684:, which ensures that all considered maps are continuous.
1447:
1336:
2544:
2542:
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2234:
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2182:
2158:
2123:
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1989:
1957:
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1827:
1803:
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1742:
1715:
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1659:
1635:
1445:
1334:
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is surjective, we compose it with both the canonical
202:
108:
1790:{\displaystyle \varepsilon =\mu \circ \varepsilon '}
2701:
2560:
2260:
2240:
2220:
2188:
2164:
2141:
2109:
2077:
2045:
2007:
1975:
1943:
1904:
1858:
1838:
1809:
1789:
1748:
1721:
1697:
1677:
1641:
1561:as the composition of the surjective homomorphism
1505:
1414:
1325:is an epimorphism if and only if the induced maps
911:In the category of algebras over commutative ring
218:
177:
2110:{\displaystyle \delta \circ \varepsilon =\alpha }
963:), and the inverse of the element represented by
2700:Lawvere, F. William; Rosebrugh, Robert (2015).
1678:{\displaystyle \varepsilon =\mu \circ \varphi }
312:and functions. To prove that every epimorphism
2648:. Dover Publications, Inc Mineola, New York.
935:and the morphism is induced by the inclusion
496:(he actually proved more, showing that every
8:
1433:. This in turn is equivalent to the induced
219:{\displaystyle \mathbb {Z} \to \mathbb {Q} }
431:β {0,1} be the characteristic function of {
328:is surjective, we compose it with both the
2500:attempted to create a distinction between
188:Epimorphisms are categorical analogues of
148:
144:
2663:Tsalenko, M.S.; Shulgeifer, E.G. (1974).
2378:Epimorphisms are used to define abstract
2253:
2233:
2201:
2181:
2157:
2122:
2090:
2058:
2020:
1988:
1956:
1924:
1885:
1851:
1826:
1802:
1765:
1741:
1714:
1690:
1658:
1634:
1586:(though not in all concrete categories).
1573:, followed by the injective homomorphism
1446:
1444:
1335:
1333:
1236:of two morphisms is an epimorphism, then
1052:is an epimorphism if and only if for all
212:
211:
204:
203:
201:
166:
153:
132:
113:
107:
2382:in general categories: two epimorphisms
2375:is a bimorphism but not an isomorphism.
2343:) is continuous and bijective but not a
2221:{\displaystyle \varepsilon \circ \mu =1}
2142:{\displaystyle \mu \circ \delta =\beta }
1126:images. For example, the inclusion map
1014:is determined entirely by its action on
676:is surjective, we proceed exactly as in
457:β {0,1} the characteristic function of {
2548:
2538:
488:. The result that every epimorphism in
1229:and an epimorphism is an isomorphism.
1118:In the category of Hausdorff spaces,
672:. To prove that every epimorphism in
7:
2248:is called a right-sided inverse for
971:is just the element represented by β
2725:"A Group Epimorphism is Surjective"
2351:. Another example is the embedding
1905:{\displaystyle \varepsilon :A\to B}
1626:of some pair of parallel morphisms.
71:in the sense that, for all objects
1290:is an epimorphism in the category
1134:, is a non-surjective epimorphism.
401:, β€) is not surjective, pick
25:
2795:Algebraic properties of elements
2581:Abstract and Concrete Categories
1846:is an epimorphism, the morphism
1594:Among other useful concepts are
275:
2422:if there exists an isomorphism
729:there is a continuous function
2741:10.1080/00029890.1970.11992448
2708:. Cambridge university press.
2665:Foundations of category theory
2627:. Cambridge University Press.
2561:Tsalenko & Shulgeifer 1974
2522:List of category theory topics
2078:{\displaystyle \delta :B\to C}
2069:
1999:
1976:{\displaystyle \alpha :A\to C}
1967:
1935:
1896:
1496:
1484:
1473:
1468:
1456:
1397:
1385:
1373:
1362:
1357:
1345:
208:
145:
1:
2729:American Mathematical Monthly
2008:{\displaystyle \beta :B\to D}
1839:{\displaystyle \varepsilon '}
1618:An epimorphism is said to be
1583:
1034:category of commutative rings
975:. Thus any homomorphism from
697:and continuous functions. If
2276:in ring theory. A morphism
2272:There is also the notion of
2261:{\displaystyle \varepsilon }
2165:{\displaystyle \varepsilon }
1749:{\displaystyle \varepsilon }
1642:{\displaystyle \varepsilon }
1516:being a monomorphism in the
190:onto or surjective functions
2688:Encyclopedia of Mathematics
2176:if there exists a morphism
1944:{\displaystyle \mu :C\to D}
1168:morphism is an epimorphism.
818:are two distinct maps from
361:β {0,1} that is constant 1.
226:is a ring epimorphism. The
2811:
2645:Category Theory in Context
2053:, there exists a morphism
1760:if in each representation
1653:if in each representation
1549:, we can define the group
1272:equivalences of categories
1263:is also an epimorphism in
234:(i.e. an epimorphism in a
29:
2723:Linderholm, Carl (1970).
2623:Borceux, Francis (1994).
2587:. John Wiley & Sons.
2484:were first introduced by
2290:full and faithful functor
1255:, then every morphism in
890:, so the restrictions of
241:is a monomorphism in the
2602:Bergman, George (2015).
1240:must be an epimorphism.
530:and group homomorphisms.
492:is surjective is due to
2274:homological epimorphism
1302:) is an epimorphism in
1274:: given an equivalence
1225:, a map that is both a
951:(note that the unit in
709:is not surjective, let
557:-linear transformations
330:characteristic function
230:of an epimorphism is a
27:Surjective homomorphism
2262:
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2222:
2190:
2166:
2143:
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2047:
2009:
1977:
1945:
1906:
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1840:
1811:
1791:
1750:
1723:
1699:
1679:
1643:
1507:
1435:natural transformation
1416:
1040:homomorphism of rings
947:generates the algebra
766:and the zero function
383:partially ordered sets
220:
179:
41:
2642:Riehl, Emily (2016).
2263:
2243:
2223:
2191:
2167:
2144:
2112:
2080:
2048:
2010:
1978:
1946:
1907:
1877:
1861:
1841:
1812:
1792:
1751:
1724:
1700:
1680:
1644:
1604:immediate epimorphism
1569:that is defined like
1508:
1417:
342:β {0,1} of the image
221:
180:
40:
2704:Sets for Mathematics
2474:The companion terms
2460:equivalence relation
2252:
2241:{\displaystyle \mu }
2232:
2200:
2189:{\displaystyle \mu }
2180:
2156:
2121:
2089:
2057:
2019:
1987:
1955:
1923:
1884:
1866:is automatically an
1859:{\displaystyle \mu }
1850:
1825:
1810:{\displaystyle \mu }
1801:
1764:
1740:
1729:is automatically an
1722:{\displaystyle \mu }
1713:
1698:{\displaystyle \mu }
1689:
1657:
1633:
1600:extremal epimorphism
1443:
1429:for every choice of
1332:
1079:or is prime, and if
998:, the inclusion map
670:continuous functions
582:module homomorphisms
286:Every morphism in a
200:
106:
2494:surjective function
2492:as shorthand for a
2323:S (thought of as a
1596:regular epimorphism
787:category of monoids
713: ∈
682:indiscrete topology
680:, giving {0,1} the
486:group homomorphisms
18:Regular epimorphism
2768:Strong epimorphism
2317:half-open interval
2294:derived categories
2258:
2238:
2218:
2186:
2162:
2139:
2107:
2075:
2043:
2005:
1973:
1951:and any morphisms
1941:
1902:
1879:
1856:
1836:
1807:
1787:
1746:
1719:
1695:
1675:
1639:
1608:strong epimorphism
1503:
1501:
1412:
1410:
1087:, the induced map
1038:finitely generated
1029:is an epimorphism.
1025:to any one of its
666:topological spaces
387:monotone functions
276:Β§ Terminology
216:
175:
75:and all morphisms
69:right-cancellative
42:
2715:978-0-521-80444-8
2615:978-3-319-11478-1
2498:Saunders Mac Lane
1612:split epimorphism
1557:) and then write
1008:ring homomorphism
992:category of rings
826:. Then for some
290:whose underlying
288:concrete category
257:universal algebra
16:(Redirected from
2802:
2744:
2719:
2707:
2696:
2678:
2659:
2638:
2619:
2598:
2586:
2564:
2558:
2552:
2546:
2488:. Bourbaki uses
2380:quotient objects
2359:in the category
2267:
2265:
2264:
2259:
2247:
2245:
2244:
2239:
2227:
2225:
2224:
2219:
2195:
2193:
2192:
2187:
2171:
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2168:
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2148:
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2114:
2113:
2108:
2084:
2082:
2081:
2076:
2052:
2050:
2049:
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2014:
2012:
2011:
2006:
1982:
1980:
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1974:
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1911:
1909:
1908:
1903:
1865:
1863:
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1702:
1701:
1696:
1684:
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1681:
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1648:
1646:
1645:
1640:
1590:Related concepts
1518:functor category
1512:
1510:
1509:
1504:
1502:
1421:
1419:
1418:
1413:
1411:
1020:commutative ring
695:Hausdorff spaces
369:binary relations
253:abstract algebra
251:Many authors in
225:
223:
222:
217:
215:
207:
194:category of sets
184:
182:
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176:
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118:
117:
98:
21:
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2722:
2716:
2699:
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2635:
2622:
2616:
2601:
2595:
2584:
2577:
2573:
2568:
2567:
2559:
2555:
2547:
2540:
2535:
2518:
2472:
2457:
2450:
2439:
2432:
2418:are said to be
2417:
2406:
2399:
2388:
2341:Euler's formula
2250:
2249:
2230:
2229:
2198:
2197:
2178:
2177:
2154:
2153:
2152:An epimorphism
2119:
2118:
2087:
2086:
2055:
2054:
2017:
2016:
1985:
1984:
1953:
1952:
1921:
1920:
1882:
1881:
1880:An epimorphism
1848:
1847:
1828:
1823:
1822:
1799:
1798:
1779:
1762:
1761:
1738:
1737:
1736:An epimorphism
1711:
1710:
1709:, the morphism
1687:
1686:
1655:
1654:
1631:
1630:
1629:An epimorphism
1592:
1584:Β§ Examples
1500:
1499:
1476:
1471:
1441:
1440:
1409:
1408:
1400:
1395:
1389:
1388:
1365:
1360:
1330:
1329:
1294:if and only if
1202:
1175:
903:
896:
873:
862:
851:
840:
822:to some monoid
817:
810:
772:
765:
746:
735:
727:Urysohn's Lemma
639:
613:
467:
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441:
426:
407:
356:
337:
284:
198:
197:
162:
149:
128:
109:
104:
103:
89:
82:
76:
46:category theory
35:
28:
23:
22:
15:
12:
11:
5:
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2797:
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2782:
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2778:
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2751:
2750:External links
2748:
2746:
2745:
2735:(2): 176β177.
2720:
2714:
2697:
2679:
2673:
2660:
2654:
2639:
2634:978-0521061193
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2565:
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2537:
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2514:
2506:epic morphisms
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2448:
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2397:
2386:
2270:
2269:
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2237:
2228:(in this case
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2185:
2172:is said to be
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1831:
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1785:
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1756:is said to be
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1649:is said to be
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1341:
1338:
1337:
1227:monic morphism
1198:
1174:
1171:
1170:
1169:
1162:directed graph
1158:
1136:
1135:
1116:
1030:
988:
931:of the monoid
909:
901:
894:
871:
860:
849:
838:
815:
808:
779:
778:
770:
763:
744:
733:
725:is closed, by
685:
657:
637:
611:
560:
531:
528:abelian groups
519:
509:
473:
465:
450:
439:
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350:) and the map
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2683:"Epimorphism"
2680:
2676:
2674:5-02-014427-4
2670:
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2655:9780486809038
2651:
2647:
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2461:
2458:. This is an
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2385:
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2358:
2354:
2350:
2346:
2345:homeomorphism
2342:
2338:
2334:
2331:) that sends
2330:
2329:complex plane
2326:
2322:
2319:[0,1) to the
2318:
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1289:
1286:, a morphism
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1067:generated by
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1027:localizations
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829:
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803:
799:
796:
795:inclusion map
792:
788:
784:
783:
782:
776:
769:
762:
758:
755:. We compose
754:
750:
743:
740:β such that
739:
732:
728:
724:
720:
717: β
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571:right modules
568:
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543:vector spaces
540:
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532:
529:
525:
524:
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517:
516:finite groups
513:
510:
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503:
499:
495:
494:Otto Schreier
491:
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397:, β€) β (
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263:simply as an
262:
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243:dual category
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2686:
2664:
2644:
2624:
2608:. Springer.
2604:
2580:
2556:
2549:Borceux 1994
2527:Monomorphism
2510:
2505:
2502:epimorphisms
2501:
2489:
2481:monomorphism
2479:
2475:
2473:
2463:
2452:
2445:
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2427:
2423:
2419:
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2332:
2310:
2305:
2301:
2297:
2285:
2281:
2277:
2273:
2271:
2173:
1918:monomorphism
1913:
1819:monomorphism
1757:
1707:monomorphism
1650:
1619:
1611:
1607:
1603:
1599:
1595:
1593:
1578:
1574:
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1199:
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1190:
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1137:
1131:
1127:
1119:
1104:
1100:
1096:
1092:
1084:
1080:
1076:
1075:) is either
1072:
1068:
1064:
1063:, the ideal
1060:
1056:
1054:prime ideals
1049:
1045:
1041:
1022:
1015:
1011:
1003:
999:
995:
984:
980:
976:
972:
968:
964:
960:
956:
955:is given by
952:
948:
944:
940:
936:
932:
924:
920:
916:
912:
908:are unequal.
905:
898:
891:
887:
883:
879:
875:
868:
864:
857:
853:
846:
842:
835:
831:
827:
823:
819:
812:
805:
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797:
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748:
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722:
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714:
710:
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702:
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673:
659:
653:
649:
645:
641:
634:
627:
623:
619:
615:
608:
606:quotient map
601:
597:
593:
589:
585:
577:
566:
562:
554:
549:
535:
533:
521:
511:
506:free product
489:
475:
469:
462:
458:
454:
447:
443:
436:
432:
428:
421:
417:
413:
409:
402:
398:
394:
390:
378:
372:
367:: sets with
364:
358:
351:
347:
343:
339:
332:
325:
321:
317:
313:
303:
285:
272:homomorphism
264:
260:
250:
245:
238:
232:monomorphism
192:(and in the
187:
95:
91:
84:
77:
72:
64:
60:
56:
49:
43:
32:Epimorphosis
2756:epimorphism
2490:epimorphism
2476:epimorphism
2470:Terminology
2321:unit circle
2300:) : D(
1916:if for any
1868:isomorphism
1731:isomorphism
1624:coequalizer
1622:if it is a
1528:coequalizer
1249:subcategory
1179:isomorphism
1115:IV 17.2.6).
1109:isomorphism
929:monoid ring
878:). Either
261:epimorphism
50:epimorphism
2784:Categories
2571:References
2420:equivalent
2367:, the map
2335:to exp(2Οi
2313:bimorphism
2196:such that
2085:such that
2015:such that
1193:such that
1173:Properties
759:with both
630:) and the
504:using the
420:) and let
296:surjective
269:surjective
259:define an
2790:Morphisms
2693:EMS Press
2667:. Nauka.
2256:ε
2236:μ
2210:μ
2207:∘
2204:ε
2184:μ
2160:ε
2137:β
2131:δ
2128:∘
2125:μ
2105:α
2099:ε
2096:∘
2093:δ
2070:→
2061:δ
2041:α
2038:∘
2035:μ
2029:ε
2026:∘
2023:β
2000:→
1991:β
1968:→
1959:α
1936:→
1927:μ
1897:→
1888:ε
1854:μ
1830:ε
1805:μ
1781:ε
1777:∘
1774:μ
1768:ε
1758:immediate
1744:ε
1717:μ
1693:μ
1673:φ
1670:∘
1667:μ
1661:ε
1637:ε
1494:−
1482:
1474:→
1466:−
1454:
1427:injective
1398:↦
1371:
1363:→
1343:
1218:split epi
1140:injective
1099:) β Frac(
751:and 1 on
502:equalizer
393: : (
209:→
146:⟹
139:∘
120:∘
2516:See also
2486:Bourbaki
2426: :
2407: :
2389: :
2371: β
2355: β
2325:subspace
1833:′
1797:, where
1784:′
1685:, where
1651:extremal
1541: :
1532:cokernel
1317: :
1278: :
1221:. In a
1185: :
1107:) is an
1044: :
923:, where
747:is 0 on
721:. Since
632:zero map
498:subgroup
453: :
427: :
292:function
282:Examples
236:category
67:that is
59: :
54:morphism
2770:at the
2758:at the
2695:, 2001
2339:) (see
2327:of the
1620:regular
1308:duality
1203:, then
1083:is not
1032:In the
990:In the
927:is the
915:, take
785:In the
692:compact
573:over a
545:over a
278:below.
2712:
2671:
2652:
2631:
2612:
2591:
2444:
2304:) β D(
1914:strong
1610:, and
1526:Every
1247:is a
1177:Every
1151:monoid
886:is in
856:), so
793:, the
512:FinGrp
500:is an
482:groups
446:} and
2585:(PDF)
2533:Notes
2440:with
2365:rings
2174:split
1817:is a
1705:is a
1553:= im(
1223:topos
1166:every
1160:If a
1149:If a
1124:dense
688:HComp
547:field
538:-Vect
468:<
389:. If
52:is a
48:, an
2710:ISBN
2669:ISBN
2650:ISBN
2629:ISBN
2610:ISBN
2589:ISBN
2478:and
2400:and
2361:Haus
2296:: D(
2117:and
1983:and
1821:and
1425:are
1306:. A
1197:= id
1155:ring
1120:Haus
1089:Frac
1036:, a
996:Ring
897:and
882:or β
867:) β
845:) β
811:and
668:and
580:and
575:ring
552:and
484:and
385:and
310:sets
265:onto
255:and
228:dual
2775:Lab
2763:Lab
2737:doi
2349:Top
2308:).
2292:on
1614:.
1521:Set
1479:Hom
1451:Hom
1368:Hom
1340:Hom
1251:of
1153:or
1113:EGA
1059:of
1010:on
983:of
967:in
959:of
904:to
830:in
791:Mon
777:β .
678:Set
674:Top
661:Top
598:Mod
596:in
563:Mod
490:Grp
477:Grp
408:in
379:Pos
373:Set
365:Rel
326:Set
324:in
305:Set
294:is
267:or
248:).
44:In
2786::
2733:77
2731:.
2727:.
2691:,
2685:,
2541:^
2466:.
2451:=
2433:β
2411:β
2393:β
2284:β
2280::
2268:).
1606:,
1602:,
1598:,
1577:β
1565:β
1545:β
1523:.
1321:β
1282:β
1234:fg
1211:β
1207::
1195:fj
1189:β
1142:.
1130:β
1048:β
1002:β
994:,
939:β
919:β
874:(β
863:(β
834:,
800:β
789:,
773::
749:fX
723:fX
719:fX
705:β
701::
690::
664::
656:).
644:β
640::
618:β
614::
592:β
588::
569::
541::
526::
523:Ab
514::
480::
461:|
442:β€
435:|
412:\
381::
357::
338::
320:β
316::
308::
99:,
94:β
90::
83:,
63:β
2773:n
2761:n
2743:.
2739::
2718:.
2677:.
2658:.
2637:.
2618:.
2597:.
2563:.
2551:.
2464:X
2456:2
2453:f
2449:1
2446:f
2442:j
2438:2
2435:Y
2431:1
2428:Y
2424:j
2416:2
2413:Y
2409:X
2405:2
2402:f
2398:1
2395:Y
2391:X
2387:1
2384:f
2373:Q
2369:Z
2357:R
2353:Q
2337:x
2333:x
2306:A
2302:B
2298:f
2286:B
2282:A
2278:f
2216:1
2213:=
2149:.
2134:=
2102:=
2073:C
2067:B
2064::
2032:=
2003:D
1997:B
1994::
1971:C
1965:A
1962::
1939:D
1933:C
1930::
1900:B
1894:A
1891::
1870:.
1771:=
1733:.
1664:=
1579:H
1575:K
1571:f
1567:K
1563:G
1559:f
1555:f
1551:K
1547:H
1543:G
1539:f
1497:)
1491:,
1488:X
1485:(
1469:)
1463:,
1460:Y
1457:(
1431:Z
1406:f
1403:g
1393:g
1386:)
1383:Z
1380:,
1377:X
1374:(
1358:)
1355:Z
1352:,
1349:Y
1346:(
1323:Y
1319:X
1315:f
1304:D
1300:f
1298:(
1296:F
1292:C
1288:f
1284:D
1280:C
1276:F
1265:D
1261:C
1257:D
1253:C
1245:D
1238:f
1213:Y
1209:X
1205:f
1200:Y
1191:X
1187:Y
1183:j
1132:R
1128:Q
1111:(
1105:Q
1103:/
1101:S
1097:P
1095:/
1093:R
1091:(
1085:S
1081:Q
1077:S
1073:P
1071:(
1069:f
1065:Q
1061:R
1057:P
1050:S
1046:R
1042:f
1023:R
1016:Z
1012:Q
1004:Q
1000:Z
987:.
985:Z
981:1
977:R
973:n
969:Z
965:n
961:Z
957:0
953:R
949:R
945:1
941:Z
937:N
933:G
925:R
921:R
917:R
913:R
906:N
902:2
899:g
895:1
892:g
888:N
884:n
880:n
876:n
872:2
869:g
865:n
861:1
858:g
854:n
852:(
850:2
847:g
843:n
841:(
839:1
836:g
832:Z
828:n
824:M
820:Z
816:2
813:g
809:1
806:g
802:Z
798:N
775:Y
771:2
768:g
764:1
761:g
757:f
753:y
745:1
742:g
738:Y
736::
734:1
731:g
715:Y
711:y
707:Y
703:X
699:f
654:X
652:(
650:f
648:/
646:Y
642:Y
638:2
635:g
628:X
626:(
624:f
622:/
620:Y
616:Y
612:1
609:g
602:R
600:-
594:Y
590:X
586:f
578:R
567:R
565:-
559:.
555:K
550:K
536:K
470:y
466:0
463:y
459:y
455:Y
451:2
448:g
444:y
440:0
437:y
433:y
429:Y
425:1
422:g
418:X
416:(
414:f
410:Y
406:0
403:y
399:Y
395:X
391:f
359:Y
355:2
352:g
348:X
346:(
344:f
340:Y
336:1
333:g
322:Y
318:X
314:f
246:C
239:C
213:Q
205:Z
173:.
168:2
164:g
160:=
155:1
151:g
142:f
134:2
130:g
126:=
123:f
115:1
111:g
96:Z
92:Y
88:2
85:g
81:1
78:g
73:Z
65:Y
61:X
57:f
34:.
20:)
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