1629:
221:
1876:
1896:
1886:
798:
which preserves limits will take terminal objects to terminal objects, and any functor which preserves colimits will take initial objects to initial objects. For example, the initial object in any
1273:
606:
Initial and terminal objects are not required to exist in a given category. However, if they do exist, they are essentially unique. Specifically, if
1204:
915:
Many natural constructions in category theory can be formulated in terms of finding an initial or terminal object in a suitable category.
1925:
1920:
1221:
1181:
1136:
1244:
216:
of sets and relations, the empty set is the unique initial object, the unique terminal object, and hence the unique zero object.
199:) is a terminal object in this category; there are no zero objects. Similarly, the empty space is the unique initial object in
204:
1266:
1209:
1123:
486:
1899:
1839:
1889:
1675:
1539:
1447:
1173:
309:
1848:
1492:
1430:
1353:
1026:(respectively, initial functor) is a generalization of the notion of final object (respectively, initial object).
737:
528:
339:
1879:
1835:
1440:
1259:
411:, there are no initial or terminal objects. However, in the subcategory of fields of fixed characteristic, the
317:
38:
1435:
1417:
771:
733:
524:
196:
1642:
1408:
1388:
1311:
644:
96:
57:
1524:
1363:
1001:
419:
344:
212:
160:
31:
1336:
1331:
1012:
505:
516:
of the ring of integers, is a terminal object. The empty scheme (equal to the prime spectrum of the
1680:
1628:
1558:
1554:
1358:
1037:
328:
1534:
1529:
1511:
1393:
1368:
920:
868:
826:
513:
408:
297:
1843:
1780:
1768:
1670:
1595:
1590:
1548:
1544:
1326:
1321:
1217:
1199:
1177:
1132:
982:
815:
799:
750:
640:
535:
382:
356:
220:
589:, the initial and terminal objects are the anonymous zero object. This is used frequently in
1804:
1690:
1665:
1600:
1585:
1580:
1519:
1348:
1316:
1227:
1187:
1142:
830:
476:
192:
1716:
1282:
1231:
1213:
1191:
1146:
45:
1753:
233:(whose objects are non-empty sets together with a distinguished element; a morphism from
1748:
1732:
1695:
1685:
1605:
932:
668:
570:
447:
386:
1914:
1743:
1575:
1452:
1378:
1170:
Categorical foundations. Special topics in order, topology, algebra, and sheaf theory
1100:
1023:
754:
466:
301:
286:
1497:
1398:
811:
807:
660:
1758:
497:(with no objects and no morphisms), as initial object and the terminal category,
1738:
1610:
1480:
803:
625:
412:
230:
171:
49:
1150:
17:
1790:
1728:
1341:
590:
1172:. Encyclopedia of Mathematics and Its Applications. Vol. 97. Cambridge:
30:"Zero object" redirects here. For zero object in an algebraic structure, see
1784:
1475:
788:
785:
517:
393:
367:
184:
546:
may be characterised as an initial object in the category of co-cones from
501:(with a single object with a single identity morphism), as terminal object.
224:
Morphisms of pointed sets. The image also applies to algebraic zero objects
37:"Terminal element" redirects here. For the project management concept, see
1853:
1485:
1383:
80:
1823:
1813:
1462:
1373:
795:
490:
360:
837:
be the discrete category with a single object (denoted by β’), and let
1818:
643:
there is an existence theorem for initial objects. Specifically, a (
825:
Initial and terminal objects may also be characterized in terms of
285:), every singleton is a zero object. Similarly, in the category of
1700:
1251:
636:
is also an initial object. The same is true for terminal objects.
429:
can be interpreted as a category: the objects are the elements of
219:
207:
and every one-point space is a terminal object in this category.
1640:
1293:
1255:
806:
will be the free object generated by the empty set (since the
304:
is a zero object. The trivial object is also a zero object in
1240:
1122:
AdΓ‘mek, JiΕΓ; Herrlich, Horst; Strecker, George E. (1990).
624:
are two different initial objects, then there is a unique
348:
for details. This is the origin of the term "zero object".
144:
If an object is both initial and terminal, it is called a
1168:
Pedicchio, Maria
Cristina; Tholen, Walter, eds. (2004).
757:(a product is indeed the limit of the discrete diagram
651:
has an initial object if and only if there exist a set
359:
with unity and unity-preserving morphisms, the ring of
385:
with unity and unity-preserving morphisms, the rig of
27:
Special objects used in (mathematical) category theory
459:. This category has an initial object if and only if
1803:
1767:
1715:
1708:
1659:
1568:
1510:
1461:
1416:
1407:
1304:
1245:
article on examples of initial and terminal objects
993:is an initial object in the category of cones from
632:is an initial object then any object isomorphic to
1125:Abstract and Concrete Categories. The joy of cats
534:may be characterised as a terminal object in the
392:is an initial object. The zero rig, which is the
770:, in general). Dually, an initial object is a
1267:
8:
753:, a terminal object can be thought of as an
931:can be defined as an initial object in the
911:Relation to other categorical constructions
562:of chain complexes over a commutative ring
1895:
1885:
1712:
1656:
1637:
1413:
1301:
1290:
1274:
1260:
1252:
749:. Since the empty category is vacuously a
469:; it has a terminal object if and only if
1205:Categories for the Working Mathematician
137:, and terminal objects are also called
435:, and there is a single morphism from
396:, consisting only of a single element
493:as morphisms has the empty category,
166:is one for which every morphism into
7:
946:. Dually, a universal morphism from
851:be the unique (constant) functor to
566:, the zero complex is a zero object.
370:consisting only of a single element
898:to β’. The functor which sends β’ to
289:, every singleton is a zero object.
129:. Initial objects are also called
119:there exists exactly one morphism
25:
1239:This article is based in part on
1040:of an initial or terminal object
695:, there is at least one morphism
1894:
1884:
1875:
1874:
1627:
187:is the unique initial object in
111:is terminal if for every object
1077:, then for any pair of objects
875:. The functor which sends β’ to
728:Terminal objects in a category
1212:. Vol. 5 (2nd ed.).
205:category of topological spaces
1:
1210:Graduate Texts in Mathematics
894:is a universal morphism from
79:, there exists precisely one
1011:is an initial object in the
784:and can be thought of as an
487:category of small categories
1569:Constructions on categories
1002:representation of a functor
628:between them. Moreover, if
156:is one with a zero object.
71:such that for every object
1942:
1676:Higher-dimensional algebra
1174:Cambridge University Press
366:is an initial object. The
310:category of abelian groups
287:pointed topological spaces
36:
29:
1926:Objects (category theory)
1870:
1649:
1636:
1625:
1300:
1289:
1131:. John Wiley & Sons.
1089:, the unique composition
687:such that for any object
542:. Likewise, a colimit of
340:category of vector spaces
195:. Every one-element set (
1921:Limits (category theory)
973:is a terminal object in
954:is a terminal object in
602:Existence and uniqueness
318:category of pseudo-rings
39:work breakdown structure
1486:Cokernels and quotients
1409:Universal constructions
989:. Dually, a colimit of
969:The limit of a diagram
822:, preserves colimits).
732:may also be defined as
724:Equivalent formulations
520:) is an initial object.
1643:Higher category theory
1389:Natural transformation
225:
774:of the empty diagram
420:partially ordered set
415:is an initial object.
400:is a terminal object.
374:is a terminal object.
345:Zero object (algebra)
223:
161:strict initial object
32:zero object (algebra)
1512:Algebraic categories
1013:category of elements
902:is right adjoint to
827:universal properties
794:It follows that any
791:or categorical sum.
736:of the unique empty
647:) complete category
591:cohomology theories.
571:short exact sequence
99:notion is that of a
1681:Homotopy hypothesis
1359:Commutative diagram
1038:endomorphism monoid
879:is left adjoint to
641:complete categories
504:In the category of
329:category of modules
229:In the category of
1394:Universal property
1200:Mac Lane, Saunders
1073:has a zero object
921:universal morphism
886:A terminal object
869:universal morphism
859:An initial object
409:category of fields
381:, the category of
342:over a field. See
298:category of groups
226:
1908:
1907:
1866:
1865:
1862:
1861:
1844:monoidal category
1799:
1798:
1671:Enriched category
1623:
1622:
1619:
1618:
1596:Quotient category
1591:Opposite category
1506:
1505:
983:category of cones
816:forgetful functor
800:concrete category
751:discrete category
536:category of cones
357:category of rings
331:over a ring, and
257:being a function
16:(Redirected from
1933:
1898:
1897:
1888:
1887:
1878:
1877:
1713:
1691:Simplex category
1666:Categorification
1657:
1638:
1631:
1601:Product category
1586:Kleisli category
1581:Functor category
1426:Terminal objects
1414:
1349:Adjoint functors
1302:
1291:
1276:
1269:
1262:
1253:
1235:
1195:
1164:
1162:
1161:
1155:
1149:. Archived from
1130:
1110:
1106:
1098:
1088:
1084:
1080:
1076:
1072:
1065:
1043:
1031:Other properties
1018:
1006:
996:
992:
988:
980:
972:
965:
953:
949:
945:
930:
926:
905:
901:
897:
893:
889:
878:
874:
866:
862:
850:
831:adjoint functors
783:
769:
748:
731:
719:
709:
694:
690:
686:
682:
666:
654:
650:
635:
631:
623:
614:
588:
553:In the category
477:greatest element
474:
464:
458:
446:
440:
434:
428:
399:
373:
284:
270:
256:
244:
210:In the category
193:category of sets
169:
165:
154:pointed category
128:
118:
114:
110:
105:terminal element
91:
78:
74:
70:
66:
62:
21:
1941:
1940:
1936:
1935:
1934:
1932:
1931:
1930:
1911:
1910:
1909:
1904:
1858:
1828:
1795:
1772:
1763:
1720:
1704:
1655:
1645:
1632:
1615:
1564:
1502:
1471:Initial objects
1457:
1403:
1296:
1285:
1283:Category theory
1280:
1224:
1214:Springer-Verlag
1198:
1184:
1167:
1159:
1157:
1153:
1139:
1128:
1121:
1118:
1108:
1104:
1090:
1086:
1082:
1078:
1074:
1070:
1063:
1045:
1041:
1033:
1016:
1004:
994:
990:
986:
974:
970:
955:
951:
947:
935:
928:
924:
923:from an object
913:
903:
899:
895:
891:
887:
876:
872:
864:
860:
838:
775:
767:
758:
740:
729:
726:
711:
704:
696:
692:
688:
684:
680:
671:
664:
652:
648:
633:
629:
622:
616:
613:
607:
604:
599:
574:
560:
470:
460:
450:
442:
436:
430:
422:
397:
387:natural numbers
371:
272:
258:
246:
234:
180:
167:
163:
120:
116:
112:
108:
101:terminal object
83:
76:
72:
68:
64:
60:
46:category theory
42:
35:
28:
23:
22:
18:Terminal object
15:
12:
11:
5:
1939:
1937:
1929:
1928:
1923:
1913:
1912:
1906:
1905:
1903:
1902:
1892:
1882:
1871:
1868:
1867:
1864:
1863:
1860:
1859:
1857:
1856:
1851:
1846:
1832:
1826:
1821:
1816:
1810:
1808:
1801:
1800:
1797:
1796:
1794:
1793:
1788:
1777:
1775:
1770:
1765:
1764:
1762:
1761:
1756:
1751:
1746:
1741:
1736:
1725:
1723:
1718:
1710:
1706:
1705:
1703:
1698:
1696:String diagram
1693:
1688:
1686:Model category
1683:
1678:
1673:
1668:
1663:
1661:
1654:
1653:
1650:
1647:
1646:
1641:
1634:
1633:
1626:
1624:
1621:
1620:
1617:
1616:
1614:
1613:
1608:
1606:Comma category
1603:
1598:
1593:
1588:
1583:
1578:
1572:
1570:
1566:
1565:
1563:
1562:
1552:
1542:
1540:Abelian groups
1537:
1532:
1527:
1522:
1516:
1514:
1508:
1507:
1504:
1503:
1501:
1500:
1495:
1490:
1489:
1488:
1478:
1473:
1467:
1465:
1459:
1458:
1456:
1455:
1450:
1445:
1444:
1443:
1433:
1428:
1422:
1420:
1411:
1405:
1404:
1402:
1401:
1396:
1391:
1386:
1381:
1376:
1371:
1366:
1361:
1356:
1351:
1346:
1345:
1344:
1339:
1334:
1329:
1324:
1319:
1308:
1306:
1298:
1297:
1294:
1287:
1286:
1281:
1279:
1278:
1271:
1264:
1256:
1250:
1249:
1236:
1222:
1196:
1182:
1165:
1137:
1117:
1114:
1113:
1112:
1069:If a category
1067:
1059:
1032:
1029:
1028:
1027:
1022:The notion of
1020:
998:
967:
933:comma category
912:
909:
908:
907:
884:
763:
725:
722:
700:
683:of objects of
676:
669:indexed family
658:
620:
611:
603:
600:
598:
595:
594:
593:
567:
556:
551:
521:
514:prime spectrum
502:
480:
448:if and only if
416:
401:
375:
349:
290:
218:
217:
208:
179:
176:
54:initial object
48:, a branch of
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1938:
1927:
1924:
1922:
1919:
1918:
1916:
1901:
1893:
1891:
1883:
1881:
1873:
1872:
1869:
1855:
1852:
1850:
1847:
1845:
1841:
1837:
1833:
1831:
1829:
1822:
1820:
1817:
1815:
1812:
1811:
1809:
1806:
1802:
1792:
1789:
1786:
1782:
1779:
1778:
1776:
1774:
1766:
1760:
1757:
1755:
1752:
1750:
1747:
1745:
1744:Tetracategory
1742:
1740:
1737:
1734:
1733:pseudofunctor
1730:
1727:
1726:
1724:
1722:
1714:
1711:
1707:
1702:
1699:
1697:
1694:
1692:
1689:
1687:
1684:
1682:
1679:
1677:
1674:
1672:
1669:
1667:
1664:
1662:
1658:
1652:
1651:
1648:
1644:
1639:
1635:
1630:
1612:
1609:
1607:
1604:
1602:
1599:
1597:
1594:
1592:
1589:
1587:
1584:
1582:
1579:
1577:
1576:Free category
1574:
1573:
1571:
1567:
1560:
1559:Vector spaces
1556:
1553:
1550:
1546:
1543:
1541:
1538:
1536:
1533:
1531:
1528:
1526:
1523:
1521:
1518:
1517:
1515:
1513:
1509:
1499:
1496:
1494:
1491:
1487:
1484:
1483:
1482:
1479:
1477:
1474:
1472:
1469:
1468:
1466:
1464:
1460:
1454:
1453:Inverse limit
1451:
1449:
1446:
1442:
1439:
1438:
1437:
1434:
1432:
1429:
1427:
1424:
1423:
1421:
1419:
1415:
1412:
1410:
1406:
1400:
1397:
1395:
1392:
1390:
1387:
1385:
1382:
1380:
1379:Kan extension
1377:
1375:
1372:
1370:
1367:
1365:
1362:
1360:
1357:
1355:
1352:
1350:
1347:
1343:
1340:
1338:
1335:
1333:
1330:
1328:
1325:
1323:
1320:
1318:
1315:
1314:
1313:
1310:
1309:
1307:
1303:
1299:
1292:
1288:
1284:
1277:
1272:
1270:
1265:
1263:
1258:
1257:
1254:
1248:
1246:
1242:
1237:
1233:
1229:
1225:
1223:0-387-98403-8
1219:
1215:
1211:
1207:
1206:
1201:
1197:
1193:
1189:
1185:
1183:0-521-83414-7
1179:
1175:
1171:
1166:
1156:on 2015-04-21
1152:
1148:
1144:
1140:
1138:0-471-60922-6
1134:
1127:
1126:
1120:
1119:
1115:
1102:
1101:zero morphism
1097:
1093:
1068:
1062:
1057:
1053:
1049:
1039:
1035:
1034:
1030:
1025:
1024:final functor
1021:
1014:
1010:
1003:
999:
984:
978:
968:
963:
959:
943:
939:
934:
927:to a functor
922:
918:
917:
916:
910:
885:
882:
870:
858:
857:
856:
854:
849:
845:
841:
836:
832:
828:
823:
821:
817:
813:
809:
805:
801:
797:
792:
790:
787:
782:
778:
773:
766:
762:
756:
755:empty product
752:
747:
743:
739:
735:
723:
721:
718:
714:
708:
703:
699:
679:
675:
670:
662:
656:
646:
645:locally small
642:
637:
627:
619:
610:
601:
596:
592:
586:
582:
578:
572:
568:
565:
561:
559:
552:
549:
545:
541:
537:
533:
530:
526:
522:
519:
515:
511:
507:
503:
500:
496:
492:
488:
484:
481:
478:
473:
468:
467:least element
463:
457:
453:
449:
445:
439:
433:
426:
421:
417:
414:
410:
406:
402:
395:
391:
388:
384:
380:
376:
369:
365:
362:
358:
354:
350:
347:
346:
341:
337:
335:
330:
326:
324:
319:
315:
311:
307:
303:
302:trivial group
299:
295:
291:
288:
283:
279:
275:
269:
265:
261:
254:
250:
242:
238:
232:
228:
227:
222:
215:
214:
209:
206:
202:
198:
194:
190:
186:
182:
181:
177:
175:
173:
162:
157:
155:
151:
147:
142:
140:
136:
132:
127:
123:
106:
103:(also called
102:
98:
93:
90:
86:
82:
63:is an object
59:
55:
51:
47:
40:
33:
19:
1824:
1805:Categorified
1709:n-categories
1660:Key concepts
1498:Direct limit
1481:Coequalizers
1470:
1425:
1399:Yoneda lemma
1305:Key concepts
1295:Key concepts
1238:
1203:
1169:
1158:. Retrieved
1151:the original
1124:
1095:
1091:
1060:
1055:
1051:
1047:
1044:is trivial:
1008:
976:
961:
957:
941:
937:
914:
880:
852:
847:
843:
839:
834:
824:
819:
812:left adjoint
808:free functor
804:free objects
793:
780:
776:
764:
760:
745:
741:
727:
716:
712:
706:
701:
697:
677:
673:
661:proper class
638:
617:
608:
605:
584:
580:
576:
573:of the form
563:
557:
554:
547:
543:
539:
531:
509:
498:
494:
482:
471:
461:
455:
451:
443:
437:
431:
424:
404:
389:
378:
363:
352:
343:
333:
332:
322:
321:
313:
305:
293:
281:
277:
273:
267:
263:
259:
252:
248:
240:
236:
231:pointed sets
211:
200:
188:
158:
153:
149:
145:
143:
138:
134:
130:
125:
121:
104:
100:
94:
88:
84:
53:
43:
1773:-categories
1749:Kan complex
1739:Tricategory
1721:-categories
1611:Subcategory
1369:Exponential
1337:Preadditive
1332:Pre-abelian
626:isomorphism
413:prime field
172:isomorphism
150:null object
146:zero object
50:mathematics
1915:Categories
1791:3-category
1781:2-category
1754:β-groupoid
1729:Bicategory
1476:Coproducts
1436:Equalizers
1342:Bicategory
1241:PlanetMath
1232:0906.18001
1192:1034.18001
1160:2008-01-15
1147:0695.18001
1116:References
871:from β’ to
597:Properties
131:coterminal
1840:Symmetric
1785:2-functor
1525:Relations
1448:Pullbacks
789:coproduct
710:for some
663:) and an
518:zero ring
394:zero ring
368:zero ring
197:singleton
185:empty set
135:universal
1900:Glossary
1880:Category
1854:n-monoid
1807:concepts
1463:Colimits
1431:Products
1384:Morphism
1327:Concrete
1322:Additive
1312:Category
1202:(1998).
1058:) = { id
1050:) = Hom(
842: :
810:, being
491:functors
361:integers
262: :
178:Examples
81:morphism
58:category
1890:Outline
1849:n-group
1814:2-group
1769:Strict
1759:β-topos
1555:Modules
1493:Pushout
1441:Kernels
1374:Functor
1317:Abelian
855:. Then
814:to the
796:functor
772:colimit
738:diagram
529:diagram
512:), the
508:, Spec(
506:schemes
1836:Traced
1819:2-ring
1549:Fields
1535:Groups
1530:Magmas
1418:Limits
1230:
1220:
1190:
1180:
1145:
1135:
1094:β 0 β
981:, the
833:. Let
734:limits
485:, the
475:has a
465:has a
407:, the
355:, the
338:, the
327:, the
308:, the
300:, any
296:, the
203:, the
191:, the
170:is an
1830:-ring
1717:Weak
1701:Topos
1545:Rings
1154:(PDF)
1129:(PDF)
1103:from
1099:is a
975:Cone(
867:is a
802:with
786:empty
569:In a
527:of a
525:limit
489:with
405:Field
398:0 = 1
372:0 = 1
336:-Vect
271:with
152:. A
139:final
56:of a
52:, an
1520:Sets
1218:ISBN
1178:ISBN
1133:ISBN
1081:and
1046:End(
1036:The
829:and
639:For
615:and
575:0 β
427:,ββ€)
418:Any
383:rigs
353:Ring
325:-Mod
316:the
280:) =
183:The
97:dual
95:The
1364:End
1354:CCC
1243:'s
1228:Zbl
1188:Zbl
1143:Zbl
1107:to
1085:in
1015:of
1009:Set
1007:to
985:to
950:to
890:in
863:in
820:Set
818:to
691:of
657:not
587:β 0
538:to
483:Cat
441:to
403:In
379:Rig
377:In
351:In
314:Rng
294:Grp
292:In
245:to
213:Rel
201:Top
189:Set
148:or
133:or
115:in
107:):
75:in
67:in
44:In
1917::
1842:)
1838:)(
1226:.
1216:.
1208:.
1186:.
1176:.
1141:.
1054:,
1000:A
960:β
940:β
919:A
846:β
779:β
744:β
720:.
715:β
705:β
659:a
583:β
579:β
555:Ch
523:A
454:β€
320:,
312:,
306:Ab
266:β
251:,
239:,
174:.
159:A
141:.
124:β
92:.
87:β
1834:(
1827:n
1825:E
1787:)
1783:(
1771:n
1735:)
1731:(
1719:n
1561:)
1557:(
1551:)
1547:(
1275:e
1268:t
1261:v
1247:.
1234:.
1194:.
1163:.
1111:.
1109:Y
1105:X
1096:Y
1092:X
1087:C
1083:Y
1079:X
1075:0
1071:C
1066:.
1064:}
1061:I
1056:I
1052:I
1048:I
1042:I
1019:.
1017:F
1005:F
997:.
995:F
991:F
987:F
979:)
977:F
971:F
966:.
964:)
962:X
958:U
956:(
952:X
948:U
944:)
942:U
938:X
936:(
929:U
925:X
906:.
904:U
900:T
896:U
892:C
888:T
883:.
881:U
877:I
873:U
865:C
861:I
853:1
848:1
844:C
840:U
835:1
781:C
777:0
768:}
765:i
761:X
759:{
746:C
742:0
730:C
717:I
713:i
707:X
702:i
698:K
693:C
689:X
685:C
681:)
678:i
674:K
672:(
667:-
665:I
655:(
653:I
649:C
634:I
630:I
621:2
618:I
612:1
609:I
585:c
581:b
577:a
564:R
558:R
550:.
548:F
544:F
540:F
532:F
510:Z
499:1
495:0
479:.
472:P
462:P
456:y
452:x
444:y
438:x
432:P
425:P
423:(
390:N
364:Z
334:K
323:R
282:b
278:a
276:(
274:f
268:B
264:A
260:f
255:)
253:b
249:B
247:(
243:)
241:a
237:A
235:(
168:I
164:I
126:T
122:X
117:C
113:X
109:T
89:X
85:I
77:C
73:X
69:C
65:I
61:C
41:.
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
