1618:
210:
1865:
1885:
1875:
787:
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
595:
Initial and terminal objects are not required to exist in a given category. However, if they do exist, they are essentially unique. Specifically, if
904:
Many natural constructions in category theory can be formulated in terms of finding an initial or terminal object in a suitable category.
1262:
1193:
1233:
1547:
205:
of sets and relations, the empty set is the unique initial object, the unique terminal object, and hence the unique zero object.
188:) is a terminal object in this category; there are no zero objects. Similarly, the empty space is the unique initial object in
1112:
1914:
1909:
1210:
1170:
1125:
1015:(respectively, initial functor) is a generalization of the notion of final object (respectively, initial object).
400:, there are no initial or terminal objects. However, in the subcategory of fields of fixed characteristic, the
193:
1255:
1198:
475:
1888:
1828:
1537:
1878:
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1528:
1436:
1162:
298:
1837:
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1342:
726:
517:
328:
505:
of the ring of integers, is a terminal object. The empty scheme (equal to the prime spectrum of the
1868:
1824:
1429:
1248:
306:
27:
1424:
1406:
760:
722:
513:
185:
1631:
1397:
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1300:
633:
85:
46:
1513:
1352:
990:
578:, the initial and terminal objects are the anonymous zero object. This is used frequently in
408:
333:
201:
149:
20:
1325:
1320:
1001:
494:
1220:
1180:
1135:
8:
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1347:
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222:(whose objects are non-empty sets together with a distinguished element; a morphism from
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909:
857:
815:
502:
397:
286:
1832:
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1757:
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1315:
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Categorical foundations. Special topics in order, topology, algebra, and sheaf theory
1121:
971:
804:
788:
739:
629:
524:
371:
345:
1793:
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1337:
1305:
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819:
465:
181:
1705:
1271:
1202:
486:(with no objects and no morphisms), as initial object and the terminal category,
34:
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1139:
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657:
559:
436:
375:
209:
1903:
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1161:. Encyclopedia of Mathematics and Its Applications. Vol. 97. Cambridge:
1089:
1012:
743:
455:
290:
275:
19:"Zero object" redirects here. For zero object in an algebraic structure, see
1486:
1387:
800:
796:
649:
535:
may be characterised as an initial object in the category of co-cones from
490:(with a single object with a single identity morphism), as terminal object.
213:
Morphisms of pointed sets. The image also applies to algebraic zero objects
26:"Terminal element" redirects here. For the project management concept, see
1747:
1727:
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1469:
792:
614:
401:
219:
160:
38:
1779:
1717:
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be the discrete category with a single object (denoted by •), and let
1773:
1464:
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774:
506:
382:
356:
173:
632:
there is an existence theorem for initial objects. Specifically, a (
1842:
1474:
1372:
814:
Initial and terminal objects may also be characterized in terms of
274:), every singleton is a zero object. Similarly, in the category of
69:
1240:
625:
is also an initial object. The same is true for terminal objects.
418:
can be interpreted as a category: the objects are the elements of
1812:
1802:
1451:
1362:
784:
479:
349:
196:
and every one-point space is a terminal object in this category.
1807:
899:
795:
will be the free object generated by the empty set (since the
293:
is a zero object. The trivial object is also a zero object in
1689:
1229:
1111:
Adámek, Jiří; Herrlich, Horst; Strecker, George E. (1990).
613:
are two different initial objects, then there is a unique
337:
for details. This is the origin of the term "zero object".
133:
If an object is both initial and terminal, it is called a
1157:
Pedicchio, Maria
Cristina; Tholen, Walter, eds. (2004).
746:(a product is indeed the limit of the discrete diagram
640:
has an initial object if and only if there exist a set
348:
with unity and unity-preserving morphisms, the ring of
1110:
374:
with unity and unity-preserving morphisms, the rig of
16:
Special objects used in (mathematical) category theory
448:. This category has an initial object if and only if
1234:
article on examples of initial and terminal objects
982:is an initial object in the category of cones from
621:is an initial object then any object isomorphic to
1114:Abstract and Concrete Categories. The joy of cats
523:may be characterised as a terminal object in the
381:is an initial object. The zero rig, which is the
1901:
1156:
759:, in general). Dually, an initial object is a
1256:
742:, a terminal object can be thought of as an
920:can be defined as an initial object in the
900:Relation to other categorical constructions
590:
551:of chain complexes over a commutative ring
1884:
1874:
1630:
1263:
1249:
738:. Since the empty category is vacuously a
712:
458:; it has a terminal object if and only if
1194:Categories for the Working Mathematician
1187:
208:
126:, and terminal objects are also called
1902:
424:, and there is a single morphism from
385:, consisting only of a single element
1629:
1282:
1244:
482:as morphisms has the empty category,
155:is one for which every morphism into
935:. Dually, a universal morphism from
840:be the unique (constant) functor to
555:, the zero complex is a zero object.
359:consisting only of a single element
1270:
1019:
887:to •. The functor which sends • to
278:, every singleton is a zero object.
13:
118:. Initial objects are also called
108:there exists exactly one morphism
14:
1926:
1228:This article is based in part on
1029:of an initial or terminal object
684:, there is at least one morphism
1883:
1873:
1864:
1863:
1616:
1283:
176:is the unique initial object in
100:is terminal if for every object
1066:, then for any pair of objects
864:. The functor which sends • to
717:Terminal objects in a category
1201:. Vol. 5 (2nd ed.).
194:category of topological spaces
1:
1199:Graduate Texts in Mathematics
1104:
883:is a universal morphism from
585:
68:, there exists precisely one
1000:is an initial object in the
773:and can be thought of as an
476:category of small categories
7:
1558:Constructions on categories
991:representation of a functor
617:between them. Moreover, if
166:
145:is one with a zero object.
60:such that for every object
10:
1931:
1665:Higher-dimensional algebra
1163:Cambridge University Press
355:is an initial object. The
299:category of abelian groups
276:pointed topological spaces
25:
18:
1915:Objects (category theory)
1859:
1792:
1756:
1704:
1697:
1648:
1638:
1625:
1614:
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1405:
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1293:
1289:
1278:
1120:. John Wiley & Sons.
1078:, the unique composition
676:such that for any object
531:. Likewise, a colimit of
329:category of vector spaces
184:. Every one-element set (
1910:Limits (category theory)
962:is a terminal object in
943:is a terminal object in
591:Existence and uniqueness
307:category of pseudo-rings
28:work breakdown structure
1475:Cokernels and quotients
1398:Universal constructions
978:. Dually, a colimit of
958:The limit of a diagram
811:, preserves colimits).
721:may also be defined as
713:Equivalent formulations
509:) is an initial object.
1632:Higher category theory
1378:Natural transformation
214:
763:of the empty diagram
409:partially ordered set
404:is an initial object.
389:is a terminal object.
363:is a terminal object.
334:Zero object (algebra)
212:
150:strict initial object
21:zero object (algebra)
1501:Algebraic categories
1002:category of elements
891:is right adjoint to
816:universal properties
783:It follows that any
780:or categorical sum.
725:of the unique empty
636:) complete category
580:cohomology theories.
560:short exact sequence
88:notion is that of a
1670:Homotopy hypothesis
1348:Commutative diagram
1027:endomorphism monoid
868:is left adjoint to
630:complete categories
493:In the category of
318:category of modules
218:In the category of
1383:Universal property
1189:Mac Lane, Saunders
1062:has a zero object
910:universal morphism
875:A terminal object
858:universal morphism
848:An initial object
398:category of fields
370:, the category of
331:over a field. See
287:category of groups
215:
1897:
1896:
1855:
1854:
1851:
1850:
1833:monoidal category
1788:
1787:
1660:Enriched category
1612:
1611:
1608:
1607:
1585:Quotient category
1580:Opposite category
1495:
1494:
972:category of cones
805:forgetful functor
789:concrete category
740:discrete category
525:category of cones
346:category of rings
320:over a ring, and
246:being a function
1922:
1887:
1886:
1877:
1876:
1867:
1866:
1702:
1701:
1680:Simplex category
1655:Categorification
1646:
1645:
1627:
1626:
1620:
1590:Product category
1575:Kleisli category
1570:Functor category
1415:Terminal objects
1403:
1402:
1338:Adjoint functors
1291:
1290:
1280:
1279:
1265:
1258:
1251:
1242:
1241:
1224:
1184:
1153:
1151:
1150:
1144:
1138:. Archived from
1119:
1099:
1095:
1087:
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1073:
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1061:
1054:
1032:
1020:Other properties
1007:
995:
985:
981:
977:
969:
961:
954:
942:
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934:
919:
915:
894:
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886:
882:
878:
867:
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855:
851:
839:
820:adjoint functors
772:
758:
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720:
708:
698:
683:
679:
675:
671:
655:
643:
639:
624:
620:
612:
603:
577:
542:In the category
466:greatest element
463:
453:
447:
435:
429:
423:
417:
388:
362:
273:
259:
245:
233:
199:In the category
182:category of sets
158:
154:
143:pointed category
117:
107:
103:
99:
94:terminal element
80:
67:
63:
59:
55:
51:
1930:
1929:
1925:
1924:
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1921:
1920:
1919:
1900:
1899:
1898:
1893:
1847:
1817:
1784:
1761:
1752:
1709:
1693:
1644:
1634:
1621:
1604:
1553:
1491:
1460:Initial objects
1446:
1392:
1285:
1274:
1272:Category theory
1269:
1213:
1203:Springer-Verlag
1173:
1148:
1146:
1142:
1128:
1117:
1107:
1097:
1093:
1079:
1075:
1071:
1067:
1063:
1059:
1052:
1034:
1030:
1022:
1005:
993:
983:
979:
975:
963:
959:
944:
940:
936:
924:
917:
913:
912:from an object
902:
892:
888:
884:
880:
876:
865:
861:
853:
849:
827:
764:
756:
747:
729:
718:
715:
700:
693:
685:
681:
677:
673:
669:
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653:
641:
637:
622:
618:
611:
605:
602:
596:
593:
588:
563:
549:
459:
449:
439:
431:
425:
419:
411:
386:
376:natural numbers
360:
261:
247:
235:
223:
169:
156:
152:
109:
105:
101:
97:
90:terminal object
72:
65:
61:
57:
53:
49:
35:category theory
31:
24:
17:
12:
11:
5:
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1685:String diagram
1682:
1677:
1675:Model category
1672:
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1623:
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1597:
1595:Comma category
1592:
1587:
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1577:
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1567:
1561:
1559:
1555:
1554:
1552:
1551:
1541:
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1529:Abelian groups
1526:
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1503:
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1276:
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1268:
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1239:
1238:
1225:
1211:
1185:
1171:
1154:
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1106:
1103:
1102:
1101:
1058:If a category
1056:
1048:
1021:
1018:
1017:
1016:
1011:The notion of
1009:
987:
956:
922:comma category
901:
898:
897:
896:
873:
752:
714:
711:
689:
672:of objects of
665:
658:indexed family
647:
609:
600:
592:
589:
587:
584:
583:
582:
556:
545:
540:
510:
503:prime spectrum
491:
469:
437:if and only if
405:
390:
364:
338:
279:
207:
206:
197:
168:
165:
43:initial object
37:, a branch of
15:
9:
6:
4:
3:
2:
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1911:
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1749:
1746:
1744:
1741:
1739:
1736:
1734:
1733:Tetracategory
1731:
1729:
1726:
1723:
1722:pseudofunctor
1719:
1716:
1715:
1713:
1711:
1703:
1700:
1696:
1691:
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1686:
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1583:
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1578:
1576:
1573:
1571:
1568:
1566:
1565:Free category
1563:
1562:
1560:
1556:
1549:
1548:Vector spaces
1545:
1542:
1539:
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1471:
1468:
1466:
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1458:
1457:
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1453:
1449:
1443:
1442:Inverse limit
1440:
1438:
1435:
1431:
1428:
1427:
1426:
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1418:
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1413:
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1401:
1399:
1395:
1389:
1386:
1384:
1381:
1379:
1376:
1374:
1371:
1369:
1368:Kan extension
1366:
1364:
1361:
1359:
1356:
1354:
1351:
1349:
1346:
1344:
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1309:
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1303:
1302:
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1252:
1247:
1246:
1243:
1237:
1235:
1231:
1226:
1222:
1218:
1214:
1212:0-387-98403-8
1208:
1204:
1200:
1196:
1195:
1190:
1186:
1182:
1178:
1174:
1172:0-521-83414-7
1168:
1164:
1160:
1155:
1145:on 2015-04-21
1141:
1137:
1133:
1129:
1127:0-471-60922-6
1123:
1116:
1115:
1109:
1108:
1091:
1090:zero morphism
1086:
1082:
1057:
1051:
1046:
1042:
1038:
1028:
1024:
1023:
1014:
1013:final functor
1010:
1003:
999:
992:
988:
973:
967:
957:
952:
948:
932:
928:
923:
916:to a functor
911:
907:
906:
905:
874:
871:
859:
847:
846:
845:
843:
838:
834:
830:
825:
821:
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812:
810:
806:
802:
798:
794:
790:
786:
781:
779:
776:
771:
767:
762:
755:
751:
745:
744:empty product
741:
736:
732:
728:
724:
710:
707:
703:
697:
692:
688:
668:
664:
659:
651:
645:
635:
634:locally small
631:
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616:
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581:
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561:
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541:
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511:
508:
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500:
496:
492:
489:
485:
481:
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470:
467:
462:
457:
456:least element
452:
446:
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428:
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406:
403:
399:
395:
391:
384:
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296:
292:
291:trivial group
288:
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95:
92:(also called
91:
87:
82:
79:
75:
71:
52:is an object
48:
44:
40:
36:
29:
22:
1813:
1794:Categorified
1698:n-categories
1649:Key concepts
1487:Direct limit
1470:Coequalizers
1459:
1414:
1388:Yoneda lemma
1294:Key concepts
1284:Key concepts
1227:
1192:
1158:
1147:. Retrieved
1140:the original
1113:
1084:
1080:
1049:
1044:
1040:
1036:
1033:is trivial:
997:
965:
950:
946:
930:
926:
903:
869:
841:
836:
832:
828:
823:
813:
808:
801:left adjoint
797:free functor
793:free objects
782:
769:
765:
753:
749:
734:
730:
716:
705:
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1762:-categories
1738:Kan complex
1728:Tricategory
1710:-categories
1600:Subcategory
1358:Exponential
1326:Preadditive
1321:Pre-abelian
615:isomorphism
402:prime field
161:isomorphism
139:null object
135:zero object
39:mathematics
1904:Categories
1780:3-category
1770:2-category
1743:∞-groupoid
1718:Bicategory
1465:Coproducts
1425:Equalizers
1331:Bicategory
1230:PlanetMath
1221:0906.18001
1181:1034.18001
1149:2008-01-15
1136:0695.18001
1105:References
860:from • to
586:Properties
120:coterminal
1829:Symmetric
1774:2-functor
1514:Relations
1437:Pullbacks
778:coproduct
699:for some
652:) and an
507:zero ring
383:zero ring
357:zero ring
186:singleton
174:empty set
124:universal
1889:Glossary
1869:Category
1843:n-monoid
1796:concepts
1452:Colimits
1420:Products
1373:Morphism
1316:Concrete
1311:Additive
1301:Category
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480:functors
350:integers
251: :
167:Examples
70:morphism
47:category
1879:Outline
1838:n-group
1803:2-group
1758:Strict
1748:∞-topos
1544:Modules
1482:Pushout
1430:Kernels
1363:Functor
1306:Abelian
844:. Then
803:to the
785:functor
761:colimit
727:diagram
518:diagram
501:), the
497:, Spec(
495:schemes
1825:Traced
1808:2-ring
1538:Fields
1524:Groups
1519:Magmas
1407:Limits
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970:, the
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289:, any
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159:is an
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1706:Weak
1690:Topos
1534:Rings
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1092:from
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856:is a
791:with
775:empty
558:In a
516:of a
514:limit
478:with
394:Field
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260:with
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128:final
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1509:Sets
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1070:and
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372:rigs
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86:dual
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1353:End
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