42:
2034:
as shorthand for an injective function. Early category theorists believed that the correct generalization of injectivity to the context of categories was the cancellation property given above. While this is not exactly true for monic maps, it is very close, so this has caused little trouble, unlike
710:
It is not true in general, however, that all monomorphisms must be injective in other categories; that is, there are settings in which the morphisms are functions between sets, but one can have a function that is not injective and yet is a monomorphism in the categorical sense. For example, in the
259:
516:
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is injective is a monomorphism; in other words, if morphisms are actually functions between sets, then any morphism which is a one-to-one function will necessarily be a monomorphism in the categorical sense. In the
255:
1792:
1022:
1856:
1543:
378:
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1967:
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287:: the intersection of anything with itself is itself. Monomorphisms generalize this property to arbitrary categories. A morphism is a monomorphism if it is idempotent with respect to
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1987:
1935:
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764:. This is not an injective map, as for example every integer is mapped to 0. Nevertheless, it is a monomorphism in this category. This follows from the implication
2007:
1911:
1502:
1399:
2272:
1619:
2137:
2156:
2217:
177:
1759:
272:(also called "one-to-one functions"); in some categories the notions coincide, but monomorphisms are more general, as in the
945:
2176:
1331:
35:
31:
1829:
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344:
1408:
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morphisms. The converse also holds in most naturally occurring categories of algebras because of the existence of a
511:{\displaystyle f\circ g_{1}=f\circ g_{2}\Rightarrow l\circ f\circ g_{1}=l\circ f\circ g_{2}\Rightarrow g_{1}=g_{2}.}
288:
1511:
577:
523:
318:
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2047:, which are monomorphisms in the categorical sense of the word. This distinction never came into general use.
1940:
1861:
1375:
683:
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between them there are monomorphisms that are not injective: consider, for example, the quotient map
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121:
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295:
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2043:, which were maps in a concrete category whose underlying maps of sets were injective, and
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on one generator. In particular, it is true in the categories of all groups, of all
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However, a monomorphism need not be left-invertible. For example, in the category
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1326:, every mono is an equalizer, and any map that is both monic and
30:
This article is about the mathematical term. For other uses, see
2249:
2237:
2149:
Handbook of
Categorical Algebra. Volume 1: Basic Category Theory
691:
the converse also holds, so the monomorphisms are exactly the
250:{\displaystyle f\circ g_{1}=f\circ g_{2}\implies g_{1}=g_{2}.}
2195:. BRICS, Computer Science Department, University of Aarhus.
2129:
An
Invitation to General Algebra and Universal Constructions
752:
the integers (also considered a group under addition), and
1787:{\displaystyle \beta \circ \varepsilon =\mu \circ \alpha }
2039:
attempted to make a distinction between what he called
1017:{\displaystyle 0\leq {\frac {h(x)}{h(x)+1}}=h(y)<1}
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Left-invertible morphisms are necessarily monic: if
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840:. Without loss of generality, we may assume that
572:has a left inverse in the category if and only if
510:
372:
268:Monomorphisms are a categorical generalization of
249:
93:
1851:{\displaystyle \delta \circ \varepsilon =\alpha }
373:{\displaystyle l\circ f=\operatorname {id} _{X}}
1430:{\displaystyle \mu =\varphi \circ \varepsilon }
1088:To go from that implication to the fact that
8:
1538:{\displaystyle \mu =\mu '\circ \varepsilon }
220:
216:
2208:Tsalenko, M.S.; Shulgeifer, E.G. (1974).
1994:
1974:
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1962:{\displaystyle \varepsilon \circ \mu =1}
1883:{\displaystyle \mu \circ \delta =\beta }
874:is a divisible group, there exists some
601:is monic if and only if the induced map
302:, that is, a monomorphism in a category
257:
2097:
2087:
521:A left-invertible morphism is called a
1255:). From the implication just proved,
7:
2056:, although this has other uses too.
262:pullback of monomorphism with itself
75:is often denoted with the notation
1989:is called a left-sided inverse for
1685:{\displaystyle \varepsilon :A\to B}
1378:of some pair of parallel morphisms.
1346:There are also useful concepts of
94:{\displaystyle X\hookrightarrow Y}
25:
2050:Another name for monomorphism is
748:is the rationals under addition,
2273:Algebraic properties of elements
1590:is an epimorphism, the morphism
1457:is an epimorphism, the morphism
1313:is a monomorphism, as claimed.
1092:is a monomorphism, assume that
779:, which we will now prove. If
104:In the more general setting of
2210:Foundations of category theory
2151:. Cambridge University Press.
2110:Tsalenko & Shulgeifer 1974
2026:were originally introduced by
1819:{\displaystyle \delta :B\to C}
1810:
1740:
1717:{\displaystyle \alpha :A\to C}
1708:
1676:
1640:
1132:is some divisible group. Then
1005:
999:
981:
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568:is always a monomorphism; but
479:
429:
217:
85:
1:
1749:{\displaystyle \beta :B\to D}
1370:A monomorphism is said to be
797:is some divisible group, and
317:is a monomorphism, and every
273:
36:Polymorphism (disambiguation)
1982:{\displaystyle \varepsilon }
1930:{\displaystyle \varepsilon }
1603:{\displaystyle \varepsilon }
1583:{\displaystyle \varepsilon }
1470:{\displaystyle \varepsilon }
1450:{\displaystyle \varepsilon }
32:Monomorphic (disambiguation)
2172:Encyclopedia of Mathematics
2035:the case of epimorphisms.
1917:if there exists a morphism
1649:{\displaystyle \mu :C\to D}
1337:Every isomorphism is monic.
2289:
1794:, there exists a morphism
1508:if in each representation
1405:if in each representation
141:such that for all objects
29:
2184:Van Oosten, Jaap (1995).
2147:Borceux, Francis (1994).
855:instead). Then, letting
325:Relation to invertibility
306:is an epimorphism in the
2126:Bergman, George (2015).
298:of a monomorphism is an
2186:"Basic Category Theory"
1660:if for any epimorphism
2003:
1983:
1963:
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1907:
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1852:
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1570:is a monomorphism and
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1471:
1451:
1431:
1395:
1356:immediate monomorphism
1018:
512:
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333:is a left inverse for
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95:
67:. A monomorphism from
45:
27:Injective homomorphism
2004:
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1563:{\displaystyle \mu '}
1540:
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1432:
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1352:extremal monomorphism
1019:
760:is the corresponding
513:
375:
261:
252:
96:
44:
2193:Brics Lecture Series
2018:The companion terms
2002:{\displaystyle \mu }
1993:
1973:
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1921:
1906:{\displaystyle \mu }
1897:
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1610:is automatically an
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1512:
1497:{\displaystyle \mu }
1488:
1477:is automatically an
1461:
1441:
1409:
1394:{\displaystyle \mu }
1385:
1348:regular monomorphism
946:
851:(otherwise, choose −
678:Every morphism in a
391:
345:
178:
127:. That is, an arrow
79:
2246:Strong monomorphism
2071:Nodal decomposition
1360:strong monomorphism
1110:for some morphisms
724:group homomorphisms
554:then the inclusion
544:group homomorphisms
321:is an epimorphism.
270:injective functions
1999:
1979:
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1903:
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1692:and any morphisms
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1364:split monomorphism
1235:, it follows that
1078:. This says that
1041:, it follows that
1014:
939:, it follows that
916:. From this, and
648:for all morphisms
508:
370:
341:is a morphism and
283:intersections are
279:In the setting of
264:
247:
147:and all morphisms
91:
48:In the context of
46:
2139:978-3-319-11478-1
2037:Saunders Mac Lane
991:
682:whose underlying
680:concrete category
578:normal complement
550:is a subgroup of
122:left-cancellative
54:universal algebra
16:(Redirected from
2280:
2223:
2204:
2190:
2180:
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2143:
2113:
2107:
2101:
2095:
2030:; Bourbaki uses
2028:Nicolas Bourbaki
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1342:Related concepts
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1085:, as desired.
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830:. Now fix some
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720:(abelian) groups
705:abelian category
689:category of sets
666:for all objects
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1893:A monomorphism
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1624:A monomorphism
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1484:A monomorphism
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1381:A monomorphism
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112:(also called a
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2228:External links
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2167:"Monomorphism"
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2158:978-0521061193
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1969:(in this case
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387:
386:
385:
384:is monic, as
383:
365:
361:
357:
354:
351:
348:
340:
336:
332:
324:
322:
320:
316:
312:
309:
308:dual category
305:
301:
297:
292:
290:
286:
282:
277:
275:
271:
260:
244:
239:
235:
231:
226:
222:
211:
207:
203:
200:
197:
192:
188:
184:
181:
174:
173:
172:
169:
165:
158:
151:
145:
139:
135:
131:
126:
123:
119:
115:
111:
107:
102:
88:
82:
66:
63:
59:
55:
51:
43:
37:
33:
19:
18:Monomorphisms
2250:
2238:
2234:monomorphism
2209:
2192:
2170:
2148:
2132:. Springer.
2128:
2105:
2098:Borceux 1994
2051:
2049:
2044:
2040:
2032:monomorphism
2031:
2023:
2020:monomorphism
2019:
2017:
1914:
1657:
1505:
1402:
1374:if it is an
1371:
1363:
1359:
1355:
1351:
1347:
1345:
1310:
1305:
1301:
1297:
1293:
1289:
1285:
1281:
1277:
1273:
1269:
1265:
1261:
1257:
1250:
1246:
1242:
1238:
1230:
1226:
1222:
1218:
1214:
1210:
1206:
1202:
1198:
1194:
1186:
1182:
1174:
1170:
1166:
1162:
1158:
1154:
1150:
1142:
1138:
1134:
1129:
1124:
1120:
1116:
1112:
1106:
1102:
1098:
1094:
1089:
1087:
1080:
1074:
1070:
1066:
1062:
1058:
1054:
1047:
1043:
1037:
1033:
1029:
1026:
935:
931:
927:
923:
919:
911:
907:
904:
900:
896:
890:
886:
880:
876:
871:
865:
861:
857:
852:
846:
842:
836:
832:
826:
822:
818:
814:
810:
803:
799:
794:
789:
785:
781:
774:
770:
766:
757:
753:
749:
745:
740:
736:
732:
728:
712:
709:
677:
667:
658:
654:
650:
644:
640:
636:
629:
622:
618:
614:
610:
609: : Hom(
603:
597:
593:
589:
586:
581:
573:
569:
564:
560:
556:
551:
547:
535:
533:
528:
522:
520:
381:
338:
334:
330:
328:
310:
303:
293:
278:
267:
167:
163:
156:
149:
143:
137:
133:
129:
117:
113:
110:monomorphism
109:
103:
65:homomorphism
58:monomorphism
57:
47:
2024:epimorphism
2014:Terminology
1612:isomorphism
1479:isomorphism
1332:isomorphism
1052:, and thus
697:free object
587:A morphism
300:epimorphism
2262:Categories
2120:References
2045:monic maps
1937:such that
1826:such that
1756:such that
1317:Properties
1179:. (Since
884:such that
524:split mono
319:retraction
285:idempotent
2268:Morphisms
2212:. Nauka.
2201:1395-2048
2177:EMS Press
2076:Subobject
2066:Embedding
2053:extension
1997:μ
1977:ε
1951:μ
1948:∘
1945:ε
1925:ε
1901:μ
1878:β
1872:δ
1869:∘
1866:μ
1846:α
1840:ε
1837:∘
1834:δ
1811:→
1802:δ
1782:α
1779:∘
1776:μ
1770:ε
1767:∘
1764:β
1741:→
1732:β
1709:→
1700:α
1677:→
1668:ε
1641:→
1632:μ
1598:ε
1578:ε
1554:μ
1533:ε
1530:∘
1523:μ
1516:μ
1506:immediate
1492:μ
1465:ε
1445:ε
1425:ε
1422:∘
1419:φ
1413:μ
1389:μ
1376:equalizer
1309:. Hence
1276:= 0 ⇔ ∀
1268:) = 0 ⇒
1157:) :
953:≤
717:divisible
711:category
693:injective
664:injective
480:⇒
467:∘
461:∘
442:∘
436:∘
430:⇒
417:∘
398:∘
352:∘
337:(meaning
289:pullbacks
218:⟹
204:∘
185:∘
86:↪
62:injective
2060:See also
1557:′
1545:, where
1526:′
1437:, where
1403:extremal
1245:) ∈ Hom(
1189:)(0) = 0
1147:, where
1128:, where
1119: :
1061:) = 0 =
934:) + 1 =
870:, since
793:, where
784: :
744:, where
731: :
684:function
674:Examples
653: :
617:) → Hom(
592: :
559: :
380:), then
313:. Every
132: :
125:morphism
2248:at the
2236:at the
2179:, 2001
1372:regular
926:) <
808:, then
538:of all
529:section
315:section
120:) is a
2216:
2199:
2155:
2136:
1658:strong
1362:, and
1330:is an
1191:, and
1027:Since
773:= 0 ⇒
576:has a
540:groups
281:posets
60:is an
2189:(PDF)
2082:Notes
1915:split
1324:topos
1322:In a
1221:) + (
1209:) = (
1145:) = 0
1069:), ∀
1050:) = 0
894:, so
868:) + 1
849:) ≥ 0
701:rings
662:, is
536:Group
527:or a
116:or a
2214:ISBN
2197:ISSN
2153:ISBN
2134:ISBN
2022:and
1858:and
1724:and
1328:epic
1300:) ⇔
1292:) =
1169:) −
1036:) ∈
1009:<
918:0 ≤
903:) =
821:, ∀
817:) ∈
722:and
639:) =
542:and
294:The
118:mono
108:, a
56:, a
34:and
2253:Lab
2241:Lab
1366:.
1260:∘ (
1137:∘ (
1083:= 0
806:= 0
777:= 0
715:of
713:Div
584:.
580:in
531:.
71:to
52:or
2264::
2191:.
2175:,
2169:,
2090:^
2009:).
1358:,
1354:,
1350:,
1304:=
1284:,
1280:∈
1272:−
1264:−
1249:,
1241:−
1229:)(
1225:−
1217:)(
1213:−
1205:+
1201:)(
1197:−
1185:−
1161:↦
1153:−
1141:−
1123:→
1115:,
1105:∘
1101:=
1097:∘
1073:∈
1065:(−
891:ny
889:=
879:∈
860:=
835:∈
825:∈
802:∘
788:→
769:∘
735:→
707:.
670:.
657:→
643:∘
621:,
613:,
596:→
563:→
362:id
291:.
276:.
171:,
166:→
162::
155:,
136:→
101:.
2251:n
2239:n
2222:.
2203:.
2161:.
2142:.
2112:.
2100:.
1957:1
1954:=
1890:.
1875:=
1843:=
1814:C
1808:B
1805::
1773:=
1744:D
1738:B
1735::
1712:C
1706:A
1703::
1680:B
1674:A
1671::
1644:D
1638:C
1635::
1614:.
1519:=
1481:.
1416:=
1334:.
1311:q
1306:g
1302:f
1298:x
1296:(
1294:g
1290:x
1288:(
1286:f
1282:G
1278:x
1274:g
1270:f
1266:g
1262:f
1258:q
1253:)
1251:Q
1247:G
1243:g
1239:f
1237:(
1233:)
1231:y
1227:g
1223:f
1219:x
1215:g
1211:f
1207:y
1203:x
1199:g
1195:f
1193:(
1187:g
1183:f
1181:(
1177:)
1175:x
1173:(
1171:g
1167:x
1165:(
1163:f
1159:x
1155:g
1151:f
1149:(
1143:g
1139:f
1135:q
1130:G
1125:Q
1121:G
1117:g
1113:f
1107:g
1103:q
1099:f
1095:q
1090:q
1081:h
1075:G
1071:x
1067:x
1063:h
1059:x
1057:(
1055:h
1048:y
1046:(
1044:h
1038:Z
1034:y
1032:(
1030:h
1012:1
1006:)
1003:y
1000:(
997:h
994:=
988:1
985:+
982:)
979:x
976:(
973:h
968:)
965:x
962:(
959:h
950:0
936:n
932:x
930:(
928:h
924:x
922:(
920:h
914:)
912:y
910:(
908:h
905:n
901:x
899:(
897:h
887:x
881:G
877:y
872:G
866:x
864:(
862:h
858:n
853:x
847:x
845:(
843:h
837:G
833:x
827:G
823:x
819:Z
815:x
813:(
811:h
804:h
800:q
795:G
790:Q
786:G
782:h
775:h
771:h
767:q
758:Z
756:/
754:Q
750:Z
746:Q
741:Z
739:/
737:Q
733:Q
729:q
668:Z
659:X
655:Z
651:h
645:h
641:f
637:h
635:(
633:∗
630:f
625:)
623:Y
619:Z
615:X
611:Z
607:∗
604:f
598:Y
594:X
590:f
582:G
574:H
570:f
565:G
561:H
557:f
552:G
548:H
506:.
501:2
497:g
493:=
488:1
484:g
475:2
471:g
464:f
458:l
455:=
450:1
446:g
439:f
433:l
425:2
421:g
414:f
411:=
406:1
402:g
395:f
382:f
366:X
358:=
355:f
349:l
339:l
335:f
331:l
311:C
304:C
245:.
240:2
236:g
232:=
227:1
223:g
212:2
208:g
201:f
198:=
193:1
189:g
182:f
168:X
164:Z
160:2
157:g
153:1
150:g
144:Z
138:Y
134:X
130:f
89:Y
83:X
73:Y
69:X
38:.
20:)
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