724:
by a two-sided ideal. To get maps which truly behave like subobject embeddings or quotients, rather than as arbitrary injective functions or maps with dense image, one must restrict to monomorphisms and epimorphisms satisfying additional hypotheses. Therefore, one might define a "subobject" to be an
671:
However, in some contexts these definitions are inadequate as they do not concord with well-established notions of subobject or quotient object. In the category of topological spaces, monomorphisms are precisely the injective continuous functions; but not all injective continuous functions are
733:
This definition corresponds to the ordinary understanding of a subobject outside category theory. When the category's objects are sets (possibly with additional structure, such as a group structure) and the morphisms are set functions (preserving the additional structure), one thinks of a
725:
equivalence class of so-called "regular monomorphisms" (monomorphisms which can be expressed as an equalizer of two morphisms) and a "quotient object" to be any equivalence class of "regular epimorphisms" (morphisms which can be expressed as a coequalizer of two morphisms)
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monomorphism in terms of its image. An equivalence class of monomorphisms is determined by the image of each monomorphism in the class; that is, two monomorphisms
625:
597:
567:
391:
368:
228:
136:
114:
An appropriate categorical definition of "subobject" may vary with context, depending on the goal. One common definition is as follows.
987:
484:
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1061:
148:
1034:
1004:
71:. Since the detailed structure of objects is immaterial in category theory, the definition of subobject relies on a
17:
992:
634:; this means that the discussion given is somewhat loose. If the subobject-collection of every object is a
370:
1026:
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99:
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753:
40:
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that describes how one object sits inside another, rather than relying on the use of elements.
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Categorical foundations. Special topics in order, topology, algebra, and sheaf theory
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itself. This is in part because all arrows in such a category will be monomorphisms.
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has a greatest element, the subobject partial order of this greatest element will be
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are equivalent if and only if their images are the same subset (thus, subobject) of
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28:
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1025:. Encyclopedia of Mathematics and Its Applications. Vol. 97. Cambridge:
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of their domains under which corresponding elements of the domains map by
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32:
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52:
650:, namely that there is a set of morphisms between any two objects).
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207:
72:
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48:
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subspace embeddings. In the category of rings, the inclusion
534:{\displaystyle u\equiv v\iff u\leq v\ {\text{and}}\ v\leq u}
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The collection of subobjects of an object may in fact be a
665:
is then an equivalence class of epimorphisms with domain
1021:
Pedicchio, Maria
Cristina; Tholen, Walter, eds. (2004).
695:{\displaystyle \mathbb {Z} \hookrightarrow \mathbb {Q} }
47:. The notion is a generalization of concepts such as
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896:, ≤), we can form a category with the elements of
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835:, or rather the collection of all maps from sets
661:" above and reverse arrows. A quotient object of
646:(this clashes with a different usage of the term
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196:{\displaystyle u:S\to A\ {\text{and}}\ v:T\to A}
1020:
798:; this explains the definition of equivalence.
702:is an epimorphism but is not the quotient of
995:, vol. 5 (2nd ed.), New York, NY:
43:that sits inside another object in the same
851:. The subobject partial order of a set in
750:. In that case there is the isomorphism
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497:
139:be an object of some category. Given two
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688:
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988:Categories for the Working Mathematician
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962:
794:, respectively, to the same element of
231:, we define an equivalence relation by
14:
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900:as objects, and a single arrow from
90:. This generalizes concepts such as
607:on the collection of subobjects of
548:on the monomorphisms with codomain
24:
25:
1073:
728:
574:of these monomorphisms are the
257:if there exists an isomorphism
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498:
451:{\displaystyle u=v\circ \phi }
410:
314:{\displaystyle u=v\circ \phi }
273:
187:
161:
109:
13:
1:
993:Graduate Texts in Mathematics
975:
779:{\displaystyle g^{-1}\circ f}
657:, replace "monomorphism" by "
717:{\displaystyle \mathbb {Z} }
419:{\displaystyle \phi :S\to T}
282:{\displaystyle \phi :S\to T}
82:concept to a subobject is a
7:
938:
801:
653:To get the dual concept of
10:
1078:
1027:Cambridge University Press
805:
394:—that is, if there exists
1062:Objects (category theory)
808:Category:Quotient objects
638:, the category is called
603:The relation ≤ induces a
250:{\displaystyle u\equiv v}
39:is, roughly speaking, an
955:
570:, and the corresponding
887:partially ordered class
471:{\displaystyle \equiv }
458:. The binary relation
343:{\displaystyle u\leq v}
324:Equivalently, we write
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945:Subobject classifier
870:, the subobjects of
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546:equivalence relation
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328:
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235:
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149:
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855:is just its subset
572:equivalence classes
983:Mac Lane, Saunders
933:subterminal object
874:correspond to the
868:category of groups
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927:A subobject of a
824:corresponds to a
820:, a subobject of
620:{\displaystyle A}
592:{\displaystyle A}
562:{\displaystyle A}
521:
517:
513:
386:{\displaystyle v}
363:{\displaystyle u}
223:{\displaystyle A}
177:
173:
169:
131:{\displaystyle A}
16:(Redirected from
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968:Mac Lane, p. 126
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818:category of sets
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978:
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929:terminal object
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742:into an object
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655:quotient object
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371:factors through
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117:In detail, let
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104:quotient graphs
100:quotient spaces
96:quotient groups
86:quotient object
85:
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29:category theory
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18:Quotient object
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729:Interpretation
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31:, a branch of
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1006:0-387-98403-8
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648:locally small
645:
644:locally small
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605:partial order
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141:monomorphisms
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92:quotient sets
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62:
58:
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50:
46:
42:
38:
34:
30:
19:
1022:
986:
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931:is called a
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795:
791:
787:
747:
743:
739:
735:
732:
670:
666:
662:
654:
652:
643:
642:or, rarely,
640:well-powered
639:
632:proper class
629:
602:
579:
575:
549:
543:
373:
323:
210:
205:
118:
116:
113:
83:
77:
61:group theory
36:
26:
950:Subquotient
659:epimorphism
478:defined by
110:Definitions
33:mathematics
1045:1034.18001
1015:0906.18001
976:References
837:equipotent
806:See also:
576:subobjects
426:such that
53:set theory
876:subgroups
771:∘
763:−
685:↪
526:≤
506:≤
499:⟺
492:≡
466:≡
446:ϕ
443:∘
411:→
402:ϕ
335:≤
309:ϕ
306:∘
274:→
265:ϕ
242:≡
188:→
162:→
65:subspaces
57:subgroups
37:subobject
1056:Category
985:(1998),
939:See also
885:Given a
847:exactly
802:Examples
208:codomain
73:morphism
69:topology
45:category
857:lattice
106:, etc.
49:subsets
1043:
1033:
1013:
1003:
866:, the
826:subset
816:, the
544:is an
520:
512:
176:
168:
63:, and
41:object
956:Notes
916:. If
845:image
843:with
289:with
206:with
67:from
59:from
51:from
1031:ISBN
1001:ISBN
908:iff
790:and
738:and
80:dual
78:The
35:, a
1041:Zbl
1011:Zbl
904:to
892:= (
878:of
864:Grp
862:In
853:Set
839:to
831:of
814:Set
812:In
636:set
578:of
516:and
350:if
172:and
27:In
1058::
1039:.
1029:.
1009:,
999:,
991:,
935:.
912:≤
882:.
859:.
667:A.
627:.
600:.
321:.
102:,
98:,
94:,
55:,
1047:.
922:P
918:P
914:q
910:p
906:q
902:p
898:P
894:P
890:P
880:A
872:A
849:B
841:B
833:A
829:B
822:A
796:T
792:g
788:f
774:f
766:1
759:g
748:T
744:T
740:g
736:f
711:Z
689:Q
681:Z
663:A
615:A
587:A
557:A
529:u
523:v
509:v
503:u
495:v
489:u
440:v
437:=
434:u
414:T
408:S
405::
381:v
358:u
338:v
332:u
303:v
300:=
297:u
277:T
271:S
268::
245:v
239:u
218:A
191:A
185:T
182::
179:v
165:A
159:S
156::
153:u
126:A
20:)
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