713:
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
660:
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
722:
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
714:
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
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357:
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125:
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An appropriate categorical definition of "subobject" may vary with context, depending on the goal. One common definition is as follows.
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60:. Since the detailed structure of objects is immaterial in category theory, the definition of subobject relies on a
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623:; this means that the discussion given is somewhat loose. If the subobject-collection of every object is a
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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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1014:. 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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639:, namely that there is a set of morphisms between any two objects).
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subspace embeddings. In the category of rings, the inclusion
523:{\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
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is then an equivalence class of epimorphisms with domain
1010:
Pedicchio, Maria
Cristina; Tholen, Walter, eds. (2004).
684:{\displaystyle \mathbb {Z} \hookrightarrow \mathbb {Q} }
36:. The notion is a generalization of concepts such as
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885:, ≤), we can form a category with the elements of
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824:, or rather the collection of all maps from sets
650:" above and reverse arrows. A quotient object of
635:(this clashes with a different usage of the term
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185:{\displaystyle u:S\to A\ {\text{and}}\ v:T\to A}
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787:; this explains the definition of equivalence.
691:is an epimorphism but is not the quotient of
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32:that sits inside another object in the same
840:. The subobject partial order of a set in
739:. In that case there is the isomorphism
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128:be an object of some category. Given two
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977:Categories for the Working Mathematician
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783:, respectively, to the same element of
220:, we define an equivalence relation by
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889:as objects, and a single arrow from
79:. This generalizes concepts such as
596:on the collection of subobjects of
537:on the monomorphisms with codomain
13:
14:
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563:of these monomorphisms are the
246:if there exists an isomorphism
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440:{\displaystyle u=v\circ \phi }
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303:{\displaystyle u=v\circ \phi }
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176:
150:
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1:
982:Graduate Texts in Mathematics
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768:{\displaystyle g^{-1}\circ f}
646:, replace "monomorphism" by "
706:{\displaystyle \mathbb {Z} }
408:{\displaystyle \phi :S\to T}
271:{\displaystyle \phi :S\to T}
71:concept to a subobject is a
7:
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790:
642:To get the dual concept of
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1016:Cambridge University Press
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383:—that is, if there exists
1051:Objects (category theory)
797:Category:Quotient objects
627:, the category is called
592:The relation ≤ induces a
239:{\displaystyle u\equiv v}
28:is, roughly speaking, an
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559:, and the corresponding
876:partially ordered class
460:{\displaystyle \equiv }
447:. The binary relation
332:{\displaystyle u\leq v}
313:Equivalently, we write
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934:Subobject classifier
859:, the subobjects of
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535:equivalence relation
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844:is just its subset
561:equivalence classes
972:Mac Lane, Saunders
922:subterminal object
863:correspond to the
857:category of groups
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916:A subobject of a
813:corresponds to a
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609:{\displaystyle A}
581:{\displaystyle A}
551:{\displaystyle A}
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375:{\displaystyle v}
352:{\displaystyle u}
212:{\displaystyle A}
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120:{\displaystyle A}
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957:Mac Lane, p. 126
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807:category of sets
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106:In detail, let
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93:quotient graphs
89:quotient spaces
85:quotient groups
75:quotient object
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718:Interpretation
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20:, a branch of
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637:locally small
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633:locally small
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594:partial order
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130:monomorphisms
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81:quotient sets
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920:is called a
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732:
728:
724:
721:
659:
655:
651:
643:
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632:
631:or, rarely,
629:well-powered
628:
621:proper class
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591:
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362:
312:
199:
194:
107:
105:
102:
72:
66:
50:group theory
25:
15:
939:Subquotient
648:epimorphism
467:defined by
99:Definitions
22:mathematics
1034:1034.18001
1004:0906.18001
965:References
826:equipotent
795:See also:
565:subobjects
415:such that
42:set theory
865:subgroups
760:∘
752:−
674:↪
515:≤
495:≤
488:⟺
481:≡
455:≡
435:ϕ
432:∘
400:→
391:ϕ
324:≤
298:ϕ
295:∘
263:→
254:ϕ
231:≡
177:→
151:→
54:subspaces
46:subgroups
26:subobject
1045:Category
974:(1998),
928:See also
874:Given a
836:exactly
791:Examples
197:codomain
62:morphism
58:topology
34:category
846:lattice
95:, etc.
38:subsets
1032:
1022:
1002:
992:
855:, the
815:subset
805:, the
533:is an
509:
501:
165:
157:
52:, and
30:object
945:Notes
905:. If
834:image
832:with
278:with
195:with
56:from
48:from
40:from
1020:ISBN
990:ISBN
897:iff
779:and
727:and
69:dual
67:The
24:, a
1030:Zbl
1000:Zbl
893:to
881:= (
867:of
853:Grp
851:In
842:Set
828:to
820:of
803:Set
801:In
625:set
567:of
505:and
339:if
161:and
16:In
1047::
1028:.
1018:.
998:,
988:,
980:,
924:.
901:≤
871:.
848:.
656:A.
616:.
589:.
310:.
91:,
87:,
83:,
44:,
1036:.
911:P
907:P
903:q
899:p
895:q
891:p
887:P
883:P
879:P
869:A
861:A
838:B
830:B
822:A
818:B
811:A
785:T
781:g
777:f
763:f
755:1
748:g
737:T
733:T
729:g
725:f
700:Z
678:Q
670:Z
652:A
604:A
576:A
546:A
518:u
512:v
498:v
492:u
484:v
478:u
429:v
426:=
423:u
403:T
397:S
394::
370:v
347:u
327:v
321:u
292:v
289:=
286:u
266:T
260:S
257::
234:v
228:u
207:A
180:A
174:T
171::
168:v
154:A
148:S
145::
142:u
115:A
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