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| In [[category theory]], a branch of [[mathematics]], a '''subobject''' is, roughly speaking, an object which sits inside another object in the same [[category (mathematics)|category]]. The notion is a generalization of the older concepts of [[subset]] from [[set theory]] and [[subgroup]] from [[group theory]].<ref name="Mac Lane">Mac Lane, p. 126</ref> Since the actual structure of objects is immaterial in category theory, the definition of subobject relies on a [[morphism]] which describes how one object sits inside another, rather than relying on the use of elements.
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| ==Definition==
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| In detail, let ''A'' be an object of some category. Given two [[monomorphism]]s
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| :''u'': ''S'' → ''A'' and | |
| :''v'': ''T'' → ''A''
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| with codomain ''A'', say that ''u'' ≤ ''v'' if ''u'' [[Mathematical jargon#factor_through|factors through]] ''v'' — that is, if there exists ''w'': ''S'' → ''T'' such that <math>u = v \circ w</math>. The binary relation ≡ defined by
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| :''u'' ≡ ''v'' if and only if ''u'' ≤ ''v'' and ''v'' ≤ ''u''
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| is an [[equivalence relation]] on the monomorphisms with codomain ''A'', and the corresponding equivalence classes of these monomorphisms are the '''subobjects''' of ''A''. If two monomorphisms represent the same subobject of ''A'', then their domains are isomorphic. The collection of monomorphisms with codomain ''A'' under the relation ≤ forms a [[preorder]], but the definition of a subobject ensures that the collection of subobjects of ''A'' is a [[partial order]]. (The collection of subobjects of an object may in fact be a [[proper class]]; this means that the discussion given is somewhat loose. If the subobject-collection of every object is a [[Set (mathematics)|set]], the category is ''well-powered''.)
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| The [[dual (category theory)|dual]] concept to a subobject is a '''quotient object'''; that is, to define ''quotient object'' replace ''monomorphism'' by ''epimorphism'' above and reverse arrows.
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| ==Examples==
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| In the category '''Sets''', a subobject of A corresponds to a subset B of A, or rather the collection of all maps from sets equipotent to B with image exactly B. The subobject partial order of a set in '''Sets''' is just its subset lattice. Similar results hold in '''Groups''', and some other categories.
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| Given a partially ordered class '''P''', we can form a category with '''P''''s elements as objects and a single arrow going from one object (element) to another if the first is less than or equal to the second. If '''P''' has a greatest element, the subobject partial order of this greatest element will be '''P''' itself. This is in part because all arrows in such a category will be monomorphisms.
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| ==See also==
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| *[[Subobject classifier]]
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| *[[Mereology]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * {{citation | last=Mac Lane | first=Saunders | authorlink=Saunders Mac Lane | title=[[Categories for the Working Mathematician]] | edition=2nd | series=[[Graduate Texts in Mathematics]] | volume=5 | location=New York, NY | publisher=[[Springer-Verlag]] | year=1998 | isbn=0-387-98403-8 | zbl=0906.18001 }}
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| * {{cite book | editor1-last=Pedicchio | editor1-first=Maria Cristina | editor2-last=Tholen | editor2-first=Walter | title=Categorical foundations. Special topics in order, topology, algebra, and sheaf theory | series=Encyclopedia of Mathematics and Its Applications | volume=97 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2004 | isbn=0-521-83414-7 | zbl=1034.18001 }}
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| [[Category:Objects (category theory)]]
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