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subobject classifier
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(Definition)
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Consider a set  and a subset
 .  can be thought of as a property of  : there is a function
 , such that
 iff  . This function can be seen to be uniquely determined by the subset  , and conversely. If we denote  the set of all subsets of  (the power set of  ), and
 the set of all functions from  to
 , then
 .
In fact, we have established a commutative diagram
where  and  are inclusion functions and  is the unique constant function. Any function  gives rise to a unique set  making the above diagram commute.
In category theory, a subobject classifier is the generalization of the above example, where  is an object of a certain given category
 and  is a subobject of  ,
 is replaced by a terminal object, and  is replaced by what is known as a subobject classifier, or a truth object. If we think of the category Set,  “classifies” elements of a given set as to whether they belong to a certain subset or not, via a characteristic function. If the value of the function is  , then the element is in that subset, otherwise it is not.
Formally, let
 be a category with a terminal object  . A subobject classifier is an object  in
 such that, for any monomorphism  , there exists a unique morphism  such that
is a pullback diagram.  is called the characteristic morphism of  and  is a truth morphism.
In a category with a terminal object 1, a subobject classifier may or may not exist. If it does, it is unique up to isomorphism. Suppose  has a terminal object  , has pullbacks, and has a subobject  . Then for any object  in
 , any morphism
 gives rise to a unique monomorphism  via the pull back of  and  :
Since  is a subobject classifier,  determines  uniquely as well. So what we have is a one-to-one correspondence
between the subobject functor and hom functor. It can be verified that the bijection is actually a natural isomorphism, so that
 is a representable functor. Conversely, it may be shown that if
 is representable, then  has a subobject classifier.
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"subobject classifier" is owned by CWoo. [ full author list (2) ]
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(view preamble | get metadata)
See Also: power object
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truth object, characteristic morphism |
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Cross-references: representable, representable functor, natural isomorphism, hom functor, subobject functor, one-to-one correspondence, isomorphism, pullback diagram, morphism, monomorphism, characteristic function, terminal object, subobject, category, object, category theory, diagram, constant function, inclusion, commutative diagram, power set, iff, function, property, subset
There are 10 references to this entry.
This is version 5 of subobject classifier, born on 2007-01-20, modified 2007-05-24.
Object id is 8805, canonical name is SubobjectClassifier.
Accessed 1548 times total.
Classification:
| AMS MSC: | 18B25 (Category theory; homological algebra :: Special categories :: Topoi) |
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Pending Errata and Addenda
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![$\displaystyle \xymatrix@+=4pc{ B \ar[r]^{k}\ar[d]_{inc}& \lbrace 1\rbrace \ar[d]^i \ A \ar[r]_{\chi_B}& 2 } $](http://images.planetmath.org:8080/cache/objects/8805/l2h/img16.png "$\displaystyle \xymatrix@+=4pc{ B \ar[r]^{k}\ar[d]_{inc}& \lbrace 1\rbrace \ar[d]^i \ A \ar[r]_{\chi_B}& 2 } $")
![$\displaystyle \xymatrix@+=4pc{ B \ar[r] \ar[d]_f & 1 \ar[d]^{\top}\ A \ar@{.>}[r]^{\chi_B} & \Omega } $](http://images.planetmath.org:8080/cache/objects/8805/l2h/img36.png "$\displaystyle \xymatrix@+=4pc{ B \ar[r] \ar[d]_f & 1 \ar[d]^{\top}\ A \ar@{.>}[r]^{\chi_B} & \Omega } $")
![$\displaystyle \xymatrix@+=4pc{ A \ar[r] \ar[d]_g & 1 \ar[d]^{\top}\ X \ar[r]^f & \Omega } $](http://images.planetmath.org:8080/cache/objects/8805/l2h/img49.png "$\displaystyle \xymatrix@+=4pc{ A \ar[r] \ar[d]_g & 1 \ar[d]^{\top}\ X \ar[r]^f & \Omega } $")
