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subobject classifier
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(Definition)
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Consider a set $A$ and a subset $B\subseteq A$ . $B$ can be thought of as a property of $A$ : there is a function $\chi_B:A\to \lbrace 0,1\rbrace$ , such that $\chi_B(x)=1$ iff $x\in B$ . This function can be seen to be uniquely determined by the subset $B$ , and conversely. If we denote $P(A)$ the set of all subsets of $A$ (the power set of $A$ ), and $2^A$ the set of all functions from $A$ to $2:=\lbrace 0,1\rbrace$ , then $P(A)\cong 2^A$ .
In fact, we have established a commutative diagram
where $inc$ and $i$ are inclusion functions and $k$ is the unique constant function. Any function $A\to 2$ gives rise to a unique set $B$ making the above diagram commute.
In category theory, a subobject classifier is the generalization of the above example, where $A$ is an object of a certain given category $\mathcal{C}$ and $B$ is a subobject of $A$ , $\lbrace 1\rbrace$ is replaced by a terminal object, and $2$ is replaced by what is known as a subobject
classifier, or a truth object. If we think of the category Set, $2$ ``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 $1$ , then the element is in that subset, otherwise it is not.
Formally, let $\mathcal{C}$ be a category with a terminal object $1$ . A subobject classifier is an object $\Omega$ in $\mathcal{C}$ such that, for any monomorphism $f:B\to A$ , there exists a unique morphism $\chi_B$ such that
is a pullback diagram. $\chi_B$ is called the characteristic morphism of $f$ and $\top$ 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 $C$ has a terminal object $1$ , has pullbacks, and has a subobject $\Omega$ . Then for any object $X$ in $\mathcal{C}$ , any morphism $f:X\to \Omega$ gives rise to a unique monomorphism $g:A\to X$ via the pull back of $f$ and $\top$ :
Since $\Omega$ is a subobject classifier, $g$ determines $f$ uniquely as well. So what we have is a one-to-one correspondence $$\Sub(X)\cong \hom(X,\Omega)$$ between the subobject functor and hom functor. It can be verified that the bijection is actually a natural isomorphism, so that $\Sub$ is a representable functor. Conversely, it may be shown that if
$\Sub$ is representable, then $C$ has a subobject classifier.
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"subobject classifier" is owned by CWoo. [ full author list (2) ]
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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, conversely, 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 2247 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 } $](http://images.planetmath.org:8080/cache/objects/8805/js/img1.png "$\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 } $](http://images.planetmath.org:8080/cache/objects/8805/js/img2.png "$\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 } $](http://images.planetmath.org:8080/cache/objects/8805/js/img3.png "$\displaystyle \xymatrix@+=4pc{ A \ar[r] \ar[d]_g & 1 \ar[d]^{\top}\ X \ar[r]^f & \Omega } $")