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As an example, the set Ω = {0,1} is a subobject classifier in the category of sets and functions: to every subset ''j'':''U'' → ''X'' we can assign the function ''χ<sub>j</sub>'' from ''X'' to Ω that maps precisely the elements of ''U'' to 1 (see [[indicator function|characteristic function]]). Every function from ''X'' to Ω arises in this fashion from precisely one subset ''U''.
The above example of subobject classifier in Sets is very usefull because it enables us to easliy prove the following axiom:
'''Axiom''': Given a category '''C''', then there exists an Isomorphisms,<math>y:Sub_C(X)\cong Hom_C(X,\Omega)</math> <math>\forall X\in C</math>
In Set this axiom can be restated as follows:
'''Axiom''': The collection of all subsets of S denoted by <math>\mathcal{P}(S)</math>, and
the collection of all maps from S to the set <math>\{0,1\}=2</math> denoted by <math>2^S</math> are isomorphic i.e. the function <math>y:\mathcal{P}(S)\rightarrow2^S</math>,
which in terms of single elements of <math>\mathcal{P}(S)</math> is <math>A\rightarrow\chi_A</math>, it is a bijection.
The above axiom implyes the alternative definition of a subobject calssifier:
'''Definition''': <math>\Omega</math> is a '''Subobject classifier''' iff there is a ``one to one" correspondence between subobject of X and morphisms
from X to <math>\Omega</math>.
== Definition ==
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