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<math>\Omega:C\rightarrow\mathcal{S}^{C^{op}}</math> such that to each object <math>A\in C</math>
there corresponds an object <math>\Omega(A)\in\mathcal{S}^{C^{op}}</math> which represents the set
of all sieves (see [[sieve]]).
of all sieves (see [[sieve]]). 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>.
on A, and to each
<math>C</math>-arrow <math>f:B\rightarrow A</math> there corresponds an
<math>\mathcal{S}^{C^{op}}</math>-arrow <math>\Omega(f):\Omega(A)\rightarrow\Omega(B)</math> such that
<math>\Omega(f)(S):=\{h:C\rightarrow B|f o h\in S\}</math> is a [[sieve]] on B, where <math>\Omega(f)(S)\equiv f^*(S)</math>
== References ==
* Robert Goldblatt: ''Topoi, the Categorial Analysis of Logic''. North-Holland, New York, 1984. (Studies in logic and the foundations of mathematics, 98.). A good start.
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