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## Elements of Subobject Classifier

The elements of the subobject classifier (see 1.7) in a Topos are derived from the following Theorem

**Theorem 1.1**

*Subpresheaves 1.10 can be identified with sieves (see definition 1.7.1)*

**Lemma 1.1**

**Preliminary**: if is a locally small category^{1.1}, then each object A of induces a natural contravariant functor 3.2 to Set called a hom-functor, i.e there exists a map which in terms of individual elements is defined as .**Yoneda lemma**: Given an arbitrary presheaf P 3.6 on there exists a bijective correspondence between natural transformations 3.5 and elements of the set P(A) defined as an arrow which in terms of individual elements is*Proof*. Let's consider to be a subobject classifier of , i.e. we want to classify subobjects in .

Consider now a presheaf . We know from axiom 1.1 that

form Yonedas lemma it follows that

, therefore the subobject classifier must be a presheaf such that

Now if is a subfunctor of , then the set

is a sieve on C. Conversely given a sieve S on C we define

which produces a preshaef which is a subfunctor of , i.e to each object Q assigns the set . Since the transformation function from Q to S is a bijection (as can be seen from the above definition) we can conclude the following:

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**Previous:**Sieve Cecilia Flori 2007-01-02