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**Definition 1.14**

*A*

**Subobject Classifier**is a presheaf such that to each object there corresponds an object which represents the set of all sieves 1.9.1 on A, and to each -arrow there corresponds an -arrow such that is a sieve on B, where*true*(T) and the

*characteristic function*(see section 1.8.1) in Topos.

**Definition 1.15**

*is the natural transformation 3.5 that has components given by = principal sieve on A*

therefore

What equation 1.2 means is that we require F(f) to be the restriction of X(f) to F(A). This condition is expressed by the following diagram

i.e. f belongs to iff X(f) maps x into F(B).

, as defined by equation 1.2,
represents a sieve on A.

*Proof*. Consider the following commuting diagram which represents subobject F of the presheaf X.

**Definition 1.16**

*is a*

**subobject classifier**iff there is a ``one to one" correspondence between subobject of X and morphisms from X to .Moreover for each morphism we have

subobject of X |

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**Previous:**Example of Yoneda's Lemma Cecilia Flori 2007-03-17