Subobject classifier: Difference between revisions

Revision as of 11:14, 18 March 2007

Revision as of 11:15, 18 March 2007

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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 [[sievessieve]]). 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>