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I don't think the definition of the subobject classifier of sheaves of sets on a space X is correct. Under the definition given, Ω would have only one global section, but the map of global sections from a sheaf A to the subobject classifier should provide information about where restrictions of the
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Yes, especially since many subobject classifiers have structure beyond the set {true,false}. As it sits, I think the article is a bit misleading, at least for people who can't follow the topology example. But unfortunately, my topos theory isn't quite good enough to improve this article yet.
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global sections of A belong to the subsheaf. For instance, for the one point set, Ω would be the one point set, not the two point set. I think the correct definition should be the sheaf that assigns an open set to it's set of open subsets, restricting by taking intersections.
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This sentence "Since in any category subobjects are identified as monic arrows, we identify the value true with the arrow: true: {0} → {0, 1} which maps 0 to 1. " has to be modified, no logical link. Moreover where are now explicitly in the category of sets.
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For instance, to get the map which represents the subsheaf H^0(X) of constant functions inside C(X), we would take each function to the subset on which it is constant. Right? I'll change the page and add this example if no one objects.
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Some other examples are Z2 (the cyclic ab. group) in AbGrp, the Be(1/2) (this is the coin) in the Prob category or a simple two vertex connected graph for FinGraph category. All of them have their own internal structure.
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You are quite right. I went ahead and made a correction. You might want to give a fuller description and add your example. But it will need to say `locally constant.'
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The references Mac Lane (1998) p.105 and
Pedicchio & Tholen (2004) p.330 both define a subobject classifier not as just a special object Ω but as a monomorphism
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Anybody think there should be more examples (at the very least, the 'classical' subobject classifier in
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Oh, I got it now, it means that the object "true" (here {0}) is identified with the arrow from 0 to 1
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Within a topos, every pair of subobject classifiers are categorically isomorphic.
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is interpreted as the "truth" map), such that every mono is a pullback of
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It intuitively represents the "true" point in the subobject classifier
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don't have time to correct this.... what is that axiom business??
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