Can a topos ever be a nontrivial abelian category? If not, where does the contradiction lie? If a topos can be an abelian category, can you give a (notrivial!) example?
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No. In fact no nontrivial cartesian closed category can have a zero object 0 (one which is both initial and final), as then for any X, 0 = 0 × X = X. (The first equality uses the fact that – × X commutes with colimits and in particular the empty colimit, and the second holds because 0 is also the final object.) |
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