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?



Can a topos ever be an abelian category?
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.)


