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?
1 Answer
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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.)
answered Dec 31, 2009 at 17:19
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3$\begingroup$ Does anybody else remember reading an n-category cafe post about how toposes and abelian categories differ only in the behavior of the initial and terminal objects...? $\endgroup$ Commented Dec 31, 2009 at 20:18
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3$\begingroup$ Reid's answer is clearly spot on, but I would just like to note that in a topos if an object has an arrow to an initial object, then it is itself initial. First use the fact that the graph of that arrow is monic, and then use the subobject classifier pullback square to get the desired isomorphism. @Qiaochu, I do remember the post, but I was unable to locate it. If anyone else can I would love to read it again! $\endgroup$ Commented Jan 1, 2010 at 1:25
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3$\begingroup$ There was a discussion about this on the categories mailing list in 1997. See mta.ca/~cat-dist/catlist/1999/atcat $\endgroup$ Commented Jan 2, 2010 at 1:22
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4$\begingroup$ In case anyone is interested, I wrote up an exposition of the discussion between Freyd and Pratt in the nLab: ncatlab.org/nlab/show/AT+category. $\endgroup$ Commented Sep 7, 2010 at 23:18