Is there a nontrivial abelian category $C$ such that both $C$ and $C^{op}$ satisfy AB3-AB5? ("nontrivial" means that there are nonzero objects in $C$)
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3$\begingroup$ No, see page 129 of Grothendieck's Tohoku paper. $\endgroup$– Fred RohrerCommented Dec 4, 2017 at 19:17
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1$\begingroup$ If $C, C^{op}$ are both AB5 and they have a generator, then they are both locally finitely presentable. Categories such that $C,C^{op}$ are both locally presentable tend to be very trivial. $\endgroup$– foscoCommented Dec 5, 2017 at 8:41
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1$\begingroup$ @FoscoLoregian Not trivial in the sense of the question. If a category and its opposite are both locally presentable, it is a complete lattice. $\endgroup$– Philippe GaucherCommented Dec 5, 2017 at 14:06
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$\begingroup$ Well, yeah, I shouldhave phrased the answer more on the lines of "although not the zero category, an AB5 and AB5* category tend to be quite trivial". $\endgroup$– foscoCommented Dec 5, 2017 at 16:23
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1 Answer
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The answer is no.
This is mentioned (with a hint for a proof) on page 129 of A. Grothendieck, Sur quelques points d'algèbre homologique, Tohoku Math. J. (2) 9, (1957), 119-221.
For a complete proof (albeit in somewhat different terminology) see Proposition III.1.10 in B. Mitchell, Theory of categories, Academic Press (1965).
answered Dec 5, 2017 at 12:16