All right, somebody should answer... Dear Dmitry, First, may I suggest that looking at category theory as merely a 'generalized abelian category theory' is not really a good idea? Specifically, in the abelian context 'right exact + preserves monos => left exact' is simple but extremely important, while for general categories it is indeed wrong, and adding conditions to make this work seems to be a strange thing to do! Now, more specifically, concerning Barr exact: Let 2 be the ordered set {0,1} considered as a category, let S be the category of sets, and let F : S --> 2 be the left adjoint of the canonical embedding 2 --> S (hence, for a set X, F(X) = 0 if and only if X is empty). Then: (a) 2 and S are Barr exact. (b) F preserves all colimits since it is a left adjoint. (c) F preserves products (moreover, the fact that it preserves all products is one of the standard forms of the Axiom of Choice). (d) F preserves monomorphisms since every morphism in 2 is a monomorphism. (e) F does not preserve all equalizers since two parallel morphisms between non-empty sets might have the empty equalizer. Best regards George Janelidze -------------------------------------------------- From: "Dmitry Roytenberg" <starrgazerr@gmail.com> Sent: Wednesday, October 26, 2011 1:59 PM To: "Categories list" <categories@mta.ca> Subject: categories: Re: when does preservation of monos imply left exactness?
I'll try again...
On Mon, Oct 24, 2011 at 11:23 AM, Dmitry Roytenberg <starrgazerr@gmail.com> wrote:
Dear category theorists,
It is well known that any functor that preserves finite limits preserves monomorphisms, and that for an additive right-exact functor between abelian categories, the converse is also true. Is it known how far this extends to the non-additive setting? In other words, what exactness properties of two categories and a functor between them would suffice to conclude that the functor preserves finite limits if and only if it preserves monos? For instance, is it enough to assume that the categories be Barr-exact and that the functor preserve all colimits, finite products and monos to conclude that it also preserves equalizers?
Any references would be extremely helpful.
Thanks, Dmitry
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