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A well known result states that, if $F:C \rightarrow D$ is a covariant functor between categories which admit finite projective limits, then $F$ is left exact if and only if it preserves finite projective limits.

I need to use this result, but unfortunately I was unable to find a reference or to prove it by myself. I would like to have one of the two.

Note: For completeness it's useful to say that the same result holds for right exact functor and finite direct limits. And that a functor preserves finite projective limits if and only if it preserves final objects and fiber products, or if and only if it preserves final objects, products and equalizers.

Edit: The definition of left exactness I suppose given is the one that can be found in wikipedia: http://en.wikipedia.org/wiki/Exact_functor. Which is: "$F$ is left exact if it brings the short exact sequence $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ of objects (and morphisms) of $C$ to an exact sequence $0 \rightarrow F(A) \rightarrow F(B) \rightarrow F(C) \rightarrow ...$ of objects of $D$". I don't think the result is untrue since also the page I linked states the result.

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    Isn't this the *very definition* of left exactness in this generality? If not, what is yours?2011-05-12
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    Your note is certainly untrue; and your first statement is probably untrue. Please, be more precise when asking questions and provide us with necessary definitions.2011-05-12
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    @Fallen: I'm not sure what exactly you're talking about. Preservation of finite limits is the only definition I know for left exactness of a functor without further assumptions. I'd love to know what you have in mind since student seems to refuse further elaboration. I agree that things need to be clarified in order to make sense of this question.2011-05-12
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    @Fallen, @Theo. I do apologize for the imprecisions in the question and the delay in the corrections. I edited the question expliciting the definition and linking to the page of wikipedia I'm referring to. I hope now it's clear.2011-05-12
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    Dear Student73, The definition in terms of exact sequences only makes sense for *abelian* categories (which is the context in which the wikipedia entry you cite is written). In more general categories there is no notion of exact sequence, and (as @Theo already mentioned) left exactness is *defined* in terms of preserving finite projective limits. This raises the question of showing the two definitions are equivalent in the abelian category case, which I'm guessing is what you mean to ask about. Regards,2011-05-13
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    @Matt, Oops! You are right, I was a little bit confused about all the matter. Now I will try to prove the equivalence with this necessary condition you explicited. Thank you for your patience.2011-05-13
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    Theo: (1) I do not know what you understand by "projective limits", but in general, preserving projective limits does not mean preserving all finite limits, (2) if you do not assume anything about categories, then the only sensible notion of "left exact functor" $F$ is that for each object $A$ the corepresentation of $F$ at $A$ --- that is: $hom(A, F(-))$ is a filtered colimit of representable functors.2011-05-13
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    @Fallen: (1) for me "limit" and "projective limit" are synonymous. What is a projective limit for you, then? Do you assume that all morphisms in the diagram ar epis? (2) for set-valued functors your definition coincides with mine, see e.g. Borceux I p. 250 (who gives "my" definition there as well). I don't know why the one you give is the only sensible one, e.g. Freyd "mine" at several places (I don't know of one off-hand but I could look it up) and that's what I understand when I read *left exact* in a non-abelian setting. (by the way: be sure to add an `@`-sign if you want me to be notified)2011-05-13
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    @Theo: (1) ok, (2) these definitions coincide for finitely complete categories, but preservation of limits is *not* about exactness; if a category does not have enough limits, then preserving existing limits says about nothing;2011-05-13
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    @Theo: the situation here is somehow similar to that with adjoint functors --- a functor from a complete category has a left adjoint if it preserves limits and satisfies some very mild conditions, but being right adjoint is about having a suitable “generalized inverse”, and *not* about preservation of limits; in line with this intuition exactness is about being a *finitary approximation* of an adjoint functor.2011-05-13
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    @Fallen: Thank you very much, that makes a lot of sense. I admit that I have never thought about these things this way but it seems to be an extremely good point. Also, I'm probably a bit spoiled by the fact that most of the categories I care about usually are finitely bicomplete.2011-05-14

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