# On a Characterization of Exact Functors

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? – t.b. May 12 '11 at 11:53
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. – Michal R. Przybylek May 12 '11 at 21:14
@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. – t.b. May 12 '11 at 22:27
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, – Matt E May 13 '11 at 0:22
@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. – Michal R. Przybylek May 13 '11 at 23:46