# Limits, Exact Sequences and Conservative Functors

Is there any relation between limits and exact sequences?

In particular, a conservative functor reflects limits. How does that imply that it reflects exact sequences?

Edited: Having clarified the relation between conservative functors and preservation/reflection of limits, i am now looking for a description of exact sequences in terms of limits and colimits.

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Exactness for functors between arbitrary categories is defined in terms of limits and colimits: if $A$ is a category with finite limits (resp. colimits), a functor $F:A\rightarrow B$ is left (resp. right) exact if it preserves finite limits (resp. colimits). When $A$ and $B$ are abelian, left or right exactness in this sense implies left or right exactness in the sense of preserving short exact sequences (including additivity), and conversely, if $F$ is additive, then the definitions of left and right exactness in terms of short exact sequences are equivalent to preservation of finite limits – Keenan Kidwell Nov 7 '12 at 22:43
(resp. colimits). – Keenan Kidwell Nov 7 '12 at 22:43
It is not true that a conservative functor reflects limits. Rather, a conservative functor that preserves limits will also reflect limits. The same is true for colimits. Thus a conservative functor that preserves exact sequences will also reflect exact sequences, because being an exact sequence is a property that can be expressed in terms of limits and colimits. – Zhen Lin Nov 8 '12 at 0:37
@KeenanKidwell: Is there an accessible reference treating this topic of exact sequences described as limits and of conservative functors? I have not been able to find one. Thanks. – Manos Nov 8 '12 at 3:13
I'm not familiar with the term conservative functor, but the general definition of exactness and the equivalence with the usual notion for abelian categories can be found in the Stacks Project. The relevant chapters are Categories and Homological Algebra. – Keenan Kidwell Nov 8 '12 at 3:23