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Let $C$ be a category with zero morphisms (equivalently, $\mathsf{Set}_*$-enriched), for example it could be a linear category. Then we can talk about kernels and cokernels of morphisms in $C$. I wonder if the following definition is already established and appears somewhere in the literature:

Definition: If $f : A \to B$ and $g : B \to C$ are morphisms, then $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$ is called exact if $f$ is a kernel of $g$ and $g$ is a cokernel of $f$.

Here, "$0 \to A$" and "$C \to 0$" are just notation; I don't require that a zero object exists.

The definition is well-known when $C$ is abelian (and simplifies a bit in that special case).

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up vote 3 down vote accepted

This appears as Definition 4.1.5 in [Borceux and Bourn, Mal'cev, protomodular, homological and semi-abelian categories]:

In a pointed category, a sequence of morphisms $$1 \longrightarrow K \stackrel{k}{\longrightarrow} A \stackrel{q}{\longrightarrow} Q \longrightarrow 1$$ is a short exact sequence when $k = \ker q$ and $q = \operatorname{coker} k$.

Note that their definition of pointed category includes a zero object (denoted by $1$).

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Great! Thanks a lot. I used this definition on my own in the last months (it will also appear in my thesis) and doubted that it is really new, but also wondered if it is "accepted" at all. It's just perfect that it appears in a well-reviewed published book. –  Martin Brandenburg Nov 29 '13 at 17:24
    
Unfortunately the definition is outside of the google and amazon preview. Do you know where I can read it online? –  Martin Brandenburg Nov 29 '13 at 17:28
    
I added the quote. –  Zhen Lin Nov 29 '13 at 17:32
    
Thank you. I also would like to read what they prove about these exact sequences (so that I may cite their results if I need them, instead of reproving them). –  Martin Brandenburg Nov 29 '13 at 17:33
    
It doesn't seem to be on SpringerLink, for some reason. I suppose it might be available elsewhere. –  Zhen Lin Nov 29 '13 at 17:37

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