# Exact sequence in a category with zero morphisms

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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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 $1$).