It seems to me that there are (at least) two notions of exact sequences in a category:
1) Let $\mathcal{C}$ be a pointed category with kernels and images. Then we call a complex (i.e. the composite of two adjacent maps is zero) $$\dotsc \to A_0 \to A_1 \to A_2 \to \dotsc$$ exact if the canonical map $\mathrm{im}(A_i \to A_{i+1}) \to \ker(A_{i+1} \to A_{i+2})$ is an isomorphism for all $i$.
2) Let $\mathcal{C}$ be a pointed category with kernels and cokernels. Then we call a complex
$$0 \to A \to B \to C \to 0$$
exact if $A \to B$ is a kernel of $B \to C$ and $B \to C$ is a cokernel of $A \to B$.
(There is also the notion of an exact category, but my question is not about such extra structures on a category.)
Surely 1) is well-known, and 2) appears as Definition 4.1.5 in Mal'cev, protomodular, homological and semi-abelian categories by Borceux and Bourn (are there other references?). I think that 2) generalizes to long exact sequences in a natural way, because we can slice them into short exact sequences and take this as a definition; is there a reference for this? But my actual question is about the equivalence between 1) and 2).
It is well-known that 1) and 2) are equivalent for abelian categories, but 2) is stronger than 1) in the category of pointed sets. Namely,
$$0 \to (X,x_0) \xrightarrow{f} (Y,y_0) \xrightarrow{g} (Z,z_0) \to 0$$
is exact in the sense of 2) if and only if $f$ is injective, $g$ is surjective, and $\mathrm{im}(f)=\ker(g)$, whereas for 1) we only need that $f$ is weakly injective, meaning that $\ker(f)=0$.
I have to admit that this appears to be a bit strange for me. In particular, it means that 1) is not a self-dual notion, since $(Y,y_0) \xrightarrow{g} (Z,z_0) \to 0$ is exact iff $g$ is surjective iff $g$ is an epimorphism, but $0 \to (X,x_0) \xrightarrow{f} (Y,y_0)$ is exact with 1) iff $f$ is weakly injective, which does not imply that $f$ is a monomorphism, i.e. injective.
So why is 1) the standard definition of an exact sequence of pointed sets? Actually I only know one prominent example, namely the long exact sequence of homotopy groups (sets in degree $0$) associated to a fibration, but this doesn't start with $0$; it ends with $0$, so that there is no problem. Are there examples which show that 1) is more useful than 2) for pointed sets? (If this is true at all.)
Best Answer
The basic reason for the appearance of 1) in the long exact sequence in homotopy is that it is exactly the kind of exactness you get if you apply $\pi_0$ to a fiber sequence $F \to E \to B$ of pointed spaces. You simply don't get 2) because $F$ cannot see what happens outside of the connected component of the basepoint of $B$.