It's known that a module P is projective if and only if every short exact $0\rightarrow M\rightarrow N \rightarrow P\rightarrow 0$ splits, and I'm wondering if this holds true for general Abelian categories.
P is projective implies the sequence splits is obvious. For the converse, I tried to substitute N's with direct sums involving P, but got stucked here.
Yes. Do you have in hands the concept of pullback in a category? Consider a diagram $$\begin{array} & &&p\\ &&\downarrow\\ a & \rightarrow & b&\rightarrow &0 \end{array}$$ in an abelian category. Let $$\begin{array} &c & \xrightarrow{f} & p & \rightarrow & 0 \\ \downarrow & & \downarrow \\ a & \rightarrow & b & \rightarrow & 0\end{array}$$
be the pullback of the initial diagram. Therefore there exists exact sequence $0\rightarrow k\rightarrow c\xrightarrow{f}p\rightarrow0$ so it splits. Let $g$ an arrow (morphism) such that $f\circ g=1_p$. It is clear that $p\xrightarrow{g}c\rightarrow a$ factors $p\rightarrow b$ as desired.