I am a bit confused regarding the definition of a split exact sequence, whose definition is for example available here (http://ncatlab.org/nlab/show/split+exact+sequence). Let's work in an abelian category, and suppose to have a s.e.s. $0 \to B \xrightarrow{f} A \oplus{ B} \xrightarrow{g} A \to 0$ where the maps $f$ and $g$ are not supposed to be the inclusion and the projection. Does the sequence split?
I have tried using the universal property of direct sum...but...mmm...I feel as I am missing something. Please, could you help me?
Thank you in advance! Cheers
A counterexample follows in the category of abelian groups. Let $A=\bigoplus_{n=0}^{\infty} \Bbb{Z}/2 \Bbb{Z}$, $B = \Bbb{Z}$.
Define $g: B \oplus A \longrightarrow A$ to be the map $$(z, x_0, x_1,x_2, \dots) \mapsto (z \bmod{2}, x_0, x_1, x_2, \dots)$$ and $f: B \longrightarrow B \oplus A$ to be the map $$z \mapsto (2z, 0, 0, 0, \dots) $$
Then $0 \to B \to B \oplus A \to A \to 0$ is exact and does not split.