I read on Ugo Bruzzo's Derived Functors and Sheaf Cohomology, in the paragraph below Definition 1.6, that in an abelian category $f:a\to a'$ is monic iff there is $l:a'\to a$ such that $l\circ f=1_a$. The "only if" doesn't seem true to me, as for example $\mathbb Z\to \mathbb Z$ defined by $1\mapsto 2$ is injective and not left-invertible in the category of abelian groups. Am I misinterpreting something, or did I miss something about the definitions in an abelian category? It seems strange that it is a mistake of the book, because it couldn't be just a typo, it is an entire paragraph. Thanks in advance
Monomorphisms in abelian categories need not be split. In fact, $A\to B$ is split if and only if the sequence $$0\to A \to B\to A/B\to 0$$ is split exact, which is the case if and only if $B = A\oplus A/B$, the first map is the inclusion of the coproduct and the second map the projection of the second factor. This is the splitting lemma. There wouldn't be a name for split exact sequences, if all short exact sequences would be split exact.
Since $\mathbb Z \neq \mathbb Z\oplus \mathbb Z/2\mathbb Z$, your example is correct. The sequence $$0\to \mathbb Z \xrightarrow{\cdot 2} \mathbb Z\to \mathbb Z/2\mathbb Z \to 0$$ is not split, but exact.
In a category of $R$-modules for $R$ a ring all monomorphisms are split if and only if the ring $R$ is semisimple.