Biproduct

Remarks

In a preadditive category, an object $X$ with maps $p_k$, $i_k$ satisfying clause 1 and $i_1 \circ p_1 + i_2 \circ p_2 = \id_X$ is a biproduct, and every biproduct has this form. In particular, in a preadditive category every binary product, and every binary coproduct, is a biproduct.

Example: for abelian groups $A$ and $B$, the direct sum $A \oplus B$ with $p_1(a, b) = a$, $p_2(a, b) = b$, $i_1(a) = (a, 0)$, $i_2(b) = (0, b)$ is a biproduct. Clause 1 is immediate, and $i_1 p_1 + i_2 p_2$ sends $(a, b)$ to $(a, 0) + (0, b) = (a, b)$.

Non-example: in the category of groups, with the trivial group as zero object, the product $\Z \times \Z$ with its inclusions satisfies clauses 1 and 2 but not clause 3. If it were a coproduct, the maps $\Z \to F_2$ onto the two generators $a, b$ of the free group $F_2$ would extend to one homomorphism $\Z \times \Z \to F_2$. Its image would be abelian and contain $a$ and $b$, but $ab \ne ba$ in $F_2$. The coproduct in groups is the free product instead.