Binary Biproduct

A binary biproduct of $X$ and $Y$ in a Category $\mathcal{C}$ is an object $X\oplus Y$ equipped with maps $X\xleftarrow{\pi_{1}}X\oplus Y\xrightarrow{\pi_{2}}Y$ and $X\xrightarrow{\iota_{1}}X\oplus Y\xleftarrow{\iota_{2}}Y$ such that

• $(X\oplus Y,\pi_{1},\pi_{2})$ is a Binary Cartesian Product of $X$ and $Y$ in $\mathcal{C}$

• $(X\oplus Y,\iota_{1},\iota_{2})$ is a Binary Cocartesian Coproduct of $X$ and $Y$ in $\mathcal{C}$

• $\pi_{1}\circ\iota_{1}=mathrm{id}{ifx..lse\par i}$

• $\pi_{2}\circ\iota_{2}=mathrm{id}{ifx..lse\par i}$

• The following octagon axiom holds:

  \[

  \begin{tikzcd}

& {X ⊕ Y}
Y && X
{X ⊕ Y} && {X ⊕ Y}
X && Y
& {X ⊕ Y} \end{tikzcd} \]

Examples

Properties

• If $\mathcal{C}$ has a Zero Object $0$, then the maps $\pi_{2}\circ\iota_{1}:X\to Y$ and $\pi_{1}\circ\iota_{2}:Y\to X$ are equal to the zero map. This sheds some light on the octagon axiom: if we examine two sides, we see that the left-hand side includes $\pi_{1}\circ\iota_{2}$ and the right hand side includes $\pi_{2}\circ\iota_{1}$; the axiom essentially encodes what the zero morphism would let you do without actually requiring a full-blown zero object!