Indexed Biproduct

An indexed biproduct of an $I$-indexed family $X_{i}:I\to\mathcal{C}$ is an object $\bigoplus_{i:I}X_{i}$ equipped with maps $\pi_{i}:\bigoplus_{i:I}X_{i}\to X_{i}$ and $\iota_{i}:X_{i}\to\bigoplus_{i:I}$ such that

• $(\bigoplus_{i:I}X_{i},\pi_{i})$ is an Indexed Cartesian Product

• $(\bigoplus_{i:I}X_{i},\iota_{i})$ is an Indexed Cocartesian Coproduct

• $\pi_{i}\circ\iota_{i}=mathrm{id}{ifx..lse\par i}$ for all $i:I$

• For all $i,j:I$, the octagon axiom holds:

Properties

• If $\mathcal{C}$ has indexed biproducts, then we can define an morphism $\delta_{i,j}=\pi_{i}\circ\iota_{j}$ that acts like the Kronecker Delta.