biproduct

Let \(\mathcal{C}\) be a pointed category.

Then a biproduct is a

Let \(\mathcal{C}\) be a category with a zero object \(0\). Suppose \(A_1,...,A_n\) is a finite (possibly empty) collection of objects \(A_i \in \mathrm{Ob}(\mathcal{C})\), a biproduct is an object \(A_1 \oplus ... \oplus A_n\) with morphisms

• \(\pi_k: A_1 \oplus ... \oplus A_n \rightarrow A_k\), called projections
• \(\iota_k: A_k \rightarrow A_1 \oplus ... \oplus A_n\), called embedding

such that:

• \(\pi_k \circ \iota_k = \mathrm{id}_{A_k}\)
• \(\pi_l \circ \iota_k = 0_{A_k \rightarrow A_l}\), the zero morphism, for \(k \neq l\)

and that

1. \((A_1 \oplus ... \oplus A_n, \pi_k)\) is the categorical product
2. \((A_1 \oplus ... \oplus A_n, \iota_k)\) is the categorical coproduct