Biproduct

Let $𝖢$ be a category with zero morphisms. A biproduct $𝐴$ of a finite collection ${𝐴 𝑖} 𝑛 𝑖 = 1$ of objects in $𝖢$ is simultaneously a product and coproduct in a compatible way. #m/def/cat If $𝜋 𝑖 : 𝐴 ↠ 𝐴 𝑖$ and $𝜄 𝑖 : 𝐴 𝑖 ↪ 𝐴$ denote the product projections and coproduct inclusions respectively, we require

𝜋 𝑖 𝜄 𝑗 = {1 𝐴 𝑖 𝑖 = 𝑗 0 𝑖 \neq 𝑗

for all $𝑖, 𝑗 \in ℕ 𝑛$ . A monoidal category whose tensor product is a binary biproduct is called a Bicartesian category.

#state/tidy | #lang/en | #SemBr