Definition 4.2.15 ((Semi)additive \(\infty \)-category). An \(\infty \)-category \(C\) is semiadditive if it has a zero object, finite products, finite coproducts, and the canonical map \[ \begin {psmallmatrix} \id _X & 0 \\ 0 & \id _Y \end {psmallmatrix} \colon X \sqcup Y \to X \times Y \] is an isomorphism. We write \(X \oplus Y\) for the resulting biproducts. We say that \(C\) is additive if, in addition, the shear map \[ \begin {psmallmatrix} \id _X & \id _X \\ 0 & \id _X \end {psmallmatrix}\colon X \oplus X \to X \oplus X \] is an equivalence.
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