Notation 7.2.1. Let \(C\) be a semiadditive \(\infty \)-category. We saw in Proposition 5.3.25 that the hom anima \(\Hom _C(X,Y)\) admits a canonical structure of a commutative monoid for all \(X,Y \in C\). In particular, there is for any \(n \in \N \) a multiplication-by-\(n\) map \[ n \cdot -\colon \Hom _C(X,Y) \to \Hom _C(X,Y), \] given explicitly by sending a map \(f\colon X \to Y\) to the composite \[ X \xrightarrow {\Delta } \bigoplus _n X \xrightarrow {\bigoplus _n f} \bigoplus _n Y \xrightarrow {\nabla } Y. \] For \(f = \id _X \colon X \to X\), we refer to this map as multiplication by \(n\) on \(X\), and denote it by \[ n\colon X \to X. \]
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