Definition 22.24.1 (09MJ)—The Stacks project

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Definition 22.24.1. Let $R$ be a ring. An $R$ -linear category $\mathcal{A}$ is a category where every morphism set is given the structure of an $R$ -module and where for $x, y, z \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ composition law

$\mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(y, z) \times \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(x, y) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(x, z)$

is $R$ -bilinear.

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