The Stacks project

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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View Definition 22.24.1 as pdf