## 22.24 Linear categories

Just the definitions.

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.

Thus composition determines an $R$-linear map

\[ \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(y, z) \otimes _ R \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(x, y) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(x, z) \]

of $R$-modules. Note that we do not assume $R$-linear categories to be additive.

Definition 22.24.2. Let $R$ be a ring. A *functor of $R$-linear categories*, or an *$R$-linear functor* is a functor $F : \mathcal{A} \to \mathcal{B}$ where for all objects $x, y$ of $\mathcal{A}$ the map $F : \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(x, y) \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {B}(F(x), F(y))$ is a homomorphism of $R$-modules.

## Comments (2)

Comment #6795 by PS on

Comment #6942 by Johan on