additive and abelian categories
(AB1) pre-abelian category
(AB2) abelian category
(AB5) Grothendieck category
left/right exact functor
equivalences in/of $(\infty,1)$-categories
If $k$ is a commutative ring, $k$-linear (infinity,1)-categories are the analogue in (∞,1)-category theory of the notion of $k$-linear category in category theory.
A $k$-linear (infinity,1)-category is an additive (infinity,1)-category $A$ whose homotopy category $ho(A)$ is a $k$-linear category.
More generally, let $R$ be a commutative ring spectrum and let $Mod(R)$ denote the symmetric monoidal (infinity,1)-category of modules over it. An $R$-linear (infinity,1)-category is an object of the (infinity,1)-category of modules over $Mod(R)$ in the symmetric monoidal (infinity,1)-category of (infinity,1)-categories.
An $R$-linear (infinity,1)-category is naturally enriched over the symmetric monoidal (infinity,1)-category of modules over $R$.
Section 6 of
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