nLab linear (∞,1)-category

Idea

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.

Definition

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.

Properties

An $R$-linear (infinity,1)-category is naturally enriched over the symmetric monoidal (infinity,1)-category of modules over $R$.

Related concepts

• additive (∞,1)-category
• stable (∞,1)-category

References

Section 6 of

• Jacob Lurie, Spectral Schemes.