# nLab linear (∞,1)-category

## Derived categories

#### $(\infty,1)$-Category theory

(∞,1)-category theory

# Contents

## 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 $R$.

## References

Section 6 of

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