#
nLab

linear (∞,1)-category

Contents
### Context

#### Additive and abelian categories

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

**(∞,1)-category theory**

## Background

## Basic concepts

## Universal constructions

## Local presentation

## Theorems

## Models

# 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

Last revised on January 25, 2015 at 15:55:43.
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