# nLab closed monoidal (infinity,1)-category

Contents

### Context

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

(∞,1)-category theory

Background

Basic concepts

Universal constructions

Local presentation

Theorems

Extra stuff, structure, properties

Models

#### Monoidal categories

monoidal categories

# Contents

## Definition

###### Definition

A symmetric monoidal (∞,1)-category $(C,\otimes)$ is closed if for each object $X \in C$ the (∞,1)-functor

$X \otimes (-) : C \to C$

given by forming the tensor product with $C$ has a right adjoint (∞,1)-functor

$(X \otimes(-)\dashv [X,-] ) : C \stackrel{\overset{X \otimes (-)}{\leftarrow}}{\underset{[X,-]}{\to}} C \,.$

(In cases where the monoidal structure is not assumed symmetric, the property of possessing a right adjoint to tensoring on the left (resp. right) is called left (resp. right) closed, while closed is used for the properties jointly (Def. Definition 4.1.1.15 of Higher Algebra).)

## Examples

Last revised on May 26, 2022 at 18:07:59. See the history of this page for a list of all contributions to it.