# 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

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

# 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.