# nLab closed monoidal (infinity,1)-category

### Context

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

(∞,1)-category theory

## Models

#### Monoidal categories

monoidal categories

## With traces

• trace

• traced monoidal category?

# Contents

## Definition

###### Definition

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

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

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

$\left(X\otimes \left(-\right)⊣\left[X,-\right]\right):C\stackrel{\stackrel{X\otimes \left(-\right)}{←}}{\underset{\left[X,-\right]}{\to }}\phantom{\rule{thinmathspace}{0ex}}.$(X \otimes(-)\dashv [X,-] ) : C \stackrel{\overset{X \otimes (-)}{\leftarrow}}{\underset{[X,-]}{\to}} \,.

## Examples

Every (∞,1)-topos with its structure of a cartesian monoidal (∞,1)-category is closed. See there for details.

Created on November 23, 2010 21:51:35 by Urs Schreiber (87.212.203.135)