nLab closed monoidal (infinity,1)-category

Context

$(\infty,1)$-Category theory

(∞,1)-category theory

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

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

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

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

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)