nLab
closed monoidal (infinity,1)-category

Context

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

Monoidal categories

Contents

Definition

Definition

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

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

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

(X()[X,]):C[X,]X(). (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)