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

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

(X()[X,]):C[X,]X()C. (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 4.1.1.17 of Higher Algebra).)

Examples

Revised on August 15, 2017 08:24:51 by David Corfield (129.12.18.176)