nLab
closed monoidal (infinity,1)-category

Contents

Context

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

Monoidal categories

monoidal categories

With symmetry

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,)(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 4.1.1.15 of Higher Algebra).)

Examples

Last revised on November 1, 2017 at 09:07:30. See the history of this page for a list of all contributions to it.