nLab closed monoidal (infinity,1)-category

Definition

Definition

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

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

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

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

(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

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

• The (∞,1)-category of (∞,1)-modules over an E-∞ ring is closed.

Related concepts

• monoidal category, monoidal (∞,1)-category

• symmetric monoidal category, symmetric monoidal (∞,1)-category

• closed monoidal category, closed monoidal $(\infty,1)$-category

• trace-class map