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

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

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

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

Related concepts

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

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

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