closed

A symmetric monoidal category is called closed when, for every object $A$, the endofunctor $- \otimes A$ has a right adjoint $[A,-]$. Thus, we have natural bijections $\Hom(B \otimes A,C) \cong \Hom(B,[A,C])$.

Relevant implications

• cartesian () implies closed
• closed andself-dual implies coclosed
• closed () implies cocomplete
• closed () implies distributive
• closed () implies finitely cocomplete
• closed () implies infinitary distributive
• coclosed andself-dual implies closed
• cocomplete () implies closed
• trivial implies closed

Examples

There are 9 symmetric monoidal categories with this property.

• cartesian symmetric monoidal category of sets
• cartesian symmetric monoidal category of small categories
• symmetric monoidal category of abelian groups
• symmetric monoidal category of finite-dimensional vector spaces
• symmetric monoidal category of finitely generated abelian groups
• symmetric monoidal category of modules over a commutative ring
• symmetric monoidal category of modules over a non-absolutely flat commutative ring
• symmetric monoidal category of modules over an absolutely flat commutative ring
• trivial symmetric monoidal category

Counterexamples

There are 3 symmetric monoidal categories without this property.

• cartesian symmetric monoidal category of topological spaces
• cocartesian symmetric monoidal category of sets
• symmetric monoidal poset of natural numbers

Unknown

There are 0 symmetric monoidal categories for which the database has no information on whether they satisfy this property.

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