CatDat

coclosed

A symmetric monoidal category is called coclosed when, for every object $A$, the endofunctor $- \otimes A$ has a left adjoint. This property is not very common, but we have included it because of its dual property.

• Dual property: closed

Relevant implications

• closed andself-dual implies coclosed
• cocartesian () implies coclosed
• coclosed andself-dual implies closed
• coclosed () implies codistributive
• coclosed () implies complete
• coclosed () implies finitely complete
• coclosed () implies infinitary codistributive
• trivial implies coclosed

Examples

There are 3 symmetric monoidal categories with this property.

• symmetric monoidal category of finite-dimensional vector spaces
• symmetric monoidal poset of natural numbers
• trivial symmetric monoidal category

Counterexamples

There are 9 symmetric monoidal categories without this property.

• cartesian symmetric monoidal category of sets
• cartesian symmetric monoidal category of small categories
• cartesian symmetric monoidal category of topological spaces
• cocartesian symmetric monoidal category of sets
• symmetric monoidal category of abelian groups
• 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

Unknown

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

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