CatDat

cocomplete

A symmetric monoidal category is called cocomplete when its underlying category is cocomplete and, for every object $A$, the endofunctor $A \otimes -$ is cocontinuous. Of course, then the endofunctor $- \otimes A$ is also cocontinuous. There is no need for a more complicated term such as "symmetric monoidally cocomplete" if one is careful to distinguish between a symmetric monoidal category and its underlying category.

• Dual property: complete
• Related properties: finitely cocomplete, infinitary distributive
• nLab Link

Relevant implications

• closed () implies cocomplete
• cocomplete andself-dual implies complete
• cocomplete implies finitely cocomplete andinfinitary distributive
• complete andself-dual implies cocomplete
• trivial implies cocomplete

Examples

There are 7 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 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 5 symmetric monoidal categories without this property.

• cartesian symmetric monoidal category of topological spaces
• cocartesian symmetric monoidal category of sets
• symmetric monoidal category of finite-dimensional vector spaces
• symmetric monoidal category of finitely generated abelian groups
• 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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