CatDat

codistributive

A symmetric monoidal category is called codistributive when its underlying category has finite products and, for every object $A$, the endofunctor $A \otimes -$ preserves finite products. There should be no confusion with the closely related notion of a codistributive category as long as we distinguish carefully between a symmetric monoidal category and its underlying category.

• Dual property: distributive
• Related properties: finitely complete, infinitary codistributive

Relevant implications

• cartesian andcodistributive implies trivial
• cocartesian () implies codistributive
• coclosed () implies codistributive
• codistributive andself-dual implies distributive
• codistributive () implies distributive
• distributive andself-dual implies codistributive
• distributive () implies codistributive
• finitely complete implies codistributive
• infinitary codistributive implies codistributive

Examples

There are 7 symmetric monoidal categories with this property.

• 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 5 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 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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