CatDat

infinitary codistributive

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

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

Relevant implications

• cocartesian () implies infinitary codistributive
• coclosed () implies infinitary codistributive
• complete implies infinitary codistributive
• infinitary codistributive andself-dual implies infinitary distributive
• infinitary codistributive implies codistributive
• infinitary distributive andself-dual implies infinitary codistributive

Examples

There is 1 symmetric monoidal category with this property.

• trivial symmetric monoidal category

Counterexamples

There are 11 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 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
• 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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