CatDat

cocartesian

A symmetric monoidal category is called cocartesian when its underlying category has finite coproducts and the symmetric monoidal structure is induced by these finite coproducts; that is, $A \otimes B = A \sqcup B$, and so on.

• Dual property: cartesian
• nLab Link

Relevant implications

• cartesian andself-dual implies cocartesian
• cocartesian anddistributive implies trivial
• cocartesian andself-dual implies cartesian
• cocartesian () implies coclosed
• cocartesian () implies codistributive
• cocartesian () implies infinitary codistributive
• trivial implies cocartesian

Examples

There are 2 symmetric monoidal categories with this property.

• cocartesian symmetric monoidal category of sets
• trivial symmetric monoidal category

Counterexamples

There are 10 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
• 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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