CatDat

trivial

A symmetric monoidal category is trivial when it is equivalent to the terminal symmetric monoidal category. This is equivalent to the condition that its underlying category is trivial.

• Dual property: trivial (self-dual)

Relevant implications

• cartesian andcodistributive implies trivial
• cocartesian anddistributive implies trivial
• trivial implies cartesian andclosed andcocomplete andself-dual
• trivial implies cocartesian andcoclosed andcomplete andself-dual
• trivial implies well-pointed

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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