CatDat

self-dual

A symmetric monoidal category $(\C,\otimes,1)$ is self-dual when it is equivalent to its dual symmetric monoidal category $(\C^{\op},\otimes,1)$.

• Dual property: self-dual (self-dual)

Relevant implications

• cartesian andself-dual implies cocartesian
• closed andself-dual implies coclosed
• cocartesian andself-dual implies cartesian
• coclosed andself-dual implies closed
• cocomplete andself-dual implies complete
• codistributive andself-dual implies distributive
• complete andself-dual implies cocomplete
• distributive andself-dual implies codistributive
• finitely cocomplete andself-dual implies finitely complete
• finitely complete andself-dual implies finitely cocomplete
• infinitary codistributive andself-dual implies infinitary distributive
• infinitary distributive andself-dual implies infinitary codistributive
• trivial implies self-dual
• trivial implies self-dual

Examples

There are 2 symmetric monoidal categories with this property.

• symmetric monoidal category of finite-dimensional vector spaces
• 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
• cocartesian symmetric monoidal category of sets
• symmetric monoidal category of abelian groups
• 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.

—