CatDat

strict

A symmetric monoidal category is called strict when its associator, symmetry, left unitor, and right unitor are identities. A strict symmetric monoidal category is simply a commutative monoid object in $(\Cat^+,\times,1)$. Warning: This property is not invariant under equivalences.

• Dual property: strict (self-dual)
• nLab Link

Relevant implications

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Examples

There are 2 symmetric monoidal categories with this property.

• symmetric monoidal poset of natural numbers
• 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 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

Unknown

There are 0 symmetric monoidal categories for which the database has no information on whether they satisfy this property.

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