CatDat

well-pointed

A symmetric monoidal category $(\C,\otimes,1)$ is called well-pointed if the monoidal unit $1$ is a generator. Equivalently, the functor $\Hom(1,-) : \C \to \Set^+$ is faithful. The terminology is taken from Categories for Quantum Theory by Heunen-Vicari, Def. 1.12.

Relevant implications

• trivial implies well-pointed
• () implies well-pointed

Examples

There are 10 symmetric monoidal categories with this property.

• cartesian symmetric monoidal category of sets
• 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
• trivial symmetric monoidal category

Counterexamples

There are 2 symmetric monoidal categories without this property.

• cartesian symmetric monoidal category of small categories
• cocartesian symmetric monoidal category of sets

Unknown

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

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