CatDat

cartesian

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

• Dual property: cocartesian
• nLab Link

Relevant implications

• cartesian andcodistributive implies trivial
• cartesian andself-dual implies cocartesian
• cartesian () implies closed
• cartesian () implies distributive
• cartesian () implies infinitary distributive
• cocartesian andself-dual implies cartesian
• trivial implies cartesian

Examples

There are 4 symmetric monoidal categories with this property.

• cartesian symmetric monoidal category of sets
• cartesian symmetric monoidal category of small categories
• cartesian symmetric monoidal category of topological spaces
• trivial symmetric monoidal category

Counterexamples

There are 8 symmetric monoidal categories without this property.

• 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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