CatDat

finitely complete

A symmetric monoidal category is called finitely complete when its underlying category is finitely complete and, for every object $A$, the endofunctor $A \otimes -$ preserves finite limits (that is, it is left exact). Of course, then the endofunctor $- \otimes A$ also preserves finite limits. There is no need for a more complicated term such as "symmetric monoidally finitely complete" if one is careful to distinguish between a symmetric monoidal category and its underlying category.

• Dual property: finitely cocomplete
• Related properties: complete

Relevant implications

• coclosed () implies finitely complete
• complete implies finitely complete
• finitely cocomplete andself-dual implies finitely complete
• finitely complete andself-dual implies finitely cocomplete
• finitely complete implies codistributive

Examples

There are 3 symmetric monoidal categories with this property.

• symmetric monoidal category of finite-dimensional vector spaces
• symmetric monoidal category of modules over an absolutely flat commutative ring
• trivial symmetric monoidal category

Counterexamples

There are 8 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 non-absolutely flat commutative ring
• symmetric monoidal poset of natural numbers

Undecidable symmetric monoidal categories

There is 1 symmetric monoidal category for which it cannot be decided if this property is satisfied or not.

• symmetric monoidal category of modules over a 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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