CatDat

finitely cocomplete

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

• Dual property: finitely complete
• Related properties: cocomplete

Relevant implications

• closed () implies finitely cocomplete
• cocomplete implies finitely cocomplete
• finitely cocomplete andself-dual implies finitely complete
• finitely cocomplete implies distributive
• finitely complete andself-dual implies finitely cocomplete

Examples

There are 9 symmetric monoidal categories with this property.

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

Counterexamples

There are 3 symmetric monoidal categories without this property.

• cartesian symmetric monoidal category of topological spaces
• cocartesian symmetric monoidal category of sets
• 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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