CatDat

coproduct-preserving

A functor $F$ preserves coproducts when for every family of objects $(A_i)$ in the source whose coproduct $\prod_i A_i$ exists, also the coproduct $\coprod_i F(A_i)$ exists in the target and such that the canonical morphism $\coprod_i F(A_i) \to F(\coprod_i A_i)$ is an isomorphism.

• Dual property: product-preserving

Relevant implications

• cocontinuous implies coequalizer-preserving andcoproduct-preserving andfinitary
• coequalizer-preserving andcoproduct-preserving implies cocontinuous *
• coproduct-preserving implies finite-coproduct-preserving
• finitary andfinite-coproduct-preserving implies coproduct-preserving *

*Those implications also require assumptions on the source or target category.

Examples

There are 3 functors with this property.

• abelianization functor for groups
• free group functor
• identity functor on the category of sets

Counterexamples

There are 3 functors without this property.

• contravariant power set functor
• covariant power set functor
• forgetful functor for vector spaces

Unknown

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

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