CatDat

product-preserving

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

• Dual property: coproduct-preserving

Relevant implications

• cofinitary andfinite-product-preserving implies product-preserving *
• continuous implies cofinitary andequalizer-preserving andproduct-preserving
• equalizer-preserving andproduct-preserving implies continuous *
• product-preserving implies finite-product-preserving

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

Examples

There are 3 functors with this property.

• contravariant power set functor
• forgetful functor for vector spaces
• identity functor on the category of sets

Counterexamples

There are 3 functors without this property.

• abelianization functor for groups
• covariant power set functor
• free group functor

Unknown

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

—