CatDat

preserves binary products

A functor $F : \C \to \D$ preserves binary products when for every pair of objects $A,B \in \C$ whose product $A \times B$ exists, also the product $F(A) \times F(B)$ exists and such that the canonical morphism $F(A \times B) \to F(A) \times F(B)$ is an isomorphism.

• Dual property: preserves binary coproducts
• Related properties: continuous, preserves finite products, preserves products
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Relevant implications

• preserves binary products andpreserves coreflexive equalizers () implies preserves equalizers
• preserves binary products andpreserves terminal objects () is equivalent to preserves finite products
• preserves finite products implies preserves binary products

Examples

There are 30 functors with this property.

• abelianization functor for groups
• binary diagonal functor on the category of sets
• binary product functor on sets
• contravariant power set functor
• discrete topology functor
• enveloping group functor
• forgetful functor for groups
• forgetful functor for rings
• forgetful functor for topological spaces
• forgetful functor for vector spaces
• forgetful functor from abelian groups to groups
• forgetful functor from groups to monoids
• forgetful functor from groups to pointed sets
• forgetful functor from rings to monoids
• functor of continuous functions
• fundamental group functor
• group of units functor
• identity functor on the category of sets
• indiscrete topology functor
• modulo p functor
• opposite category functor
• opposite monoid functor
• p-torsion functor
• path components functor
• sequences functor on sets
• squaring functor on sets
• torsion functor
• trivial functor from the category of groups
• trivial functor from the category of sets
• walking isomorphism object inclusion

Counterexamples

There are 6 functors without this property.

• binary coproduct functor on sets
• countable copower functor on sets
• covariant power set functor
• doubling functor on sets
• free group functor
• monoid ring functor

Unknown

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

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