binary product functor on sets
- notation:
- Source: category of pairs of sets
- Target: category of sets
- Left adjoint functor:
- Related functors: ,
This functor maps a pair of sets to their product . It is an example of a right-invertible right adjoint functor which is not a coreflector.
Satisfied Properties
Assigned properties
- is right-invertible
- preserves initial objects
- is representable
- is finitary
- preserves reflexive coequalizers
Deduced properties
- is continuous
- is essentially surjective
- preserves regular epimorphisms
- is a right adjoint
- is cofinitary
- is left exact
- preserves products
- is dominant
- preserves epimorphisms
- preserves finite products
- preserves equalizers
- preserves monomorphisms
- is regular
- preserves binary products
- preserves terminal objects
- preserves coreflexive equalizers
- preserves regular monomorphisms
Unsatisfied Properties
Assigned properties
- is not full
- is not faithful
- is not essentially injective
- does not preserve coequalizers
- is not a coreflector
Deduced properties*
- is not fully faithful
- is not conservative
- is not left-invertible
- is not full on isomorphisms
- is not pseudomonic
- is not monadic
- is not an equivalence
- is not right exact
- does not preserve binary coproducts
- is not comonadic
- is not an isomorphism
- is not exact
- is not cocontinuous
- does not preserve finite coproducts
- is not coregular
- is not a left adjoint
- does not preserve coproducts
- is not a reflector
*This also uses the deduced satisfied properties.
Unknown properties
—