binary diagonal functor on the category of sets
- notation:
- Source: category of sets
- Target: category of pairs of sets
- Left adjoint functor:
- Right adjoint functor:
- Related functors:
- nLab Link
Every category has a (binary) diagonal functor , . Here, we specify that is the category of sets.
Satisfied Properties
Assigned properties
- is left-invertible
- is a left adjoint
- is a right adjoint
Deduced properties
- is continuous
- is conservative
- is essentially injective
- is faithful
- is cocontinuous
- is cofinitary
- is left exact
- preserves products
- is finitary
- preserves coproducts
- is right exact
- preserves finite products
- is exact
- preserves equalizers
- preserves monomorphisms
- preserves finite coproducts
- preserves coequalizers
- preserves epimorphisms
- preserves binary products
- preserves terminal objects
- preserves coreflexive equalizers
- preserves regular monomorphisms
- preserves binary coproducts
- preserves initial objects
- preserves reflexive coequalizers
- preserves regular epimorphisms
- is regular
- is monadic
- is coregular
- is comonadic
Unsatisfied Properties
Assigned properties
- is not full on isomorphisms
- is not dominant
Deduced properties*
- is not full
- is not pseudomonic
- is not essentially surjective
- is not an equivalence
- is not fully faithful
- is not right-invertible
- is not a reflector
- is not an isomorphism
- is not a coreflector
- is not representable
*This also uses the deduced satisfied properties.
Unknown properties
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