empty functor to the category of sets
- Notation:
- Domain: empty category
- Codomain: category of sets
- Related functors:
Every category has a unique functor . Here, we specify , but most of the properties do not depend on the choice of , as long as is non-empty. This is the simplest example of a functor to that is both continuous and cocontinuous, but is neither representable nor a left or right adjoint.
Satisfied Properties
Assigned properties
- is fully faithful
- is continuous
- is cocontinuous
Deduced properties
- is cofinitary
- is left exact
- preserves products
- preserves equalizers
- preserves coreflexive equalizers
- is faithful
- is full
- is conservative
- is finitary
- preserves coproducts
- is right exact
- preserves coequalizers
- preserves reflexive coequalizers
- preserves finite products
- is exact
- preserves monomorphisms
- preserves regular monomorphisms
- is full on isomorphisms
- preserves finite coproducts
- preserves epimorphisms
- preserves regular epimorphisms
- preserves binary products
- preserves terminal objects
- is regular
- is essentially injective
- is pseudomonic
- preserves binary coproducts
- preserves initial objects
- is coregular
Unsatisfied Properties
Assigned properties
- is not left-invertible
- is not dominant
- is not representable
Deduced properties*
- is not an equivalence
- is not essentially surjective
- is not a right adjoint
- is not a left adjoint
- is not a reflector
- is not an isomorphism
- is not right-invertible
- is not monadic
- is not a coreflector
- is not comonadic
*This also uses the deduced satisfied properties.
Unknown properties
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Indistinguishable functors
These functors in the database currently have exactly the same properties as the empty functor to the category of sets. This indicates that the data may be incomplete or that a distinguishing property may be missing from the database.