Functor implications

Found 48 implications*

cofinitary andleft exact implies continuous
cofinitary andpreserves finite products implies preserves products
conservative andfull implies full on isomorphisms
conservative andpreserves equalizers implies faithful
conservative andpreserves reflexive coequalizers andright adjoint implies monadic
continuous implies cofinitary andleft exact andpreserves products
continuous implies right adjoint
equivalence is equivalent to left-invertible andright-invertible
equivalence implies monadic andreflector
essentially surjective andfully faithful is equivalent to equivalence
essentially surjective implies dominant
exact is equivalent to left exact andright exact
faithful andfull is equivalent to fully faithful
faithful implies conservative
full on isomorphisms implies essentially injective
fully faithful implies conservative
isomorphism implies equivalence
left exact andpreserves regular epimorphisms is equivalent to regular
left exact implies preserves equalizers andpreserves finite products andpreserves monomorphisms
left-invertible implies conservative andessentially injective andfaithful
monadic implies conservative andfaithful andright adjoint
preserves binary products andpreserves coreflexive equalizers implies preserves equalizers
preserves binary products andpreserves terminal objects implies preserves finite products
preserves coreflexive equalizers implies preserves regular monomorphisms
preserves equalizers andpreserves finite products implies left exact
preserves equalizers andpreserves products implies continuous
preserves equalizers implies preserves coreflexive equalizers andpreserves regular monomorphisms
preserves finite products implies preserves binary products andpreserves terminal objects
preserves finite products implies preserves finite coproducts
preserves monomorphisms implies preserves regular monomorphisms
preserves products implies preserves finite products
preserves regular monomorphisms implies preserves monomorphisms
preserves terminal objects implies preserves initial objects
pseudomonic is equivalent to faithful andfull on isomorphisms
reflector implies left adjoint andright-invertible
representable implies continuous
representable implies right adjoint
right adjoint implies continuous
right-invertible implies essentially surjective
implies conservative
implies continuous
implies coreflector
implies essentially injective
implies essentially surjective
implies full on isomorphisms
implies full
implies preserves equalizers
implies right-invertible

*Deductions from these implications are automatically incorporated into each functor whenever applicable. For instance, if a functor is identified as monadic, the property of being continuous is automatically inferred and added.

Moreover, implications are automatically dualized when the corresponding dual properties exist. For example, the statement that a right adjoint is continuous automatically implies that a left adjoint is cocontinuous.

For results that do not quite fit the implication model, content pages are used instead.