CatDat

reflector

A functor $F : \C \to \D$ is a reflector if it is left adjoint to a functor $G : \D \to \C$ which is fully faithful. Hence, $G$ is equivalent to the inclusion of a full reflective subcategory. The condition that $G$ is fully faithful can also be expressed by the condition that the counit $\varepsilon : F \circ G \to \id_{\D}$ is an isomorphism (Prop. 3.4 at the nLab).

• Dual property: coreflector
• Related properties: left adjoint, right-invertible
• nLab Link

Relevant implications

• equivalence implies reflector
• reflector implies left adjoint andright-invertible
• reflector implies preserves terminal objects
• () implies reflector

Examples

There are 10 functors with this property.

• abelianization functor for groups
• enveloping group functor
• forgetful functor for topological spaces
• identity functor on the category of sets
• opposite category functor
• opposite monoid functor
• Stone-Čech compactification functor
• trivial functor from the category of groups
• trivial functor from the category of sets
• walking isomorphism object inclusion

Counterexamples

There are 46 functors without this property.

• binary coproduct functor on sets
• binary diagonal functor on the category of sets
• binary product functor on sets
• Brauer group functor
• contravariant power set functor
• countable copower functor on sets
• covariant power set functor
• discrete topology functor
• doubling functor on sets
• empty functor to the category of sets
• forgetful functor for groups
• forgetful functor for rings
• forgetful functor for vector spaces
• forgetful functor from abelian groups to groups
• forgetful functor from commutative rings to rings
• forgetful functor from finite abelian groups to abelian groups
• forgetful functor from finite groups to groups
• forgetful functor from finite sets to sets
• forgetful functor from groups to monoids
• forgetful functor from groups to pointed sets
• forgetful functor from Hausdorff spaces to topological spaces
• forgetful functor from rings to monoids
• forgetful functor from torsion abelian groups to abelian groups
• forgetful functor from torsion-free abelian groups to abelian groups
• free group functor
• functor of continuous functions
• fundamental group functor
• group of units functor
• inclusion functor from extended natural numbers to ordinal numbers
• indiscrete topology functor
• modulo p functor
• monoid ring functor
• morphism endpoints inclusion
• nerve functor
• p-torsion functor
• path components functor
• rational product functor
• ring idempotents functor
• sequences functor on sets
• simple-group probing functor
• span endpoints inclusion
• squaring functor on sets
• torsion functor
• trivial functor from the delooping
• trivial functor from the walking idempotent
• walking morphism representation

Unknown

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

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