The reflection of an object in a category along a functor is a morphism in , playing the role of a would-be unit of an adjunction (which may not exist in general), satisfying that part of the universal property of the unit for that component. If a reflection of each object in along exists and is chosen then the left adjoint to does exists.
Similarly, one can define a coreflection along a functor by a morphism in playing the role of the would-be counit of an adjunction .
Let be a functor and an object of the category .
A reflection of along (synonym: universal arrow from to ) is a pair of
which is universal in the sense that for any object and morphism there is a unique such that . In other words, it is an initial object in the comma category .
In other words this means that a reflection is an adjoint relative to the functor which picks out .
Dually, a coreflection of along is a pair of
which is universal in the sense that for any and a morphism there is a unique such that .
In other words this means that a coreflection is a coadjoint relative to the functor which picks out .
If is a pseudofunctor among bicategories (homomorphism of bicategories) and an object of , then a biuniversal arrow from to is an object in and a morphism such that for every object in the functor given by , (where and is a 2-cell) is an equivalence of categories.
For the bicategorical case see for example around def. 9.4 in
Last revised on September 22, 2024 at 14:25:06. See the history of this page for a list of all contributions to it.