A smothering 2-functor is the 2-categorical analogue of a smothering functor.
A 2-functor is smothering if
Smothering 2-functors reflect equivalences: if is in and is an equivalence, then is an equivalence. For by fullness on 1-cells, any inverse of lifts to , by fullness on 2-cells, the comparison 2-cells lift to , and then by conservativity those lifts are again isomorphisms.
Applying fullness on 1-cells again, if and in , then in .
Smothering 2-functors lift and reflect adjoints: any adjunction in is the image of some adjunction in , and the lift can make use of any specified lifts of the objects, 1-cells, and either the unit or the counit 2-cell. This is RV, 4.5.2; the idea is to lift the missing 2-cell arbitrarily and then “fix” it by composing with the inverse of one triangle identity composite (which is an isomorphism by 2-cell conservativity).
Created on April 13, 2016 at 17:21:04. See the history of this page for a list of all contributions to it.