nLab
smothering 2-functor

Smothering 2-functors

Idea

A smothering 2-functor is the 2-categorical analogue of a smothering functor.

Definition

A 2-functor F:CDF:C\to D is smothering if

  1. It is surjective on objects, and
  2. It is locally smothering, i.e. each hom-functor C(x,y)D(Fx,Fy)C(x,y) \to D(F x, F y) is a smothering functor. Thus, FF is full on 1-cells, full on 2-cells, and conservative on 2-cells.

Properties

  • Smothering 2-functors reflect equivalences: if f:xyf:x\to y is in CC and F(f)F(f) is an equivalence, then ff is an equivalence. For by fullness on 1-cells, any inverse of F(f)F(f) lifts to CC, by fullness on 2-cells, the comparison 2-cells lift to CC, and then by conservativity those lifts are again isomorphisms.

  • Applying fullness on 1-cells again, if x,yCx,y\in C and FxFyF x \simeq F y in DD, then xyx\simeq y in CC.

  • Smothering 2-functors lift and reflect adjoints: any adjunction in DD is the image of some adjunction in CC, 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).

Examples

  • For any ∞-cosmos? KK, there is a smothering 2-functor (K/A) 2K 2/A(K/A)_2 \to K_2/A, where K 2K_2 denotes the homotopy 2-category of an ∞-cosmos?. In the special case when K=QCatK=QCat consists of quasi-categories, this is RV, 3.4.7.

References

Created on April 13, 2016 13:22:54 by Mike Shulman (192.195.154.58)