# Smothering 2-functors

## Idea

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

## Definition

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

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

## Properties

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

• Applying fullness on 1-cells again, if $x,y\in C$ and $F x \simeq F y$ in $D$, then $x\simeq y$ in $C$.

• Smothering 2-functors lift and reflect adjoints: any adjunction in $D$ is the image of some adjunction in $C$, 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? $K$, there is a smothering 2-functor $(K/A)_2 \to K_2/A$, where $K_2$ denotes the homotopy 2-category of an ∞-cosmos?. In the special case when $K=QCat$ consists of quasi-categories, this is RV, 3.4.7.