# Smothering functor

## Idea

A smothering functor is a functor that is “almost an equivalence of categories” except that it may not be faithful. Smothering functors tend to arise when comparing “different homotopy categories” of the same category that impose more or less refined notions of homotopy equivalence. Frequently they can be treated more or less like equivalences.

The notion is due to Riehl and Verity.

## Definition

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

If instead of being surjective on objects $F$ is essentially surjective on objects, we may say that $F$ is weakly smothering.

## Properties

• Each (strict) fiber of a smothering functor is an inhabited connected groupoid. (RV, 3.3.2)
• If $F:C\to D$ is smothering and $x,y\in C$ satisfy $F x \cong F y$ in $D$, then $x \cong y$ in $C$, by fullness combined with conservativity. In other words, $F$ is “full on isomorphisms” (but since it is not faithful, it is not pseudomonic).
• Further combining this with surjectivity on objects, every smothering functor is an isofibration.

## Examples

• For any model category $C$, the functor $Ho(C^{\mathbf{2}}) \to Ho(C)^{\mathbf{2}}$ is weakly smothering, where $\mathbf{2}$ denotes the interval category. This property appears in the axiom (Der5) for derivators.