nLab
smothering functor

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:CDF:C\to D is smothering if it is

  1. surjective on objects,
  2. full, and
  3. conservative.

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

Properties

  • Each (strict) fiber of a smothering functor is an inhabited connected groupoid. (RV, 3.3.2)
  • If F:CDF:C\to D is smothering and x,yCx,y\in C satisfy FxFyF x \cong F y in DD, then xyx \cong y in CC, by fullness combined with conservativity. In other words, FF 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 CC, the functor Ho(C 2)Ho(C) 2Ho(C^{\mathbf{2}}) \to Ho(C)^{\mathbf{2}} is weakly smothering, where 2\mathbf{2} denotes the interval category. This property appears in the axiom (Der5) for derivators.

References

Last revised on August 17, 2017 at 12:27:48. See the history of this page for a list of all contributions to it.