nLab smothering functor

Idea

The terms epivalence (Keller 1990 p. 18), detecting functor (Baues 1995) and smothering functor (Riehl & Verity 2015) refer to functors that are at least close to be “equivalences of categories” except that they 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.

Definition

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

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

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.

Related pages

• smothering 2-functor

• derivator

References

Use of the term epivalence:

• Bernhard Keller; p. 18 of: Chain Complexes and Stable Categories, Manus. Math. 67 (1990) 379–417 [doi:10.1007/BF02568439, pdf]

Use of the term “detecting functor”:

• Hans-Joachim Baues; p. 25 of: Homotopy types, in: Ioan Mackenzie James (ed.) Handbook of Algebraic Topology, North Holland (1995) [doi:10.1016/B978-0-444-81779-2.X5000-7, ISBN:9780080532981, pdf]

Use of the term smothering functor:

• Emily Riehl, Dominic Verity: The 2-category theory of quasi-categories, Advances in Mathematics 280 (2015) 549–642 [arXiv:1306.5144]

Discussion of Examples:

• Claus Michael Ringel; section 1 of: On the representation dimension of Artin algebras [pdf]