### Pullbacks That Preserve Weak Equivalences

#### Posted by Mike Shulman

The following concept seems to have been reinvented a bunch of times by a bunch of people, and every time they give it a different name.

**Definition:** Let $C$ be a category with pullbacks and a class of weak equivalences. A morphism $f:A\to B$ is a *[insert name here]* if the pullback functor $f^\ast:C/B \to C/A$ preserves weak equivalences.

In a right proper model category, every fibration is one of these. But even in that case, there are usually more of these than just the fibrations. There is of course also a dual notion in which pullbacks are replaced by pushouts, and every cofibration in a left proper model category is one of those.

What should we call them?

The names that I’m aware of that have so far been given to these things are:

*sharp map*, by Charles Rezk. This is a dualization of the terminology*flat map*used for the dual notion by Mike Hopkins (I don’t know a reference, does anyone?). I presume that Hopkins’ motivation was that a ring homomorphism is flat if tensoring with it (which is the pushout in the category of commutative rings) is exact, hence preserves weak equivalences of chain complexes.However, “flat” has the problem of being a rather overused word. For instance, we may want to talk about these objects in the canonical model structure on $Cat$ (where in fact it turns out that every such functor is a cofibration), but flat functor has a very different meaning. David White has pointed out that “flat” would also make sense to use for the monoid axiom in monoidal model categories.

*right proper*, by Andrei Radulescu-Banu. This is presumably motivated by the above-mentioned fact that fibrations in right proper model categories are such. Unfortunately, proper map also has another meaning.*$h$-fibration*, by Berger and Batanin. This is presumably motivated by the fact that “$h$-cofibration” has been used by May and Sigurdsson for an intrinsic notion of cofibration in topologically enriched categories, that specializes in compactly generated spaces to closed Hurewicz cofibrations, and pushouts along the latter preserve weak homotopy equivalences. However, it makes more sense to me to keep “$h$-cofibration” with May and Sigurdsson’s original meaning.*Grothendieck $W$-fibration*(where $W$ is the class of weak equivalences on $C$), by Ara and Maltsiniotis. Apparently this comes from unpublished work of Grothendieck. Here I guess the motivation is that these maps are “like fibrations” and are determined by the class $W$ of weak equivalences.

Does anyone know of other references for this notion, perhaps with other names? And any opinions on what the best name is? I’m currently inclined towards “$W$-fibration” mainly because it doesn’t clash with anything else, but I could be convinced otherwise.

## Re: Pullbacks That Preserve Weak Equivalences

It’s tempting to call these “quasifibrations”. Unfortunately, actual quasifibrations (in the sense of Dold and Thom) need not have this property (though all maps of this kind are quasifibrations).