# Morphisms every pushout of which is a weak equivalence

Let $M$ be a category equipped with a class of weak equivalences $W$. Is there a name for a morphism $f$ such that every pushout of $f$ (including, of course, $f$ itself) is a weak equivalence?

For example, if $(M,W)$ underlies a model structure, then any acyclic cofibration in the model structure is such a map. But the concept as defined depends only on the weak equivalences. For example, we may be primarily considering a particular model structure, but if it so happens that there is another model structure with the same (or smaller) class of weak equivalences, then the acyclic cofibrations in that other model structure will also be examples of this concept.

A name for a the dual concept (a morphism every pullback of which is a weak equivalence) would of course be just as good, since then I could stick a "co" in front of it.

• One obvious name for the dual notion would be "stable weak equivalence", but that unfortunately has totally different connotations. (-: One could say "pullback-stable weak equivalence" but that's a bit of a mouthful and doesn't dualize so well. Jul 11 '14 at 21:58

Maps $f$ such that every pushout along $f$ is a weak equivalence were called couniversal weak equivalences in the preprint Homotopy theory for algebras over polynomial monads by Michael Batanin and Clemens Berger. In left proper model categories such maps are characterized in Lemmas 1.5 and 1.6.
I'm not sure I agree with calling these maps 'flat' as that word is already so over-used. For example, this terminology could easily cause confusion in examples like $Ch(R)$ or the stable module category. Furthermore, a common axiom for monoidal model categories is that whenever $X$ is cofibrant and $f$ is a weak equivalence, $X\otimes f$ is a weak equivalence. Motivated by the examples above, this axiom has sometimes been called the axiom that 'cofibrant objects are flat.' I discussed this axiom a bit at this mathoverflow thread. I don't know if that terminology will stick (in the paper above this axiom is called the Resolution Axiom, and in a preprint of Pavlov and Scholback it's called the left cow axiom), but it's more evidence to avoid using the already-saturated 'flat.'