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.