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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.

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  • $\begingroup$ 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. $\endgroup$
    – Mike Shulman
    Commented Jul 11, 2014 at 21:58

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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.'

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  • $\begingroup$ I agree that "flat" is overused. (Although "sharp" isn't!) Unfortunately "couniversal weak equivalence" is very long, and I cordially dislike the use of "universal" to mean "pullback-stable" (I would prefer to reserve it for "universal properties".) Thanks for the reference though! $\endgroup$
    – Mike Shulman
    Commented Jul 15, 2014 at 3:19
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    $\begingroup$ I may end up using "couniversal weak equivalence" despite its disadvantages, though, just because someone else has already used it. (The problems with "flat" are also clear in my other question, since "flat functor" means something very different!) $\endgroup$
    – Mike Shulman
    Commented Jul 15, 2014 at 3:27
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Here's a suggestion. Hopkins uses the name flat map for a morphism such that every pushout along it is a homotopy pushout. Rezk uses the name sharp map for the dual notion. (Roughly speaking, I think one has to be a little careful with such definitions in "non-proper" situations.) So by analogy with acyclic (co)fibrations one could use the name acyclic flat maps for weak equivalences that are stable under pushouts and acyclic sharp maps for the dual notion. I don't think I've seen these names in use, but they seem to be perfectly consistent with the established nomenclature.

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    $\begingroup$ Haha, I had about the same idea after asking the question. I'll wait a bit and see if anyone else has a suggestion. $\endgroup$
    – Mike Shulman
    Commented Jul 12, 2014 at 7:06

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