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If $f:C\to D$ is an equivalence of categories that is injective on objects, then every pushout of $f$ is also an equivalence. This follows, for instance, because such a functor is an acyclic cofibration in the canonical model structure on Cat.

Suppose conversely that every pushout of $f$ is an equivalence of categories (in the terminology suggested by Karol here, $f$ is an "acyclic flat" functor). Does it follow that $f$ is injective on objects? (I'd be equally happy with an answer to the corresponding question in Gpd.)

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Every flat functor is injective on objects.

In this blog post Chris Schommer-Pries proves that there is a unique model structure on $\mathsf{Cat}$ with categorical equivalences as weak equivalences. The argument of his "Somewhat Less Trivial Lemma" also shows that if there is a flat functor that is not injective on objects, then $E \to *$ is also flat where $E$ is the walking isomorphism. All we need to know is that flat functors are closed under composition, pushout and retracts.

Now, let $C_2$ be the group of order $2$ and let $E \to C_2$ classify the nontrivial element. The resulting pushout of $E \to *$ is $C_2 \to *$ which is not an equivalence and hence $E \to *$ is not flat.

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  • $\begingroup$ I noticed that you opted against the name "flat functor". I agree that there is a collision here, but I'm not a fan of "h-cofibrations" either. I don't know how I could edit this answer to make the names less confusing... $\endgroup$ Jul 15 '14 at 8:20
  • $\begingroup$ In view of the fact that pushouts along "h-cofibrations" or "flat morphisms" are already homotopy pushouts, why not call them "homotopically coquadrable"? $\endgroup$
    – Zhen Lin
    Jul 15 '14 at 9:45
  • $\begingroup$ I think that the generic term "flat morphism" is a very good one. As far as I understand it is inspired by an analogy with "flat resolutions" in homological algebra (perhaps indirectly via "flat symmetric spectra"). It's unfortunate that there is a conflict when we specialize to functors, but it would be a pity to throw this name away just for this reason. $\endgroup$ Jul 15 '14 at 11:16
  • $\begingroup$ I'm not in favor of "h-cofibration" either, partly because I've gotten used to the May-Sigurdsson use of that to refer to the categorical generalization of Hurewicz cofibrations. At the moment I'm leaning towards "couniversal weak equivalence" just because it's already been used by someone else, and it's no worse than any of the other options. That doesn't give us a name for the non-acyclic morphisms every pushout of which is a homotopy pushout, though. What about something like "h-flat"? Or a "virtual cofibration"? $\endgroup$ Jul 15 '14 at 18:52
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The notion of a map $f$ such that every pushout of $f$ is a weak equivalence has been considered in the recent preprint Homotopy theory for algebras over polynomial monads by Michael Batanin and Clemens Berger. They refer to such a map $f$ as a couniversal weak equivalence. They also introduce the notion of an $h$-cofibration as a map $g:A\to B$ such that in any diagram as follows where both squares are pushout squares then the map $w':X'\to Y'$ is a weak equivalence as soon as $w:X\to Y$ is:

\begin{array}{} A & \to & X & \to & Y \\ \downarrow & & \downarrow & & \downarrow \\ B & \to & X' & \to & Y'\end{array}

Say $g$ is trivial if it's additionally a weak equivalence. The authors prove in Lemma 1.6 that every couniversal weak equivalence is a trivial $h$-cofibration. In a left proper model category couniversal weak equivalences are precisely trivial $h$-cofibrations. A further characterization of such maps in left proper model categories can be found in Lemma 1.5, including an equivalent definition which I independently introduced in my thesis (namely: every pushout square with $g$ as one of the legs is a homotopy pushout square). Properties of these maps can be found throughout section 1 of the paper and also in the last section of my paper on monoidal Bousfield localizations.

What you're asking is whether every trivial $h$-cofibration is a trivial cofibration. I doubt this is true in general or even in a setting where all objects are cofibrant. However, it might still be true in your setting. In Top, the $h$-cofibrations are like the closed neighborhood deformation retracts. In general it's hard to explicitly describe the class of $h$-cofibrations, but the relationship between $h$-cofibrations and cofibrations in Top gives some hope that what you're asking might be true.

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    $\begingroup$ It's certainly not true in general, even if all objects are cofibrant. For instance, in the model structure on Set where all morphisms are weak equivalences, the cofibrations are the monomorphisms, and the fibrations are the epimorphisms, every map is a couniversal weak equivalence, but only monos are trivial cofibrations. But the canonical model structure on Cat (or Gpd) is so well-behaved that I thought there was a chance. $\endgroup$ Jul 15 '14 at 3:14
  • $\begingroup$ Hi. Thanks for the example. I mentioned your question to Batanin today and he found it interesting. I am glad to see Karol's answer and to know that you got what you needed. I wasn't aware of the lemma due to Chris Schommer-Pries. $\endgroup$ Jul 15 '14 at 10:40

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