consider a pushout diagram in a model category where all objects are cofibrant and the maps are cofibrations. consider a morphism of such things which is levelwise a weak equivalence: the induced map of pushouts is a weak equivalence. Now, picture the situation (before taking pushouts) as a ladder diagram: I have a situation where one of the two squares is only homotopy commutative, I think (and moreover, the vertical map in the side is an identity -- possibly not useful).

Can I still get an induced map of pushouts which is a weak equivalence?

out of curiosity - when you folks ask questions like the above, is it usually the case that you have some example in mind you'd like to apply it to and try to abstract to the precise level of generality you think is relevant? or is it often just that you need something for arguments about model categories themselves or whatnot (ie, it's more of the form of a technical lemma)?

well, I can only answer for myself and the above question, and it's the first option

I almost never ask about model categories, but when I ask similar questions about simplicial methods it is the first option too

that's what I assumed, but it's sometimes hard to see from outside :)

Anyway, I have nothing useful to add

@JonathanBeardsley Yes, I just meant to point out that this condition is necessary, but it's missing from the statement of the corollary

@AlexanderCampbell I feel like this ought to be fairly doable nowadays, but so far I'm not aware of a proof

Anyone aware of a source for the proof that the Serre-Quillen model structure with the weak homotopy equivalences has the same htpy category as a localization of the Strom model structure?

Cisinski told me the outline of the proof but said he didn't know a source

@HarryGindi Hmm... cannot you just say that they're both localizations of Top?

@DenisNardin Denis-Charles said it involves a right-bousfield localization of Strom

Ah Bousfield localization

That changes everything

yeah, heh

I have a reason to see it. I'd like to see what techniques are used

for a different application

Hmm... aren't weak equivalences exactly the maps that are colocal for spheres?

I guess that's what you'd like to see proved

precisely

Is it known whether the Strøm model structure presents the ∞-category associated to the obvious simplicial enrichment on Top?

Because I think that would be enough

somehow a very similar situation seems to appear in model structures on strict omega cats

Hrmm.. the nlab claims it is true (that the Strøm model structure is a simplicial model category in the obvs way)

Having that it shouldn't be hard to prove that the weak equivalences are exactly those for which Map(S^n,-) is a homotopy equivalence of Kan complexes

interesting

thanks! Will look into it

@HarryGindi Shouldn't this be the mixed model structure of Cole, which was corrected by Barthel and Riehl?

arxiv.org/abs/1204.5427

Anyway, Barthel and Riehl construct a number of "mixed" model structures like this, so the techniques might be what you're after.

Maybe also relevant -- Barthel, May, and Riehl

@HarryGindi regarding steenrod squares and strict n-categories, you should talk to Anibal Medina

Thanks!

Good answers to all of my oustanding questions

@TimCampion I guess I should look at Medina's paper on Steenrod things before I email him

Oh, it's not available

Anyway, yeah, Tim, this mixed model structute theorem looks like the ticket

thanks!

@AlexanderCampbell i'm almost certain this is in the appendix of gaitsgory--rozenblyum. the difficult thing about it is that you can't be in the usual fibrational context where we've so far been getting away with talking about left-lax or right-lax stuff, because this statement mixes the two handednesses

@HarryGindi isn't the Quillen--Serre model structure a left bousfield localization of the Strom model structure? thats what i always imagined, but maybe you're saying it's the other direction. in any case, if either direction is true then this is immediate: quillen adjunctions induce adjunctions on homotopy categories, and in a left bousfield localization the derived right adjoint on homotopy categories will be fully faithful (and dually)

Hurewicz fibrations and Serre fibrations are also different

^

They're not directly comparable, you need to use the mixed model structure somewhere in the argument

I believe

The mixed model structure is equivalent to the Serre-Quillen, and I believe you can show it's a right-Bousfield localization of the Strom one. I think that's how the argument goes

If you see the mixed model structure theorem, we see that homotopy equivalences are a subclass of weak homotopy equivalences, check. Hurewicz fibrations are always Serre fibrations as well, so there exists a mixed model structure equivalent to the Quillen-Serre