I have another out-there question for the chat. Quick claims to construct a model category of simplicial profinite sets (arxiv.org/pdf/0803.4082.pdf). But profinite simplicial sets don't have all colimits, which is usually about axiom 0 for a model category. On the other hand, his model category is fibrantly generated and constructed through a "co-small object argument", so I think he doesn't need to take a colimit to, say, factorize a morphism.

My question is: can someone put my mind at ease about this? Does one ever come across, for example, a category that doesn't have all limits but that satisfies the rest of the axioms of a cofibrantly generated model category? Can one construct an infinity-category in one of the usual ways? (Quick's category is simplicially enriched and probably satisfies SM7, but it's not tensored over sSet because it's not tensored over Set, for the exact same reason...)

@PaulVanKoughnett The category of profinite sets has all limits and colimits (this is the case for the pro-category of any category with finite limits). Then simplicial profinite sets also have all limits and colimits as those are computed objectwise.

@GeoffroyHorel One only needs C to have finite limits to get that Pro(C) has all limits and colimits?

@PaulVanKoughnett Yes, because if C has all finite limits then Pro(C)^op=Ind(C^op) is presentable and so has all limits and colimits (an easy way to see this is to identify Ind(C^op) as those functors C->Set preserving finite limits)

@User1236262625 @CharlesRezk I searched for a bit and found this paper: link.springer.com/article/10.1007/BF02762004 by Dweyer and Farjoun. I believe this might be the paper he was discussing with me but the construction doesn't look the same as what I remember from our conversation. Perhaps it's equivalent though, I don't know...

@PaulVanKoughnett If you haven't looked into it, Appendix E of SAG provides a lot of nice foundational material on profinite homotopy theory. §§E.1.5–E.1.7 also essentially give a condensed version of Quick's theory (cf. Remark E.0.7.13).

That's very weird, but thanks everyone. I was thinking about taking colimits in Top, but there's no reason that inclusion should preserve colimits.

I now think a countably infinite coproduct of a point in ProfSets is the Stone-Cech compactification of the natural numbers. Horrifying!

I think that topics related to ETCS are sometimes discussed here too. In case somebody has some feedback on the proposal to change the name of the (etcs) tag so that it is not in the abbreviated form, leaving some comments on MO editors' lounge would be more than welcome.